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Singular heteroclinic orbits for degenerate modulation equations
__ -p Ed
s
PHYSlCl [ol Physica
EISEVIEK
Singular heteroclinic
I)epariment
of Mathematics,
Received
11 June
D 82 (1995)
36-59
orbits for degenerate equations Todd
Kapitula
University
of Utah,
Salt Lake
modulation
City. UT 84112. USA
lYY3: revised 5 October 1994: accepted Communicated by C.K.R.T. Jones
5 October
1994
Abstract The problem of finding heteroclinic orbits for the quintic Ginzburg-Landau equation in the case that the When x = x, , where x is a distinguished imaginary parts of the coefficients are O(E), with 0 < ??+ 1, is considered. parameter in the equation, and E = 0 there exists a singular heteroclinic orbit connecting a stable finite amplitude state to the stable zero state. It is shown that for a certain region of parameter space this orbit persists for E small, with a wave speed C(E) which is O(E). Furthermore, using an approach reminiscent of the Melnikov method. the sign of c’(O) is calculated. This calculation is crucial in settling the persistence question. Finally, it is shown that for this wave speed it is possible for additional heteroclinic orbits to exist. These additional orbits connect two finite amplitude states.
1. Introduction Modulation equations have been used to study the behaviour Probably the most well known is the complex Ginzburg-Landau supercritical
bifurcations
A, = (1 + ip)A,,
is given,
+ [l - (1 +
in canonical
form,
of patterns at near critical conditions. (CGL) equation, which in the case of
by
iv)lAl*]A,
(1.1)
where (x, t) E R x [w’. In this equation the zero solution is unstable in favor of finite amplitude solutions. There are instances, such as in binary fluid convection [17], Taylor-Couette tlow with counterrotating cylinders, and Jeffrey-Hamel how [5,6], in which the CGL model is invalid, and an extension. hereafter called the degenerate modulation equation, must be considered. For completeness, let me briefly recall the manner in which this case arises. The following discussion can also be found in Doelman and Eckhaus [4]. In the circumstance of bifurcations of space-periodic solutions, the dynamics are governed by the Landau equation 0167-278Y/Y5/$09.50 (Q 1995 Elsevier SSDI Olh7-2789(94)00223-l
Science
B.V. All rights
reserved
37
T. Kapitula I Physica D 82 (199s) 36-59
Jg=q(k) where
a, +
P(k)(a,l’a,+ y(k)la,14 + .. ,
k is the wavenumber.
the degenerate convenient
(1.2)
If Re P(k) is of order one, then the modulation
case, in which
Re P(k) is small,
the corresponding
equation
modulation
(1.1) appears.
equation
resealings,
@, = i&I@]‘@ In the above
+
?? {(h(]@])
equation
If p,, is of order
oscillations
of @ happen
phenomena
discussed
equation,
which
+ iw(]@]))4
A and w are quartic
criticality.
unity,
which
on a much
+ i(/3,Qt]@]’
+ &@T@‘)
polynomials,
and E is a small parameter,
is the case that was studied
faster
time
scale
in [6]. If p,, is small or vanishing,
is given
in canonical
+ (1 + ip)p,,}
form
than
by Eckhaus
the evolution
then one arrives
+ O(E”)
(1.3)
due to the near and IOOSS [6], then
of (@I. This
leads
at the degenerate
to the
modulation
by
A,+(~+~~)A,,+(~(~A~)+~W(IAI))A+~(P,A~IAI~+P~A~A~).
(1.4)
h(r) =x + Y’- rJ, W(T) = d,r’ + d2r’, and p, = b, + ia,. When x < 0, as will be assumed in this (1.4) corresponds to a subcritical bifurcation. In this scenario both the zero solution and a finite
where paper,
amplitude state will be stable solutions for (1.4). If the underlying physical system possesses a reflection has purely parts
In
is, after some
real coefficients.
of the
coefficients
As such,
when
will be small.
symmetry,
the underlying
A physical
system
circumstance
the resulting is nearly in which
modulation
symmetric, the
equation
the imaginary
coefficients
degenerate modulation equation are real appears in binary fluid convection [17]. The aim of this paper is to find frontal solutions to (1.4) in the case that the imaginary
for
the
parts of the
coefficients are small. To accomplish this, the case in which all the coefficients are purely real is first considered, and then a determination is made as to which frontal solutions persist under perturbation. For convenience, the diffusion coefficient p will be set to zero, and the imaginary parts will be resealed by Ed,, and a, is substituted by ?? a, for 0 5 E + 1. Upon this resealing, w will be so that d, is substituted replaced by EW and p, will be replaced by b, + ea,. Letting z =x - ct, where c is the (as yet) undetermined solutions
front speed,
the question
of finding
solutions
is exactly
that of finding
steady
state
of
A, = A,, + cAZ + (~(1~1) + im(lAI))A Some results Eckhaus solutions. solutions
frontal
concerning
the existence
+ i(P,A,IAI’+
of steady
&AzA')
state solutions
.
(1.5)
to (1 S) can be found
in Doelman
and
[4]. They studied the existence and persistence of periodic, quasi-periodic, and homoclinic In the special case that p, = p, = 0, results about the existence of homoclinic and heteroclinic can be found in Jones, Kapitula, and Powell [9], De Mottoni and Schatzman [2], Malomed
and Nepomnyashchy Slow time periodic
[15], and Van Saarloos and Hohenberg [16], to name frontal solutions to (1.5) are of the form
A(z,t) = r(z) exp where (T is real. Substitution of the ansatz (1.6) into (1.5) framework, the question of fronts (resp. pulses) is reduced connections between appropriate fixed points.
a few.
(1.6) yields the system of ODE (2.2). In this to finding heteroclinic (resp. homoclinic)
T. Kapitula I Physica D 82 (1995) 36-59
38
The interest are given A(z,t)
When
here is to find connections
involving
stable
plane
waves.
Plane
wave solutions
to (1.5)
by = r,, exp[-i(a,,z
- E(T~)]
c = E = 0, the amplitude (Yf - B&q,
where
B=b,
paper,
(1.7)
Following
r. satisfies
the relation
- h(r,,) = 0 , -bz.
Assuming
describes
Kapitula
(1.7) that B*-4~0
and x>l/(B”-4), which will be done throughout this Now, define the parameter H by 4H = b, + b,. in the (ri, a,)-plane.
an ellipse
[12], a plane
wave will be nonlinearly
(BH-l)r~+~r~-2Hr~a,,+a~
Assuming
Thus,
relative
to (1.5)
if
af-rff-,y
that BH < 1, which will be done
the (ri, a,,)-plane.
stable
the intersection
throughout
of (1.8)
(see Fig. 3). (ri, a,,)-space Let the stable plane waves be parameterized
(1.8)
this paper,
with (1.7)
defines
Eq. (1.8)
describes
a hyperbola
in
a curve
of stable
plane
in
by the pair (r, a). Define
the parameter
waves
Q by
Q=H’-BH+l,
(1.9)
and note that BH < 1 implies a heteroclinic equation
orbit
connecting
that Q > 0. It turns out that when x = -3116Q the pair (r,,, Hri)
to (0,O).
Here
and c = E = 0, there
r,) is the largest
positive
root
_ &+Qr”-Q2r4=0,
exists of the
(1.10)
(r,,, Hri), so that this orbit Furthermore, there exists a curve of rest points for (2.2) containing singular (see Fig. 3). This curve represents the stable plane waves given by (1.7) and (1.8).
is
It is natural to wonder about the fate of this singular orbit for c and E nonzero. At first glance, one may expect that for c and E nonzero there exists an orbit which is 0( ]c] + ?? ) to the singular one. However,
careful
thought
for the wave to persist likely
that c’(0) will depend
must
satisfy
the additional
(c’(0) - a_r’)a
reveals
that this does not necessarily
c = C(C) is smooth,
on the parameters constraint
have to be true.
so that c(e) = c’(0) E + C~(E’). Under ai, d,, i = 1,2,
Suppose
that in order
this assumption,
and o. Now, for E nonzero
plane
it is waves
to (1.7)
= w(r) - (T + D(E) ,
(1.11)
where a, =a,
*a,,
(1.12)
Therefore, unless the parameters a_, d,, d,, and CT in (1.11) can be chosen correctly, plane wave solutions to (1.5) may not exist for E nonzero. Unfortunately, there is no a priori guarantee that these parameters can be so chosen. The purpose of this paper is to study this singular orbit and make a determination as to its persistence. The primary difficulty in this problem is showing that the intersection of (1.11) and (1.7) can be satisfied, as one must first find an expression for c’(0). It turns out that this can be accomplished via a method reminiscent of that of Melnikov. Malomed and Nepomnyashchy [15] considered the case in which B, = 0. Assuming that w(r) - u = 6(e)
,
39
T. Kapitula I Physica D 82 (1995) 36-59
the problem was solved via the use of formal asymptotics. They determined that the wave speed is 0(e2), which is not the case when /?, # 0. The primary difference between their work and the present work is that the front constructed in their paper is nonsingular. The
main
theorem,
result
of this paper
the function
is summarized
W(Y) is given
in the following
theorem.
In the
statement
W(Y) = O(I) + a_Hr” In the following however, generality, Theorem
that 2H - B # 0. If 2H - B = 0, then much
be modified
1.1. Let Q be as defined Suppose
(a) BH
a bit. In the following
statement
of what follows it is assumed,
is still true,
without
loss of
r,, as defined
and W(r) be as defined
in (l.lO),
in (1.13).
Let
(2H-B)HzO Oirsr,, - B)a).
= -sgn((2H
if CC > 0 is sufficiently
C(E) = O(E), which sgn(c’(0))
in (1.9),
that
G(r)-v#O,
(c) sgn(a+) Then
must
that Y(Z) + 0 as z + ~0.
x = -3116Q.
(b)
(1.13)
it is assumed
the proofs
of the
by
large.
for 0
is O(E) close to the singular
= -sgn((2H
orbit.
exists
a heteroclinic
orbit,
with
wavespeed
Furthermore,
- B)~T) .
Remark
define 1.2. The hypotheses of this theorem CT,d,, d2) (see Fig. 1 for the restrictions (B,H,c,a+,
an unbounded
domain
in (B, H)-space).
H
B
Fig. 1. Permissable
parameter
range
in (B, H)-space.
in the parameter
space
40
T. Kapitula I Physica D 82 (1995) 36-59
Remark
under
1.3.
Since
(1.4)
the hypotheses
Remark
result
1.4.
Remurk
under there
Since the theorem
continues
conclusion
is invariant
of this theorem
to hold
remains 1.5.
is proved
by showing
for x within
unchanged
C(E’)
if sgn(x*)
Since the theorem
the change
is proven
an expression
expression One
that certain - B)a).
by showing
that certain
it is seen
the
existence
that
manifolds
In fact.
= -sgn((2H
and
-p,.
c--t -c, as z+
intersect
if x = -3116Q
manifolds
intersect
and
z--t --z,
--x. transversely,
the
+ EX”, then
the
transversely,
the
so that one can say there exists a 6 > 0 so that holds. In the Appendix a forma1 perturbation the
condition
(2H - B)H 2 0 allows
a certain
to be inverted.
can
conjecture
for c’(O),
p,+
such that r(z)+0
of -3116Q.
requirement that (2H - B)H 2 0 can be relaxed slightly, if (2H - B)H 2 -6, then the conclusion of the theorem yields
of variables
also exists fronts
also that,
wonder
about
in addition
of multiple
to the orbit guaranteed
close to the curve of stable
plane
waves.
says that if the hypotheses
of Theorem
heteroclinic
by Theorem
Theorem
5.3 gives an answer
1.1 are satisfied
orbits.
1.1, there
Specifically,
exists orbits
to this conjecture.
and a_ > 0 is in a certain
one
can
which are O’(E) It essentially
interval,
sufficiently small such a scenario is possible, with the wavespeed satisfying c’(0) > 0. One may also wonder as to the stability of these travelhng waves for the PDE (1.4).
then
for H
For the slow
waves guaranteed by Theorem 5.3, it may be possible to adapt the techniques presented in Kapitula [l l] to show that these waves are stable. In the case of the orbit guaranteed by Theorem 1.1, the problem is a little more delicate, due to the singularity of the orbit. Here the results and ideas first presented
in Jones
[X] and Alexander,
Gardner,
and Jones
[l] may apply.
Using
their ideas,
it may be
possible to show that the singular orbit is also stable. This matter will be pursued in a later paper. This paper is organized in the following manner. In Section 2 the relevant system of ODES, Eq. (2.4),
is derived,
and the basic geometric
and heteroclinic
solutions
objects
to (1.5) exist when
stable and unstable manifolds. Section showing that the relevant center-stable
are discussed.
c = E = 0. These
In Section orbits
3 it is shown
correspond
that homoclinic
to the intersection
of
4 is devoted to tracking the invariant manifolds of (2.4), and and center-unstable manifolds intersect transversely, with the
wave speed c = C(E) satisfying the conclusion of Theorem 1.1. Finally, Section 5 is given to showing that (I. 11) intersects (1.7), upon which the proof of Theorem 1.1 will be complete. In the Appendix an expression for the wave speed is derived via formal asymptotics.
2. The equations Substitution
of the ansatz
A” + CA’ + (h(lA\)
(1.6)
+ iE(w(lAl)
into (1.5) - a))A
yields
+ i(/?,A’IA\’
where A(Z) = r(z) exp[-i ]’ a(s) ds] and ’= didz. u = r’Ir, one arrives at the first order system
u’ = -cu - u2 + iy7 - A(r) - Br’a a’ = -(c + 2~4)a + 4Hr’u
+ E(u(r)
the second-order
+ Ea+r’u
After
complex
+ &(A*)‘A’)
equating
valued
ODE
= 0,
real and imaginary
(2.1) parts,
and setting
.
- u + a_r’a)
.
(2.2)
41
T. Kapitula I Physica D 82 (199.5) 36-59
In the above
(Y, is as in Eq. (1.12).
u = r’ir is used so that the resulting
The transformation
to (2.2) is smooth near r = 0 [9]. In Doelman [4] it was noticed that when
c = E = 0, (2.2)
possesses
vector
field
a first integral
0,) = r2a - Hr’ It turns
out
singular
as r-+0.
that
upon
rewriting
As such,
(2.2)
with
it will be more
the variable convenient
R,, instead
of (Y. the vector
R=a-Hr’.
When
(2.3)
c = E = 0, it turns
in Eq. (2.4),
field becomes
to use the quantity
out that R = 0 is also a first integral
the vector
Substitution
of (2.3)
field is not singular into (2.2)
of the equations;
however,
as one can see
r = 0.
when
yields the system
r’ = ru .
u’ = -cu ~ u3 + R’ - h(r) + (2H ~ B)r’R 0’ = -(c + 2u)fl
- cHr’ + c(O(r)
+ cu+r’u
- CT+ a r20)
,
,
(2.4)
where h(r) = ,y + r’ - Qr” ,
(2.5)
and W(r) is defined in (1.13). It will be necessary (2.4). When this is done, the resulting equation
at times to append the equations c’ = 0 and E’ = 0 to will be represented as (2.4),. where .$ will be the
appended variable. Let us now study what is known of the dynamics of (2.4). For the invariant subspace (r = O}, it is shown in Landman [14] and Jones [9] that the flow here corresponds to a flow on S’. with critical points (u-, , 0,)
satisfying
the algebraic
LIZ+ CU + x - RZ = 0 , For E sufficiently [14] that solutions
(c + 2u)fl
words,
(2.6)
the attractor to (2.4)
(r(z), U(Z), 0(z))+((). In other
+ ECr= 0
pair in {r = O}, > 0 and ~1, < 0. The critical points form an attractor/repeller it is shown in [9] and and (u,, 0,) being the repeller. Furthermore,
small u
with (U , O_) being
relation
satisfy u,,
(0, U, . 0,)
(r(z), r’(z))+
n,)
,
possesses
and (0, u_, RJ has a one dimensional When c = ??= 0. (2.4) reduces to
(0,O)
as z -+ 2~
if and only if
z-+ 202 a one dimensional stable manifold coming out of the r = 0 plane, unstable manifold coming out of this plane (see Fig. 2).
r’ = ru . u’=-u’+fl’-h(r)+(2H-B)rzL?, R’ = -2uR.
Note
(2.7)
that Q = 0 is invariant,
so that in this subspace
(2.7)
reduces
to the planar
system
r’=ru,
u’ = -u2 - h(r) If x is further
restricted
(2.8) so that
T. Kapitula I Physica D 82 (1995) 36-59
42
Fig. 2. Flow near
(r = 0).
1 -4&
solutions,
which implies
in Section 1, Y()will denote the largest positive When u = 0, (2.7) has a curve of rest points Q2 - h(r) + (2H - B)r%
that the nonlinearity
for (2.8) is bistable.
As stated
root. given by
= 0.
(2.9)
Eq. (2.9) represents an ellipse in the (Y’, R)-plane. Let .M,, denote some open subset of this curve. M,, is said to be a normally hyperbolic manifold if for each p E A0 the linearization of (2.7) about p yields one zero eigenvalue zero. It is a routine
and two eigenvalues y,(p) with IRe y,(p)] b em g uniformly bounded calculation to show that this condition on yZ holds if for 0 < 6 G 1
4flZ + r/V(r) < -6 where
(Y, 0)
satisfies
,
away from
(2.10)
(2.9).
The curve
4R2 + rh'(r)= 0
(2.11)
describes a hyperbola in the (Y’. R)-plane, and the intersection of (2.11) with (2.9) determines the boundary of A,,. Because /\‘(r(,) < 0, ( Y,,, 0,O) E A,,, and hence so is some neighborhood (see Fig. 3). It is important to note here that A,, corresponds to the stable plane waves given by (1.7) and (1.8). Since 3/t(p) < 0 and y_(p) < 0 for each p E A,,, there exists a one dimensional stable manifold W’(p) and a one dimensional unstable manifold W”‘(p). Set w;, =
u WS(P), /JE(‘0
w;y = .g,
W”“(p) 0
Wf,~“s are invariant two dimensional manifolds. If follows from the work of Fenichel [7] that for each fixed c and E sufficiently small there is an invariant manifold, .,,!I,.,, of (2.4) nearby to JU,,, with the flow
43
T. Kapitula I Physica D 82 (1995) 36.~9
Fig. 3. The critical
on A,.,
begin
0( Ic] + 6). Furthermore,
nearby to WX”“, with the solutions Define w;; where
= l._,. w,“: , <.t
is taken
here.
For (2.4),.,,,
for each fixed c and E there
on W:;“’ approaching
over c and E sufficiently
“& = [M’(.n, c, E), M”(fl,
small.
& is a three-dimensional c, E), LJ, c,
and WY is a four dimensional WY = [Y, 9T2”(r,n,
.I(,,.
A,,
manifold
c, E), R,c,
See Remark manifold
manifolds
fast as z+
W:,;“’
5~.
on Ju,., must satisfy
graph
for my choice
of
is
?? ]) whose
graph
is given
by
e]
that c = C(E) with c(0) = 0, and that the dependence points
invariant
2.1 for the reason
whose
Similarly, a four dimensional manifold Wi could be defined. It is quite natural to wonder if any of the critical points constituting critical
exists
exponentially
./ii = t._. Je (‘.t Cc’
the union
notation
manifold
J/J,, persist
for c, E # 0. Suppose
of c on E is at least C*. Under
this scenario,
any
u = 0 and
(a) R’ - h(r) + (2H - B)r*L? = 0 (b)
R =
O(r) - m - c’(O)Hr’ c’(0) -u-r*
Eq. (2.12)(a) is a representation (2.12)(b) can be rewritten as R=
W(r) - (T - c’(O)Hr’ c’(O)( 1 - pr’)
+ Q(C) for A,,.
+ Q(E) >
(2.12) Assuming
that
c’(0)
can be chosen
independently
of a_,
(2.13)
where p = a_ /c’(O) can be arbitrarily chosen. If p is sufficiently large, then (2.13) will intersect JH,, at one point. If H is sufficiently small and p = Q(l), then the parameters d, , d, = d, + amH, and IT can be
44
T. Kapitula i Physica D 82 (199.5) 36-59
manipulated
so that
(2.13)
points. The maximality explored in Section 5. Now
consider
possesses
a one
manifold,
W”‘.
the
Remark
unstable from
critical
coming
When
manifold
?? ), P’(r.
at minimally
is due to the fact that
points
(0. c[,, 0,).
stable
manifold,
and their
graphs
c, E), c, E] ,
the R = 0 plane
one point.
W is quartic.
It is easy
and at maximally
This
to compute
idea
that
for
will be more (2.4).
four fully
(0, CI_, Q_)
W’,
and (0. 14_, L?) has a one dimensional unstable For Eq. (2.4)C.,t these then become three dimensional
arc given by wp
is viewed
from .24,, is coming
the left. Hence.
3. Existence
.,&,, transversely
out of the {Y = 0} plane.
manifolds,
= [Y, Y’(Y. c, 2.1.
condition
dimensional
center-(un)stable w:
intersects
= [r,
from a vantage
point
from the right and the stable
c, E). (‘. E] .
in which R > 0. it appears
as if the
from (0, ~1,. 0, ) is coming
manifold
the %! and Y notation.
of fronts when c = E = 0
Set c = E = 0 in (2.4).
so that Eq. (2.7)
is considered.
In the invariant
space
{0 = 0}, i.e.,
for Eq.
(2.X). the function
E(r,
14) =
f(n4)’
+
I 0
s/i(s)
ds
is a first integral. Thus, the phase portrait for (2.8) is relatively portrait is depicted for different values of x. When Actually. manifold transverse. persists.
x = -3/16Q,
WY
intersects
W)
in a nontrivial
easy to determine.
fashion,
i.e..
In Fig. 4 the phase
a heteroclinic
orbit
exists.
the fact that (Y,,. 0,O) E &,, implies that the orbit is singular. Since W”,\ is a three dimensional and W\/ is a two dimensional manifold for (2.4), , it is possible that the intersection is If so, by the implicit Unfortunately,
function
this is not sufficient
theorem
there
to determine
exists
a c = c(e) such
that the heteroclinic
that
the intersection
orbit persists,
on JZdC.tmust be understood. For instance, from the discussion of the previous section any critical points exist on JL!,.,. The remaining focus of this paper will be on these
as the flow
it is yet unclear issues.
if
When x E (-3116Q). W\, intersects WY nontrivially, so that a homoclinic orbit to Y = 0 exists. Setting C?= E(T. W> and Wl,F are each three dimensional manifolds for (2.4),,,;, so that the intersection can possibly be transverse. It can be shown, using the methods presented in this paper and [9] that these manifolds do indeed intersect transversely. As such, there will exist a c = c(e), i = C?(E) so that for E nonzero and small the intersection persists. which implies that the homoclinic orbit will continue to exist. The proof of this assertion will be postponed until a later date. It has recently been shown in de Mottoni and Schatzman [2]. however. that in the special case of B = H = a, = d2 = 0, the pulse continues to exist for d, sufficiently small (the parameter d, is the perturbation parameter in their paper). Finally. if x E (-l/4(2,-31 16Q), then W’i intersects W’# nontrivially, so that a homoclinic orbit to r = r,, exists. The persistence of this orbit will not be studied in this paper, as an excellent treatment has already been done in Doelman and Eckhaus [4] and Doelman [3]. it is possible. however, that the methods and ideas presented in this paper can be used to duplicate some of their results. In these papers it is shown that in a certain region of parameter space the homoclinic solution breaks into a
T. Kapitula
I Physica
4
_,
45
D b’2 (1995) 36-59
Y E C-3/16Q,O)
“.‘,,
Fig. 4. Flow on {R = 0)
heteroclinic present
cycle, which consists
on Al,,,
and the fast wave corresponds
It is of some interest scenario, more
the subspace
can
of a fast travelling
be said
to note the structure (0 = 0) is invariant
about
the
existence
wave and a slow travelling
to the intersection of solutions
for (2.4) when and
stability
of IV:
in the special
wave. The slow wave is
with Wk.
case that H = O(E). Under
E = 0 and for all real valued
of heteroclinic
solutions.
c. Thus,
Basically,
the
this much
results
presented wavespeed
in [9] hold. In that paper, it is shown that if x E (-l/4&, -3/16Q), then a wave exists with C(E) < 0. Furthermore, it is shown in Kapitula [ 131 that this wave is stable, modulo a spatial and rotational translation. For x E (-3/16Q, 0) a wave also exists; however, here it is necessary that
c = C(E) and manifold of not yet been no unstable
C?= G(C). This change in dimension in parameter space is due to the fact that the unstable (Y,,, 0,O) changes dimension from two to one as c crosses zero. The stability of this front has determined, although the proof presented in [13] can be modified to show that there exists eigenvalues. The case of x = -3116Q was not pursued in [9].
4. Persistence
of manifold
intersection
Following heteroclinic (2.4),,e I=
the discussion of the previous section, the initial step in showing that the singular orbit that exists for x = -3116Q persists is to show that WT intersects . W$ transversely for when E = 0. Set 0 < Y, < Y(, to be the other positive root of A(r) = 0, and let {(Y, u, f1, c, E):Y = r,, u < 0, /i-q, (cl, (El
)
(4.1)
46
T. Kapitula I Physica D 82 (199.5) 36-59
where
Y’< 0 for all 0 < r < r. and
0 < 6 + 1. Since
dimensional
manifold
c, E), n, c, E] )
w:,\ t-l I = [K’(fl, Since
the orbit
the function
0) )
Iv;. n I = [P(c, E), P(c,
are
smooth,
WY n I is a three
Let E), c, E]
2?(0,0)
= 0.
F(c, E) by
F(c, E) = K@?(c, and note
manifolds
manifold.
exists for 0 = c = E = 0,
!%‘(O, 0,O) = Y(O, Define
the
and W> n I is a two dimensional
e) )
E), c, E) - P’(c,
that F(0, 0) = 0. It is clear that F(c,
?? )= 0
implies
that W z intersects
IV;
nontrivially.
0) # 0 )
F,(O, 0) = (& :: - 9,” + a;P;)(o,
If (4.2)
then by the implicit function theorem there exists a smooth function c = C(E), with c(0) = 0, such that F(c(E), E) = 0 for E sufficiently small. Thus, assuming that (4.2) holds, the manifolds intersect transversely.
As a side remark,
c’(0) = -FJO,
O)/F,.(O, 0) .
Theorem 4.1. Suppose Proof.
showing
(4.3)
that (2H - B)H 2 0. Then
This is an immediate
Before
note that
consequence
that (4.2) holds,
of Lemmas
it is necessary
{r = O}. This can be accomplished
near
(4.2)
holds,
with F,.(O, 0) > 0.
4.8-4.10.
to derive
0
approximations
for Wg near A, and for IV>
by using the fact that each of these manifolds
the flow generated by (2.4),,,. Before making any statements is necessary. For notational ease, for the rest of the paper
about set
Wz, however,
is invariant
under
the following
lemma
b=2H-B.
(4.4)
Lemma 4.2. The manifold
& satisfies
(a) Mii(O, 0, 0) = 6
b 0
(b) M:(O,O,O)=
-&bH
(c) M~(O,O,O)=&b(W(r,,)-ir). 0
Proof.
When
g(R)
c = E = 0 the critical
= f12 + b(M)*
- h(M)
manifold
&,, exists,
so that the function
Mr(fl,
0,O) satisfies
= 0.
Differentiating g and setting R = 0 yields part (a) of the lemma. Set 77= (a, c, E). Since M”(0, 0,O) = 0, the functions comprising M’(R,
c, E) = r&
+ r,c + r,E + 0( 1~1~),
M”(R,
c, 6) = u,.c + U,E + O( 1#)
)
J& can be written
T. Kapitula
where
the coefficients
I Physica
are to be determined.
invariant under the how generated ut determines (c).
This will be accomplished
by (2.4), .~. Note that U, determines
The coefficient
yn is already given in (a). To determine so that (M’)’ = M’M”,
implies
47
D 82 (199-f) 36-59
by using
the fact that
part (b) of the lemma,
the other coefficients,
note that, on J&, Y’= TU
rJ2’ + f7(/~1’) = r,,(u,.c + U,E) + 0(1q12) .
Similarly,
Substituting
+ (O(r(,)
(4.6)
proves
4.3. The
proof
satisfies,
for r near r,,,
b(r - r,)) + O((r - r,,)‘)
bH + 0(r - r,,)
b( W(r,,) - a) + U(r - r,,) .
of (b) and (c) follows
immediately
all that is left is to prove (a). set u = %!“(r, 0, O,O), so that when
= [r, u(r, fl),
0,0,0]
and u = 0 for points
rh’(r)
the fact that
c = E = 0, WF
J&!C WY
of
has the graph
theorem
,
about
each point p E A0 satisfy
= 0,
with the associated eigenvector being (-r, -y, 0). Here r represents the coordinate critical manifold A,,. Let y+ denote the positive eigenvalue, so that y&(O) = +rh’(r)
Then
and the results
p E A(,, it is clear that uI(r,,, 0) = 0. By Taylor’s
0) = Urrl(ro, O)(r - r,,) + C((r - rJ)
R’+
from
.
so all that is left is to calculate u,.fj(rg, 0). The eigenvalues for the linearization of (2.4) y’+
yields
(_r,~c~i,,3f2
0,0,O)= &
Since A,, C WT Gr,
WG
manifold
Lemma 4.2. Thus. For convenience, Wz
then
0
.R:l(r,O,O.O)=-$&
The
the coefficients
the lemma.
(C) 92:I@, Proof.
and equating
(4.6)
(W(rd - a>r, = r,,q ,
(4 %Xr, (),O,0) = (b)
- CT)C+ 0(17~/‘) .
into (4.5)
= rouC ,
-Hr,?,r,,
Lemma
(45)
on A
R’ = -Hrf,c
which
J& is
and that
u, = y+ lr, which
- 0’.
yields
M’(f2,
0,O) of the
T. Kapitula I Physica D 82 (1995) 36-59
48
In this calculation leads
the result given in Lemma
to the conclusion
4.2(a) is explicitly
used.
Further
manipulation
of the lemma.
Remark 4.4. If H = 0, then
in [9] that 3 fY(r,0, 0,O) > 0 for 1 % r,, - r > 0.
it is shown
Remark 4.5. If b = 0, then %~j(r, 0, 0,O) = 0, as in this case (2.4) is invariant R+ -0. For the next lemma,
recall
Lemma 4.6. The manifold (a) Z:‘(r,
Y;p:'(O,0,O)
=
-
Recall
which describes DC@+,
for r sufficiently
small,
O(r)
+ +
bcr r’ + Q(r’) .
u2+cu+~-f12 i
(c+2u)Q+Ea
the critical
0
1
point
=
(u, , 0,)
(b) and (c) will only entail 2u+), and
II 0
)
of (2.4) when r = 0. This critical
showing
that u+(c, E) and n+(c,
point is contained
E) behave
in W.>, so
appropriately.
Since
0, 0,O) = diag(2u+,
G,(u+,O,O,O)=[u+,O]‘, the result
c = E = 0, U, = -fi.
Eq. (2.6),
G(u,fl*c,E)= that proving
the transformation
+ 0(r)
+
(d) 2’z(r, 0,O) = 2a+%$Proof.
under
r2 + 0(r’)
+
(b) iYf(r, 0, 0) = - $ (C)
that when
IV> satisfies,
0,O) = - $
of the above 0
now follows
G,(u+,O,0,0)=[0,(~]~-, via an application
of the implicit
function
theorem.
Note
also that
(R+)(
=
(u+), = 0. For the rest of this proof define 77= (r, c, E). The proof of (a) and (d) is a bit more complicated, and the ideas of Lemma 4.2 need to be used. Part (a) will be proven first; therefore, set E = 0 in (2.4). Note that for (2.4),. the manifold W> is tangent to the subspace spanned by the vectors [l, 0, 0, 01, [0,0, 0, l] as r-0. Thus, the functions comprising W’$ have the expansion Y7”(r, c, 0) = u, -+c
+ S(lql’)
,
Y”(Y, c, 0) = .C2,,r’+ Q,.rc + f&c’+ In this expansion
the results
CyIql”) .
of the previous
paragraph
are being
used.
49
T. Kapitula I Physica D 82 (1995) 36-59
It will now be shown order.
that Q, = R,,. = Q,. = 0, so that the expansion
for 3”
will necessarily
be third
of W,b,,
By the invariance (3?“)’ = -(c + 2Y’)3?
- cHr’ ,
(4.7)
so that 2&Y
+ L?,,r’c + a( ln13) = -2U+ (0,,Y2 + Q,YC + @)
r’ = r9”
Since
2f&u+r’ Equating
on WL, this further + &u+rc
coefficients
The function LP(r, where found.
implies
that
+ O(lql’) = -2u+(f&r’ now proves
+ C( InI’)
+ QCrc + i&c’)
+ 6((q13) .
the claim.
3”’ now has the expansion
c, 0) = R,r3 + &r’c
+ fiyrc2 + R,c’ + fT(I~I”) ,
the coefficients 0, need to be determined. Again using (4.7), it can be seen that
u+(IfI,r’
+ 2R,r2c + &rc’)
To prove
+ tY([ql”) = -Hr’c
part
(a) of the lemma,
only
(2.4),.
As r-0,
Y’(r,
to the subspace
the manifold
0, E) = u, + u,,r2 + up
?? )= -
L!?“(r, 0, The result
W,k, is tangent
comprising
of part
T$
+
spanned
and r’ = rY,
U II
+ u,,e2 + 0(171’) ,
E + 0( 1771’).
(b) is used here. that u,, = 0. The invariance - h(r) + br’9”
+
+ 0(1~1’) = -2u+u,,re
+
+ (3”)’
u,p)
the coefficients =
_-
[l, O,O, 01, [0,0,O. 11, so that
of WL implies
?? z+r’5’f”
(4.8)
so that
u+(2u,,r’+ Equating
by the vectors
then
SU,
yields
CT2
h”(0)
so that the claim
’
u,,
=jg
1
U,,
= 0,
is proved.
To prove (d), the 0(/7~]‘) t erms of .Y’ must be determined. calculated. Furthering the expansion for 9’ and .L@’yields
Specifically,
the coefficient
iY?(r, 0, E) = u, + u,,r2 + u,,~~ + u,r3 + uzr2E + u3rE2 + u4rfz3 + fT(l~l’) , LP(r,
and consider
have the expansion
It will first be shown (3”)’ = -(Z’)’
be
- 2u+(L?,r” + Rzr’c + R,rc’ + fl.,c3) + 0(1~/“) .
Thus, 0, = fi, = 0, = 0 and 4u+R, = -H, which yields the result. To finish the proof of the lemma, all that is left to show is part (d). Set c = 0 in (2.4), the functions
a2 need
0, E) = - 7j$- E + f&re + .Le,e2 + 0(l71’) I
u,,, must be
T. Kapitula I Physica D 82 (1995) 36-59
50
Recall
that it has already
the results
of the previous
u+(3u,?
+ 2u2A
4u+u2 =a+u+ finishes
Remark
4.7.
that Q2,, = 0. The coefficient
paragraph,
then
(4.8) and
that
-&,
the proof.
0
If H = 0, then .Zp(r, 0,O) = 0, as in this case (0 = 0} is invariant
Now that the behavior follow
it is seen after some manipulation
Using
gives
of Wz
the tangent
intersect
hyperplanes
shall be done via the use of 2-forms. Kopell [lo], among others. For each coordinate satisfy the variational
transversely,
as they are carried This approach
for all c in Eq. (2.4).
near ,&,, and {r = 0}, respectively,
and W$ have been characterized
must be shown that these manifolds must
u2 will now be calculated.
+ $YE?) + O(l#)
the r’e coefficients
Equating
which
shown
i.e., that (4.2) holds. To accomplish along
by the flow generated
has been
used in Jones
it
this, one
by (2.4),.,,.
et al. [9] and Jones
This and
l-form 6x. These l-forms x of (2.4),.,, (x = r, u, etc.) there exists an associated equations associated with (2.4),.,,. When linearizing about the heteroclinic orbit
that exists for c = E = 0, the variational
equations
take the form
6r’ = uSr + r6u , 6~’ = -A’(r)Gr - 2uSu + br%R - USC + a+r’u& Ml’ = -2uSR
,
- Hr26c + (W(r) - (T)& ,
&‘=O, &‘=O. From
these
(4.9) l-forms
exterior
products
of forms
of any degree
can be constructed;
however,
as already
to a 2-plane T a stated, the interest here is in 2-forms P,, = 6x A Sy. Each such 2-form associates number that is the area of the projection of a unit square of T onto the coordinate planes of the two coordinates specified by P. In order to evaluate P, consider the following. Let N represent a k-dimensional manifold, k 2 2, and let T,,N represent the tangent space to N at a point p E N. Furthermore, let {a,(p)} represent a basis for TpN. Now suppose that S(p) C TpN represents a two dimensional subspace spanned by the vectors {a,(p), a,(p)}, 1 5 i, i 5 k. These vectors may be thought of as rows to a 2 x m matrix, A(p). If x and y represent two of the m coordinates of TpN, then Pxy is the determinant of the 2 x 2 submatrix obtained by considering the xth and yth columns of A(p). Note that this implies that PxY = -PYx and that Px, = 0. Finally, the evolution equation for Px4. is given by the product rule, i.e., P:, = 6x’ A sy + sx A Sy’ . The proof
of Theorem
4.1 will be established
in a series
of lemmas.
In the proofs
of these
lemmas
51
T. Kapitula I Physica D 82 (1995) 36-59
T(Z) and U(Z) will represent
the wave at a point
z. Furthermore,
Eq. (4.9)
calculating the appropriate 2-form equations. A key observation vector field is in the tangent space of both WY and W>>. It will be assumed behavior
in the following
of the manifolds
Proof.
as they hit the section
in the
so that r(0) = r,, i.e., the
&r(O, 0,O) > 0. was already
proved
in [9].
52 = [O, g’:‘, 0, 1901 3
and note that 5, E TWF. The satisfies the evolution equation Pi, = -UP,,
- ru2 )
2-form
PJO)
P,, , when
applied
to the subspace
sgn(P&))
spanned
by these
vectors,
= r’(0) 82; .
It is clear that if P,,(z) < 0 for z + 0, then Pr,(0) < 0. But Lemma 0, sgn(% r) = sgn(bH),
and since the vector
field is tangent
to [-1,
4.3(b) -3
ensures
that for 1 P r,, - r >
r, 0, 0, 01, it follows
that
+ -sgn(W
for z <<0. Thus, the wave.
if bH > 0, then
Consider
5, = [r’, u’,
the conclusion
as r’ < 0 along
of the lemma,
0,O)) = sgn(6).
the vectors
0, 0, 01,
s~=[o>~~,~,o,ol~
Pru, when applied
P:, = -UP,,
Pr,(0) < 0, which implies
0
Lemma 4.9. sgn(&F)(O,
The 2-form equation
is that
the vectors
5, = [r’, U’, O,O,O] ,
Proof.
proofs
I (see Eq. (4.1)).
here that bH > 0, for if H = 0, the result
It will be assumed
Consider
proofs that the wave has been translated
will be observed
Lemma 4.8. If bH 2 0, then
will be used extensively
in the following
+ br’u ,
of TWY spanned
to the subspace
by these vectors,
satisfies
the evolution
Pr,(0) = r’(0) &!y2
If b > 0, then P,,(z) < 0 for z G 0 implies implies Pr,(0) > 0.
that
PrL1(0)< 0. Similarly,
b < 0 and
P,,(z) > 0 for z 40
By Lemma 4.3, sgn (%yJ) = sgn(b) for 1% r. - r > 0. Since the vector field is tangent to [ -1, %!r, O,O, 0] for r near rO, for z < 0 it is true that sgn(P,,(z)) = -sgn(b). Since r’ < 0, this concludes the proof. 0 From the remark following Lemma 4.6, if H = 0 then pf(O, 0,O) = 0. Furthermore, in this case it has been shown in [9] that &‘r(O, 0,O) < 0. Thus, in the proof of the next lemma it will be assumed that H ZO. Lemma 4.10. If bH 2 0, then
sgn(W.
&y(O, 0) < 0. Furthermore,
the manifold
pfi
satisfies
sgn(pF(O,
0)) =
52
T. Kapitula I Physica D 82 (1995) 36-59
Proof.
Consider
the vectors
5, =[o,~:,~~,o,l], and note vectors,
that
s,=[o,~;,9~,1,0],
5, E TW”,.
The 2-forms
P,,, and Pfl,, when
applied
to the subspace
spanned
by these
satisfy
PI, = -2uP,,
+ br’P,,,
PJO)
)
= -Lq
By Lemma
P& = -~uP,,~ + Hr2 P&(O) = -9; and Z;p:’< 0 for 0 < r G 1. Thus,
4.6, sgn(_!G!?:‘)= sgn(H)
> 0 1 wR&N
P,,(z)
+ u ,
for z % 0
= -w(H)
First consider Pfj,. It is clear that H > 0 implies that Pn,(z) < 0, and that H < 0 implies z 2 0, so that sgn(P,,,(z)) = -sgn(H) for positive z and sgn( 9:)
= sgn(H)
and bH > 0, P,,, = 0 implies Now consider P,,,. Since sgn(P(,,) = -sgn(H) for large z implies that Pl,e(0) > 0, which concludes the proof. Now that
Pn,(z) > 0 for
it has been
characterized in order discussion surrounding
determined to prove equations
that the manifolds or disprove (2.12) and
intersect
F,(O, 0) = (&;1:’- 9Yir:’ + &;$?;)(o,
,
the flow on A,., persists. to show
of Theorem
P,,,(z) > 0 0 must
be
Specifically, by the that c’(0) does not
4.1. To show that c’(0) is
0)
does not depend on this parameter. For the following lemmas, in order sgn( W(r) - a) = -sgn(a)
transversely,
that the heteroclinic orbit (2.13) 1 ‘t will be necessary
depend on the parameter a_. Recall Eq. (4.3) and the statement independent of a ~, it is sufficient to show that
that Pit, < 0. Thus,
to make
a definitive
statement
it is required
that
0 5 r 5 r, ,
W(r,,) - (T # 0 ,
(4.10)
which leaves open the possibility (4.10) is strengthened to sgn( W(r) - 0) = -sgn(a)
,
for O(r) - (T to have zeros between 0 I r 5 r(, ,
r, and r,,. In the special
case that (4.11)
the statement of the next theorem is considerably simplified. As such, (4.11) will be assumed to be restrictions on d, , d,, throughout the rest of this paper. Both (4.10) and (4.11) can be considered and (T in parameter space, with the admissible domain for (4.11) being a subset of that for (4.10). It should be noted here that (4.10) and (4.11) are not necessary conditions to show that F, is independent of a_. These conditions are required only so that a more precise characterization of the wave speed c(e) can be achieved. Theorem
4.11.
sgn(a+) Then
Assume
= -sgn(ba)
sgn(F,(O,
that (4.11) .
0)) = sgn(ba).
holds,
and that
T. Kapirula I Physica D 82 (1995) 36.59
The proof of this theorem, lemmas. depends
As before, these on the parameters
that the behavior
Consider
that sgn(a+)
and
note
will be accomplished
in a series of
been
determined
= -sgn(bv).
Then
in Lemma
4.9.
sgn( &r(O, 0,O)) = sgn(ba).
the vectors
5,=[r’,u’,O,O,O],
satisfies
theorem,
lemmas will state how the behavior of each of the functions comprising P, associated with (2.4), and they will be proved via the use of 2-forms. Recall
of &yj has already
Lemma 4.12. Suppose Proof.
as in the proof of the previous
53
Sz=[O,%~,O,O,l],
5, E TW’I,“. The
that
the evolution
Pi,, = -UP,,
P,,, , when
2-form
applied
to the subspace
spanned
by these
vectors,
equation
+ u+r3u2 ,
Pr,,(0) = r’(0) &I .
Furthermore, by Lemma 4.3(c) and (4.11), sgn(% z) = sgn(ba) for 0 < r. - r 6 1, so that sgn(P,,(z)) sgn(ba) for 2 GO. Since r3u2 2 0 for 2 5 0, P,,, = 0 implies that sgn(Pi,) = sgn(a+). The fact sgn( & z) = -sgn(P,,,(O)) finishes the proof. Remark
4.13. If (4.10)
sgn(b(w(r,,)
holds,
As in Lemma
hypothesis
of this
lemma
must
be changed
to sgn(a+)
=
0)) = sgn(a).
4.10, consider
~,=[O,~p:‘,,ip~,O,l], and note satisfies
the
that 0
- a)).
Lemma 4.14. sgn(&:)(O, Proof.
then
= -
that
the vectors
~,=[O,=Y~,~~,l,O],
5, E TWF.
The 2-form
Pilc = -2uP,,,. + W(r) -
(T
P,,(O)
,
Pclr, when
applied
to the subspace
spanned
by these
vectors,
= 9fl .
of the lemma will It will be shown that sgn(P,,,.(z)) = sgn(a) f or z 2 0, from which the conclusion follow. Using (4.11), it is clear that Pflc = 0 implies that sgn(Pi,,.) = -sgn(a). Thus, if sgn(P,,,(z)) = sgn(a) for z +O, the claim is proved. However, by Lemma 4.6(b), this is necessarily true. 0 Lemma 4.15. Suppose Proof.
As in Lemma
that sgn(a+)
applied
to the subspace
+ br’P,,, + u,r’u
,
0)) = -sgn(bc).
= -sgn(2u+X
+ ba)
spanned
by these
vectors,
satisfies
P[,,(O) = 9:
Recall that in the previous lemma it was shown In the statement of Lemma 4.6(d) it is stated sgn(Zz)
sgn(pr(O,
52=[O,~:,~:.I,Ol.
P,,,-, when
P:, = -2uP,,
Then
4.14, let
5,=[O,~:,~:,O,I], The 2-form
= -sgn(bu).
that sgn(P&)) that
= sgn(cT) for z 2 0.
54
T. Kapitula I Physica D 82 (1995) 36-59
for 0 < r + 1. Thus,
for z 9 0, P,,,.(z) exhibits
as x < 0 and the hypotheses under
these
same
sgn(P:,,)
hypotheses,
= sgn(br”P,,, = sgn(ba
This
when
the same behavior.
implies
The proof
of the lemma
that for z + 0, sgn(Pi,,(z))
= -sgn(ba).
is now clear, Furthermore,
PUC= 0
+ a+?~)
- a +)
= sgn(ba) This yields
of the lemma
. 0
the result.
section
will be closed
previous
two theorems
Theorem
4.16.
intersects
W>, transversely;
sgn(c’(0))
with the following
of this section
Suppose
theorem,
which
is just a simple
consequence
of the
and Eq. (4.3).
that bH ~0,
that
furthermore,
Eq.
(4.11)
the resulting
holds,
and that
wave speed
sgn(a+)
= -sgn(ba).
Then
WY;
C(E) satisfies
= -sgn(ba)
Remark 4.17.
Note
that a_ has no influence
on c’(O).
It might be of some interest to consider the effect of perturbing Then the statement of Lemma 4.6(d) becomes
and the equations Pi,, = -UP,,
in Lemma
4.12 and Lemma
+ (a+r’u - x*)ru
,
4.15 become,
PI’,<= -2uP,,,
+ br’P,,,
x. Suppose
that x = -3116Q
+ EX*.
respectively, + a+r2u - ,y* .
Since ru < 0 along the wave, it can be easily checked that if sgn(x*) = sgn(a+), then the conclusion each of these lemmas is unchanged. This yields the proof for the remark following Theorem 1.1.
of
5. Flow on A, W>> transversely. The In the previous section conditions were derived under which Wp intersects task is to now determine the flow on the slow manifold A,,, so that the existence of a heteroclinic orbit for E nonzero can be proved or disproved. Since only the wavespeed guaranteed by Theorem 4.16 will be used, the manifold will hereafter be referred to as A,. For the rest of this section, the hypotheses of Theorem 4.16 will be assumed. The strategy in this section is twofold. First, it must be shown that in a suitable domain of parameter space critical points exist in A,. Following the discussion in Section 2, this is tantamount to showing that the curves given by (2.12) and (2.13), i.e., by OTC,+ br’@, R,=
- h(r) = 0 ,
W(r) - (+ - c’(0)Hr2 c’(O)( 1 - /?r’)
+ O(E) >
(5.1)
55
T. Kapitula I Physica D 82 (1995) 36-59
intersect
for (r, L?) sufficiently
on the critical
manifold
close to (r,,, 0). In the above
expression,
0,, represents
the IZ coordinate
JY~,. and
p = a_ /c’(O) . It is important arbitrarily.
to note
Also,
W(ro) - (T - c’(0)
Finally,
here
solutions Hri
4.16 and
Remark 4.17 guarantee (r,,, 0) unless
that
p can be chosen
will not be D(E) from
E = 0, (5.1)
has a solution
and
dfiR,
then by the implicit assumed
Theorem
= 0 .
note that if, when
dR,. --F#O, dr
that
of (5.1)
without
function
theorem
loss of generality
the intersection
persists
for E sufficiently
small.
As such, it can be
that E = 0 in (5.1).
of that point must be Once a critical point has been shown to exist on A,, the stability characteristics determined. Let (r,, 0, 0,) represent the critical point on Jtl, closest to (r,, 0,O). Linearizing (2.4) about this point manipulation, D =~c’(O)(l
and
taking
-@rz)(bri
the
determinant,
+20.+)(d$%)
D, of the
resulting
3 x 3 matrix
yields,
after
some
+ 0(~‘).
In the above. dR,. dr
W’(r)(I
- pr*)
+ 2pr( W(r) - CT)- 2c’(O)Hr
c’(O)( 1 - pr’)’
+ Q(E) 3
(5.2)
dfl, ---_ dr Note
i’(r) - 2brfl,il br2 + 20,
that at (r, 0) = (r,,, 0) E Al,,,
d@ti Qro> -= dr
br(?l ’
(5.3)
which is nonzero and finite, as b f 0 and h’(ro) < 0. Recall a basic fact of linear algebra, which states that D is the product of the eigenvalues associated with the linearization of (2.4) about (r*, 0,fi.J. Since it is already known that one of these eigenvalues is positive and one is negative, it is clear that D < 0 implies that (r*, 0, 0,) is a repeller on &,, and that D > 0 implies the critical point is an attractor on the manifold. Thus, if D < 0, the (r*, 0, 0,) possesses must coincide with Wz in a a two dimensional unstable manifold, W”“, which by uniqueness neighborhood of the critical point. Therefore, since this critical point is the closest to (r,, 0,O) on .M,, by invoking Theorem 4.3 one can conclude that the singular heteroclinic orbit persists for E nonzero. If D < 0, then the question remains unanswered, as in this case the one dimensional unstable manifold, does not W”“, of (r,, 0,0 ,) is a submanifold of W>, and the fact that WY intersects W> transversely W>. imply that W u’ intersects Now, in showing that one, and possibly more, critical points exist on AE, two cases will be considered. It will first be assumed that H is arbitrary, modulo the fact that bH 2 0 be satisfied. Under
T. Kapitula
56
this assumption,
it will be necessary
0 5 IHI G 1, then the second W(r) -
0,. = Under
(T
c’(O)( 1 - @Y?)
this scenario
Jtlt. Before
stating
sgn(%
Theorem
5.1.
the appropriate
any results,
sgn(bc) then
condition
some notation
to
on p yields
the existence
of multiple
reduces
to the requirement
for all rE [r_, r,]. f&.(r*) and
E A,.
of Theorem
Furthermore,
orbit
p = a_ /c’(O)
to (2.4)
and
Thus,
sgn(c’(0))
if IpI is sufficiently
dR - +J(r*))
4.16 hold.
Then
for lb1 sufficiently
= -sgn(ba),
= -sgn(2
of (5.1)
large
the hypothesis
D (0,
by
4.16 implies
which concludes
actually
and (5.2)
there
exists
shows that by making
an r, E (r_, r,)
1~ I sufficiently
such that fl,r,(rs) =
(r*)) ,
that sgn(c’(0))
sgn(P)
orbit.
large.
w(D) = -w(c’(O)) w(P) w(b) w(b) = -sgn(c’(O)) sgn( /?) .
Thus,
on
large there
in the theorem
which proves the first part of the theorem. To prove the second part of the theorem, it must now be shown that the hypotheses For ]p] sufficiently large, sgn(1 - prz) = -sgn(p). Thus, for 0~ E + 1,
= sgn(ba)
on
if
exists which is 0(e) close to the singular
that a_ > 0 be sufficiently
= sgn(b)
sgn(D)
points
(5.4)
For each given 6 > 0, an examination
But Theorem
critical
,
5.2.
Since
that
must be set. By (5.3)
that the hypotheses (r,, 0, 0,)
point
= -sgn(P)
dR. 2
If it is assumed
function theorem yields the existence of r_ < r(, < r, with Q,, existing S’mce a,,,(O) = 0, it is clear that $2, have opposite signs; furthermore,
Remark
sgn
result.
,
for 0 < E G 1 a heteroclinic
Proof. large,
reduces
a definitive
of the implicit
Assume
a critical
in (5.1)
36-59
+a(lHI+E).
[r_, y,]. Let 0, = fi,(r,). (5.4) sgn(0,) = -sgn(b).
exists
D 82 (1995)
for I/3/ +O to achieve
equation
(r,,)) = -sgn(b)
so an invocation
I Physica
= -sgn(ba),
imply that D < 0.
(5.5) so that
. the proof.
0
T. Kapitula
Now
that
the
existence
question
has
I Physica
been
settled,
heteroclinic orbits will be addressed. Specifically, not only have an orbit close to the singular orbit, be given, and suppose that the parameters can clearly be done, as W(r) is a quartic.
D 82 (1995) 36-59
the
question
57
of the
existence
of multiple
it will be shown that if IHI G 1, then it is possible to but also an orbit existing on &,. Let r. < Y, < r2 < r+
d, , dr, (T have been manipulated Since p can be chosen arbitrarily,
so that W(r,) - v = 0. This r, E (r,, r,) can be given,
1 - prf = 0. Now, for IHI < 1,
with
W(r) - fl
f12,.=
+ 0( pf
c’(O)( 1 - pr’)
so by making
r2 - r, sufficiently
Furthermore,
at these
Recall
Eq. (4.1). G’(r,))
sgn(
+ E) )
small 0,. will intersect
lj,, transversely
at points
r(, < r’r < rC < rT < r+.
points
It is a trivial
= sgn(cT) .
consequence
sgn( i’(r?))
of this condition
= -sgn(o)
that
.
(5.6)
Since dR,. dY(r,) the choice
w’(r,) =
of rC implies
all this together
sgn(c’(0))
then
sgn(b) sgn(a)
sgn(D) Thus,
=
1
-sgn(b)
+ Q(lHI + c> ,
that
(r: )I = sgn(a)
sgn/% Putting
c’(O)( 1 - /3rf)
sgn(a)
yields that for 0 < E 6 1, ,
r = rT
,
r = rt
if bu < 0. the point (ry ,O, KIT) is a repeller has th e reverse properties. The following
(rz, 0, C) Theorem
on A,;
otherwise,
theorem
it is an attractor.
has now been
The
point
proved.
5.3. Assume
r,, < r,
that the hypothesis of Theorem 4.16 hold, and that IHI =G1. Suppose that are chosen so that r2 - r, is sufficiently small, and that the parameters d,, dz, (T have
been chosen so that W(r,) - cr = 0. Finally, let rC E (r,, r7), and let p > 0 be chosen so that 1 - pr’ = 0. Then if ba < 0 and 0 < E G 1, there exists two heteroclinic orbits to (2.4), one of which is close to the singular orbit, and the other of which lies on the slow manifold .M,. Remark
5.4. Since
bcr < 0 implies
that c’(0) > 0, the requirement
that p > 0 implies
that u_ > 0.
As a final remark, it is possible to have two, or even three, heteroclinic orbits existing on Ju,. This can be achieved by making A,, sufficiently large, so that the curve 0,. intersects .&,, at more than two points. Since A,, is just an arc contained in an ellipse in (r’, CI)-space, this can be achieved by making b sufficiently small. For example, it can be checked that if IHI @ 1 and b’
T. Kapitula I Physica D 82 (1995) 36-59
58
transversely in at least 3 places, slow manifold A,.
which will yield the existence
of at least two heteroclinic
orbits
on the
Acknowledgements I would
like to thank
application suggestions,
Jim Keener
for his assistance
in helping
me to understand
of formal asymptotics. I would also like to thank the referees which helped to markedly improve the quality of this paper.
for their
the theory
and
comments
and
Appendix In this appendix wave
is singular
an expression in nature,
for the wave speed
the machinery
of matched
will be derived asymptotics
via formal must
asymptotics.
be used.
Since the
However,
since
interest here is only in finding the wave speed, this issue will not be pursued further. Let p(z) denote the wave that exists when E = 0, and recall that in this case 0 = 0. Writing perturbations as
the the
Y = p + EY, + 2Y* + . . . , n = dl,
+ E2L$ + . . . ,
c=c,E+C2E2+-., it is found
that at O(E) the equations
r; + (h(p) + pA’(p))r, = -clp’ must be satisfied. Since p+O as z*
(P2G)(4 Note
= c,H
that unless
expression
1
p’(s) ds -
the parameters
1P’(S)(P'(S))' 08
a+p’p’+ bp(p’0,) , (p%,) = -c,Hp4 + p’(W(p) -
x, the second equation T z
into the equation
a+
+
I z
can be integrated,
$(.Y) (O(p(s))
are chosen
ds - b
1P(S) P’(S) 1 R
so that
- a) ds
correctly,
for Y, and invoking
a)
0, is unbounded
the Fredholm
P’@>
(4p(t>>
as z*
alternative
- (7) dt ds
--x.
then
Substitution
yields
of this
that
.
5
It might be of some interest to compare this expression to the note that the condition bH 2 0 is not unreasonable, as the fact multiplying c, to be inverted. Furthermore, the conditions on a+ Thus, there is good agreement between the asymptotics and the remark, note that the theorem does not yield, as the asymptotics
conditions given in Theorem 1.1. First that p’ < 0 then allows the expression and W(Y) - u guarantee the sign of c, conditions of the theorem. As a final do, the potentially useful information
T. Kapitula I Physica D 82 (1995) 36-59
that for fixed b and H the derivative
of the wave speed depends
linearly
59
on the parameters
a,,
d, , d2,
IT.
References (11 J. Alexander, R. Gardner and C.K.R.T. Jones, A topological invariant arising in the stability of travelling waves. J. Reine Angew. Math. 410 (1990) 167-212. [2] P. de Mottoni and M. Schatzman, Existence and stability of the Thual-Fame pulse, preprint. [3] A. Doelman, Travelling waves in the complex Ginzburg-Landau equation, J. Nonlinear Sci. 3 (1993) 225-266. [4] A. Doelman and W. Eckhaus, Periodic and quasi-periodic solutions of degenerate modulation equations, Physica D 53 (1991) 249-266. [5] M.P. Eagles, Supercritical flow in a diverging channel, J. Fluid Mech. 57 (1973) 149. [6] W. Eckhaus and G. Iooss, Strong selection or rejection of spatially periodic patterns in degenerate bifurcations, Physica D 39 (1989) 124. [7] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana U. Math. J. 21 (1973) 193-226. [8] C.K.R.T. Jones, Stability of the travelling wave solutions of the Fitzhugh-Nagumo system, Trans. AMS, 286(2) (1984) 431-469. [9] C.K.R.T. Jones, T. Kapitula and J. Powell, Nearly real fronts in a Ginzburg-Landau equation, Proc. R. Sot. Edinburgh A116 (1990) 193-206. [lo] C.K.R.T. Jones and N. Kopell, Tracking invariant manifolds with differential forms, J. Diff. Eq. 108(l) (1994) 64-88. [ll] T. Kapitula. Stability of weak shocks in h-w systems, Indiana U. Math. J. 40(4) (1991) 1193-1219. [12] T. Kapitula, On the nonlinear stability of plane waves for the Ginzburg-Landau equation, Commun. Pure Appl. Math. 47(6) (1994) 831-841. [13] T. Kapitula. On the stability of travelling waves in weighted L^ spaces, J. Diff. Eq. 112(l) (1994) 179-215. [14] M. Landman, Solutions of the Ginzburg-Landau equation of interest in shear flow transition. Stud. Appl. Math. 76 (1987) 187-237. [15] B. Malomed and A. Nepomnyashchy, Kinks and solitons in the generalized Ginzburg-Landau equation, Phys. Rev. A 42( 10) (1990) 6009-6014. [16] W. van Saarloos and P. Hohenberg, Fronts, pulses, sources, and sinks in the generalized complex Ginzburg-Landau equation, Physica D 56 (1992) 303-367. [17] W. Schopf and W. Zimmerman, Multicritical behaviour in binary fluid convection, Europhys. Lett. 8(5) (1989) 41.
s
PHYSlCl [ol Physica
EISEVIEK
Singular heteroclinic
I)epariment
of Mathematics,
Received
11 June
D 82 (1995)
36-59
orbits for degenerate equations Todd
Kapitula
University
of Utah,
Salt Lake
modulation
City. UT 84112. USA
lYY3: revised 5 October 1994: accepted Communicated by C.K.R.T. Jones
5 October
1994
Abstract The problem of finding heteroclinic orbits for the quintic Ginzburg-Landau equation in the case that the When x = x, , where x is a distinguished imaginary parts of the coefficients are O(E), with 0 < ??+ 1, is considered. parameter in the equation, and E = 0 there exists a singular heteroclinic orbit connecting a stable finite amplitude state to the stable zero state. It is shown that for a certain region of parameter space this orbit persists for E small, with a wave speed C(E) which is O(E). Furthermore, using an approach reminiscent of the Melnikov method. the sign of c’(O) is calculated. This calculation is crucial in settling the persistence question. Finally, it is shown that for this wave speed it is possible for additional heteroclinic orbits to exist. These additional orbits connect two finite amplitude states.
1. Introduction Modulation equations have been used to study the behaviour Probably the most well known is the complex Ginzburg-Landau supercritical
bifurcations
A, = (1 + ip)A,,
is given,
+ [l - (1 +
in canonical
form,
of patterns at near critical conditions. (CGL) equation, which in the case of
by
iv)lAl*]A,
(1.1)
where (x, t) E R x [w’. In this equation the zero solution is unstable in favor of finite amplitude solutions. There are instances, such as in binary fluid convection [17], Taylor-Couette tlow with counterrotating cylinders, and Jeffrey-Hamel how [5,6], in which the CGL model is invalid, and an extension. hereafter called the degenerate modulation equation, must be considered. For completeness, let me briefly recall the manner in which this case arises. The following discussion can also be found in Doelman and Eckhaus [4]. In the circumstance of bifurcations of space-periodic solutions, the dynamics are governed by the Landau equation 0167-278Y/Y5/$09.50 (Q 1995 Elsevier SSDI Olh7-2789(94)00223-l
Science
B.V. All rights
reserved
37
T. Kapitula I Physica D 82 (199s) 36-59
Jg=q(k) where
a, +
P(k)(a,l’a,+ y(k)la,14 + .. ,
k is the wavenumber.
the degenerate convenient
(1.2)
If Re P(k) is of order one, then the modulation
case, in which
Re P(k) is small,
the corresponding
equation
modulation
(1.1) appears.
equation
resealings,
@, = i&I@]‘@ In the above
+
?? {(h(]@])
equation
If p,, is of order
oscillations
of @ happen
phenomena
discussed
equation,
which
+ iw(]@]))4
A and w are quartic
criticality.
unity,
which
on a much
+ i(/3,Qt]@]’
+ &@T@‘)
polynomials,
and E is a small parameter,
is the case that was studied
faster
time
scale
in [6]. If p,, is small or vanishing,
is given
in canonical
+ (1 + ip)p,,}
form
than
by Eckhaus
the evolution
then one arrives
+ O(E”)
(1.3)
due to the near and IOOSS [6], then
of (@I. This
leads
at the degenerate
to the
modulation
by
A,+(~+~~)A,,+(~(~A~)+~W(IAI))A+~(P,A~IAI~+P~A~A~).
(1.4)
h(r) =x + Y’- rJ, W(T) = d,r’ + d2r’, and p, = b, + ia,. When x < 0, as will be assumed in this (1.4) corresponds to a subcritical bifurcation. In this scenario both the zero solution and a finite
where paper,
amplitude state will be stable solutions for (1.4). If the underlying physical system possesses a reflection has purely parts
In
is, after some
real coefficients.
of the
coefficients
As such,
when
will be small.
symmetry,
the underlying
A physical
system
circumstance
the resulting is nearly in which
modulation
symmetric, the
equation
the imaginary
coefficients
degenerate modulation equation are real appears in binary fluid convection [17]. The aim of this paper is to find frontal solutions to (1.4) in the case that the imaginary
for
the
parts of the
coefficients are small. To accomplish this, the case in which all the coefficients are purely real is first considered, and then a determination is made as to which frontal solutions persist under perturbation. For convenience, the diffusion coefficient p will be set to zero, and the imaginary parts will be resealed by Ed,, and a, is substituted by ?? a, for 0 5 E + 1. Upon this resealing, w will be so that d, is substituted replaced by EW and p, will be replaced by b, + ea,. Letting z =x - ct, where c is the (as yet) undetermined solutions
front speed,
the question
of finding
solutions
is exactly
that of finding
steady
state
of
A, = A,, + cAZ + (~(1~1) + im(lAI))A Some results Eckhaus solutions. solutions
frontal
concerning
the existence
+ i(P,A,IAI’+
of steady
&AzA')
state solutions
.
(1.5)
to (1 S) can be found
in Doelman
and
[4]. They studied the existence and persistence of periodic, quasi-periodic, and homoclinic In the special case that p, = p, = 0, results about the existence of homoclinic and heteroclinic can be found in Jones, Kapitula, and Powell [9], De Mottoni and Schatzman [2], Malomed
and Nepomnyashchy Slow time periodic
[15], and Van Saarloos and Hohenberg [16], to name frontal solutions to (1.5) are of the form
A(z,t) = r(z) exp where (T is real. Substitution of the ansatz (1.6) into (1.5) framework, the question of fronts (resp. pulses) is reduced connections between appropriate fixed points.
a few.
(1.6) yields the system of ODE (2.2). In this to finding heteroclinic (resp. homoclinic)
T. Kapitula I Physica D 82 (1995) 36-59
38
The interest are given A(z,t)
When
here is to find connections
involving
stable
plane
waves.
Plane
wave solutions
to (1.5)
by = r,, exp[-i(a,,z
- E(T~)]
c = E = 0, the amplitude (Yf - B&q,
where
B=b,
paper,
(1.7)
Following
r. satisfies
the relation
- h(r,,) = 0 , -bz.
Assuming
describes
Kapitula
(1.7) that B*-4~0
and x>l/(B”-4), which will be done throughout this Now, define the parameter H by 4H = b, + b,. in the (ri, a,)-plane.
an ellipse
[12], a plane
wave will be nonlinearly
(BH-l)r~+~r~-2Hr~a,,+a~
Assuming
Thus,
relative
to (1.5)
if
af-rff-,y
that BH < 1, which will be done
the (ri, a,,)-plane.
stable
the intersection
throughout
of (1.8)
(see Fig. 3). (ri, a,,)-space Let the stable plane waves be parameterized
(1.8)
this paper,
with (1.7)
defines
Eq. (1.8)
describes
a hyperbola
in
a curve
of stable
plane
in
by the pair (r, a). Define
the parameter
waves
Q by
Q=H’-BH+l,
(1.9)
and note that BH < 1 implies a heteroclinic equation
orbit
connecting
that Q > 0. It turns out that when x = -3116Q the pair (r,,, Hri)
to (0,O).
Here
and c = E = 0, there
r,) is the largest
positive
root
_ &+Qr”-Q2r4=0,
exists of the
(1.10)
(r,,, Hri), so that this orbit Furthermore, there exists a curve of rest points for (2.2) containing singular (see Fig. 3). This curve represents the stable plane waves given by (1.7) and (1.8).
is
It is natural to wonder about the fate of this singular orbit for c and E nonzero. At first glance, one may expect that for c and E nonzero there exists an orbit which is 0( ]c] + ?? ) to the singular one. However,
careful
thought
for the wave to persist likely
that c’(0) will depend
must
satisfy
the additional
(c’(0) - a_r’)a
reveals
that this does not necessarily
c = C(C) is smooth,
on the parameters constraint
have to be true.
so that c(e) = c’(0) E + C~(E’). Under ai, d,, i = 1,2,
Suppose
that in order
this assumption,
and o. Now, for E nonzero
plane
it is waves
to (1.7)
= w(r) - (T + D(E) ,
(1.11)
where a, =a,
*a,,
(1.12)
Therefore, unless the parameters a_, d,, d,, and CT in (1.11) can be chosen correctly, plane wave solutions to (1.5) may not exist for E nonzero. Unfortunately, there is no a priori guarantee that these parameters can be so chosen. The purpose of this paper is to study this singular orbit and make a determination as to its persistence. The primary difficulty in this problem is showing that the intersection of (1.11) and (1.7) can be satisfied, as one must first find an expression for c’(0). It turns out that this can be accomplished via a method reminiscent of that of Melnikov. Malomed and Nepomnyashchy [15] considered the case in which B, = 0. Assuming that w(r) - u = 6(e)
,
39
T. Kapitula I Physica D 82 (1995) 36-59
the problem was solved via the use of formal asymptotics. They determined that the wave speed is 0(e2), which is not the case when /?, # 0. The primary difference between their work and the present work is that the front constructed in their paper is nonsingular. The
main
theorem,
result
of this paper
the function
is summarized
W(Y) is given
in the following
theorem.
In the
statement
W(Y) = O(I) + a_Hr” In the following however, generality, Theorem
that 2H - B # 0. If 2H - B = 0, then much
be modified
1.1. Let Q be as defined Suppose
(a) BH
a bit. In the following
statement
of what follows it is assumed,
is still true,
without
loss of
r,, as defined
and W(r) be as defined
in (l.lO),
in (1.13).
Let
(2H-B)HzO Oirsr,, - B)a).
= -sgn((2H
if CC > 0 is sufficiently
C(E) = O(E), which sgn(c’(0))
in (1.9),
that
G(r)-v#O,
(c) sgn(a+) Then
must
that Y(Z) + 0 as z + ~0.
x = -3116Q.
(b)
(1.13)
it is assumed
the proofs
of the
by
large.
for 0
is O(E) close to the singular
= -sgn((2H
orbit.
exists
a heteroclinic
orbit,
with
wavespeed
Furthermore,
- B)~T) .
Remark
define 1.2. The hypotheses of this theorem CT,d,, d2) (see Fig. 1 for the restrictions (B,H,c,a+,
an unbounded
domain
in (B, H)-space).
H
B
Fig. 1. Permissable
parameter
range
in (B, H)-space.
in the parameter
space
40
T. Kapitula I Physica D 82 (1995) 36-59
Remark
under
1.3.
Since
(1.4)
the hypotheses
Remark
result
1.4.
Remurk
under there
Since the theorem
continues
conclusion
is invariant
of this theorem
to hold
remains 1.5.
is proved
by showing
for x within
unchanged
C(E’)
if sgn(x*)
Since the theorem
the change
is proven
an expression
expression One
that certain - B)a).
by showing
that certain
it is seen
the
existence
that
manifolds
In fact.
= -sgn((2H
and
-p,.
c--t -c, as z+
intersect
if x = -3116Q
manifolds
intersect
and
z--t --z,
--x. transversely,
the
+ EX”, then
the
transversely,
the
so that one can say there exists a 6 > 0 so that holds. In the Appendix a forma1 perturbation the
condition
(2H - B)H 2 0 allows
a certain
to be inverted.
can
conjecture
for c’(O),
p,+
such that r(z)+0
of -3116Q.
requirement that (2H - B)H 2 0 can be relaxed slightly, if (2H - B)H 2 -6, then the conclusion of the theorem yields
of variables
also exists fronts
also that,
wonder
about
in addition
of multiple
to the orbit guaranteed
close to the curve of stable
plane
waves.
says that if the hypotheses
of Theorem
heteroclinic
by Theorem
Theorem
5.3 gives an answer
1.1 are satisfied
orbits.
1.1, there
Specifically,
exists orbits
to this conjecture.
and a_ > 0 is in a certain
one
can
which are O’(E) It essentially
interval,
sufficiently small such a scenario is possible, with the wavespeed satisfying c’(0) > 0. One may also wonder as to the stability of these travelhng waves for the PDE (1.4).
then
for H
For the slow
waves guaranteed by Theorem 5.3, it may be possible to adapt the techniques presented in Kapitula [l l] to show that these waves are stable. In the case of the orbit guaranteed by Theorem 1.1, the problem is a little more delicate, due to the singularity of the orbit. Here the results and ideas first presented
in Jones
[X] and Alexander,
Gardner,
and Jones
[l] may apply.
Using
their ideas,
it may be
possible to show that the singular orbit is also stable. This matter will be pursued in a later paper. This paper is organized in the following manner. In Section 2 the relevant system of ODES, Eq. (2.4),
is derived,
and the basic geometric
and heteroclinic
solutions
objects
to (1.5) exist when
stable and unstable manifolds. Section showing that the relevant center-stable
are discussed.
c = E = 0. These
In Section orbits
3 it is shown
correspond
that homoclinic
to the intersection
of
4 is devoted to tracking the invariant manifolds of (2.4), and and center-unstable manifolds intersect transversely, with the
wave speed c = C(E) satisfying the conclusion of Theorem 1.1. Finally, Section 5 is given to showing that (I. 11) intersects (1.7), upon which the proof of Theorem 1.1 will be complete. In the Appendix an expression for the wave speed is derived via formal asymptotics.
2. The equations Substitution
of the ansatz
A” + CA’ + (h(lA\)
(1.6)
+ iE(w(lAl)
into (1.5) - a))A
yields
+ i(/?,A’IA\’
where A(Z) = r(z) exp[-i ]’ a(s) ds] and ’= didz. u = r’Ir, one arrives at the first order system
u’ = -cu - u2 + iy7 - A(r) - Br’a a’ = -(c + 2~4)a + 4Hr’u
+ E(u(r)
the second-order
+ Ea+r’u
After
complex
+ &(A*)‘A’)
equating
valued
ODE
= 0,
real and imaginary
(2.1) parts,
and setting
.
- u + a_r’a)
.
(2.2)
41
T. Kapitula I Physica D 82 (199.5) 36-59
In the above
(Y, is as in Eq. (1.12).
u = r’ir is used so that the resulting
The transformation
to (2.2) is smooth near r = 0 [9]. In Doelman [4] it was noticed that when
c = E = 0, (2.2)
possesses
vector
field
a first integral
0,) = r2a - Hr’ It turns
out
singular
as r-+0.
that
upon
rewriting
As such,
(2.2)
with
it will be more
the variable convenient
R,, instead
of (Y. the vector
R=a-Hr’.
When
(2.3)
c = E = 0, it turns
in Eq. (2.4),
field becomes
to use the quantity
out that R = 0 is also a first integral
the vector
Substitution
of (2.3)
field is not singular into (2.2)
of the equations;
however,
as one can see
r = 0.
when
yields the system
r’ = ru .
u’ = -cu ~ u3 + R’ - h(r) + (2H ~ B)r’R 0’ = -(c + 2u)fl
- cHr’ + c(O(r)
+ cu+r’u
- CT+ a r20)
,
,
(2.4)
where h(r) = ,y + r’ - Qr” ,
(2.5)
and W(r) is defined in (1.13). It will be necessary (2.4). When this is done, the resulting equation
at times to append the equations c’ = 0 and E’ = 0 to will be represented as (2.4),. where .$ will be the
appended variable. Let us now study what is known of the dynamics of (2.4). For the invariant subspace (r = O}, it is shown in Landman [14] and Jones [9] that the flow here corresponds to a flow on S’. with critical points (u-, , 0,)
satisfying
the algebraic
LIZ+ CU + x - RZ = 0 , For E sufficiently [14] that solutions
(c + 2u)fl
words,
(2.6)
the attractor to (2.4)
(r(z), U(Z), 0(z))+((). In other
+ ECr= 0
pair in {r = O}, > 0 and ~1, < 0. The critical points form an attractor/repeller it is shown in [9] and and (u,, 0,) being the repeller. Furthermore,
small u
with (U , O_) being
relation
satisfy u,,
(0, U, . 0,)
(r(z), r’(z))+
n,)
,
possesses
and (0, u_, RJ has a one dimensional When c = ??= 0. (2.4) reduces to
(0,O)
as z -+ 2~
if and only if
z-+ 202 a one dimensional stable manifold coming out of the r = 0 plane, unstable manifold coming out of this plane (see Fig. 2).
r’ = ru . u’=-u’+fl’-h(r)+(2H-B)rzL?, R’ = -2uR.
Note
(2.7)
that Q = 0 is invariant,
so that in this subspace
(2.7)
reduces
to the planar
system
r’=ru,
u’ = -u2 - h(r) If x is further
restricted
(2.8) so that
T. Kapitula I Physica D 82 (1995) 36-59
42
Fig. 2. Flow near
(r = 0).
1 -4&
solutions,
which implies
in Section 1, Y()will denote the largest positive When u = 0, (2.7) has a curve of rest points Q2 - h(r) + (2H - B)r%
that the nonlinearity
for (2.8) is bistable.
As stated
root. given by
= 0.
(2.9)
Eq. (2.9) represents an ellipse in the (Y’, R)-plane. Let .M,, denote some open subset of this curve. M,, is said to be a normally hyperbolic manifold if for each p E A0 the linearization of (2.7) about p yields one zero eigenvalue zero. It is a routine
and two eigenvalues y,(p) with IRe y,(p)] b em g uniformly bounded calculation to show that this condition on yZ holds if for 0 < 6 G 1
4flZ + r/V(r) < -6 where
(Y, 0)
satisfies
,
away from
(2.10)
(2.9).
The curve
4R2 + rh'(r)= 0
(2.11)
describes a hyperbola in the (Y’. R)-plane, and the intersection of (2.11) with (2.9) determines the boundary of A,,. Because /\‘(r(,) < 0, ( Y,,, 0,O) E A,,, and hence so is some neighborhood (see Fig. 3). It is important to note here that A,, corresponds to the stable plane waves given by (1.7) and (1.8). Since 3/t(p) < 0 and y_(p) < 0 for each p E A,,, there exists a one dimensional stable manifold W’(p) and a one dimensional unstable manifold W”‘(p). Set w;, =
u WS(P), /JE(‘0
w;y = .g,
W”“(p) 0
Wf,~“s are invariant two dimensional manifolds. If follows from the work of Fenichel [7] that for each fixed c and E sufficiently small there is an invariant manifold, .,,!I,.,, of (2.4) nearby to JU,,, with the flow
43
T. Kapitula I Physica D 82 (1995) 36.~9
Fig. 3. The critical
on A,.,
begin
0( Ic] + 6). Furthermore,
nearby to WX”“, with the solutions Define w;; where
= l._,. w,“: , <.t
is taken
here.
For (2.4),.,,,
for each fixed c and E there
on W:;“’ approaching
over c and E sufficiently
“& = [M’(.n, c, E), M”(fl,
small.
& is a three-dimensional c, E), LJ, c,
and WY is a four dimensional WY = [Y, 9T2”(r,n,
.I(,,.
A,,
manifold
c, E), R,c,
See Remark manifold
manifolds
fast as z+
W:,;“’
5~.
on Ju,., must satisfy
graph
for my choice
of
is
?? ]) whose
graph
is given
by
e]
that c = C(E) with c(0) = 0, and that the dependence points
invariant
2.1 for the reason
whose
Similarly, a four dimensional manifold Wi could be defined. It is quite natural to wonder if any of the critical points constituting critical
exists
exponentially
./ii = t._. Je (‘.t Cc’
the union
notation
manifold
J/J,, persist
for c, E # 0. Suppose
of c on E is at least C*. Under
this scenario,
any
u = 0 and
(a) R’ - h(r) + (2H - B)r*L? = 0 (b)
R =
O(r) - m - c’(O)Hr’ c’(0) -u-r*
Eq. (2.12)(a) is a representation (2.12)(b) can be rewritten as R=
W(r) - (T - c’(O)Hr’ c’(O)( 1 - pr’)
+ Q(C) for A,,.
+ Q(E) >
(2.12) Assuming
that
c’(0)
can be chosen
independently
of a_,
(2.13)
where p = a_ /c’(O) can be arbitrarily chosen. If p is sufficiently large, then (2.13) will intersect JH,, at one point. If H is sufficiently small and p = Q(l), then the parameters d, , d, = d, + amH, and IT can be
44
T. Kapitula i Physica D 82 (199.5) 36-59
manipulated
so that
(2.13)
points. The maximality explored in Section 5. Now
consider
possesses
a one
manifold,
W”‘.
the
Remark
unstable from
critical
coming
When
manifold
?? ), P’(r.
at minimally
is due to the fact that
points
(0. c[,, 0,).
stable
manifold,
and their
graphs
c, E), c, E] ,
the R = 0 plane
one point.
W is quartic.
It is easy
and at maximally
This
to compute
idea
that
for
will be more (2.4).
four fully
(0, CI_, Q_)
W’,
and (0. 14_, L?) has a one dimensional unstable For Eq. (2.4)C.,t these then become three dimensional
arc given by wp
is viewed
from .24,, is coming
the left. Hence.
3. Existence
.,&,, transversely
out of the {Y = 0} plane.
manifolds,
= [Y, Y’(Y. c, 2.1.
condition
dimensional
center-(un)stable w:
intersects
= [r,
from a vantage
point
from the right and the stable
c, E). (‘. E] .
in which R > 0. it appears
as if the
from (0, ~1,. 0, ) is coming
manifold
the %! and Y notation.
of fronts when c = E = 0
Set c = E = 0 in (2.4).
so that Eq. (2.7)
is considered.
In the invariant
space
{0 = 0}, i.e.,
for Eq.
(2.X). the function
E(r,
14) =
f(n4)’
+
I 0
s/i(s)
ds
is a first integral. Thus, the phase portrait for (2.8) is relatively portrait is depicted for different values of x. When Actually. manifold transverse. persists.
x = -3/16Q,
WY
intersects
W)
in a nontrivial
easy to determine.
fashion,
i.e..
In Fig. 4 the phase
a heteroclinic
orbit
exists.
the fact that (Y,,. 0,O) E &,, implies that the orbit is singular. Since W”,\ is a three dimensional and W\/ is a two dimensional manifold for (2.4), , it is possible that the intersection is If so, by the implicit Unfortunately,
function
this is not sufficient
theorem
there
to determine
exists
a c = c(e) such
that the heteroclinic
that
the intersection
orbit persists,
on JZdC.tmust be understood. For instance, from the discussion of the previous section any critical points exist on JL!,.,. The remaining focus of this paper will be on these
as the flow
it is yet unclear issues.
if
When x E (-3116Q). W\, intersects WY nontrivially, so that a homoclinic orbit to Y = 0 exists. Setting C?= E(T. W> and Wl,F are each three dimensional manifolds for (2.4),,,;, so that the intersection can possibly be transverse. It can be shown, using the methods presented in this paper and [9] that these manifolds do indeed intersect transversely. As such, there will exist a c = c(e), i = C?(E) so that for E nonzero and small the intersection persists. which implies that the homoclinic orbit will continue to exist. The proof of this assertion will be postponed until a later date. It has recently been shown in de Mottoni and Schatzman [2]. however. that in the special case of B = H = a, = d2 = 0, the pulse continues to exist for d, sufficiently small (the parameter d, is the perturbation parameter in their paper). Finally. if x E (-l/4(2,-31 16Q), then W’i intersects W’# nontrivially, so that a homoclinic orbit to r = r,, exists. The persistence of this orbit will not be studied in this paper, as an excellent treatment has already been done in Doelman and Eckhaus [4] and Doelman [3]. it is possible. however, that the methods and ideas presented in this paper can be used to duplicate some of their results. In these papers it is shown that in a certain region of parameter space the homoclinic solution breaks into a
T. Kapitula
I Physica
4
_,
45
D b’2 (1995) 36-59
Y E C-3/16Q,O)
“.‘,,
Fig. 4. Flow on {R = 0)
heteroclinic present
cycle, which consists
on Al,,,
and the fast wave corresponds
It is of some interest scenario, more
the subspace
can
of a fast travelling
be said
to note the structure (0 = 0) is invariant
about
the
existence
wave and a slow travelling
to the intersection of solutions
for (2.4) when and
stability
of IV:
in the special
wave. The slow wave is
with Wk.
case that H = O(E). Under
E = 0 and for all real valued
of heteroclinic
solutions.
c. Thus,
Basically,
the
this much
results
presented wavespeed
in [9] hold. In that paper, it is shown that if x E (-l/4&, -3/16Q), then a wave exists with C(E) < 0. Furthermore, it is shown in Kapitula [ 131 that this wave is stable, modulo a spatial and rotational translation. For x E (-3/16Q, 0) a wave also exists; however, here it is necessary that
c = C(E) and manifold of not yet been no unstable
C?= G(C). This change in dimension in parameter space is due to the fact that the unstable (Y,,, 0,O) changes dimension from two to one as c crosses zero. The stability of this front has determined, although the proof presented in [13] can be modified to show that there exists eigenvalues. The case of x = -3116Q was not pursued in [9].
4. Persistence
of manifold
intersection
Following heteroclinic (2.4),,e I=
the discussion of the previous section, the initial step in showing that the singular orbit that exists for x = -3116Q persists is to show that WT intersects . W$ transversely for when E = 0. Set 0 < Y, < Y(, to be the other positive root of A(r) = 0, and let {(Y, u, f1, c, E):Y = r,, u < 0, /i-q, (cl, (El
)
(4.1)
46
T. Kapitula I Physica D 82 (199.5) 36-59
where
Y’< 0 for all 0 < r < r. and
0 < 6 + 1. Since
dimensional
manifold
c, E), n, c, E] )
w:,\ t-l I = [K’(fl, Since
the orbit
the function
0) )
Iv;. n I = [P(c, E), P(c,
are
smooth,
WY n I is a three
Let E), c, E]
2?(0,0)
= 0.
F(c, E) by
F(c, E) = K@?(c, and note
manifolds
manifold.
exists for 0 = c = E = 0,
!%‘(O, 0,O) = Y(O, Define
the
and W> n I is a two dimensional
e) )
E), c, E) - P’(c,
that F(0, 0) = 0. It is clear that F(c,
?? )= 0
implies
that W z intersects
IV;
nontrivially.
0) # 0 )
F,(O, 0) = (& :: - 9,” + a;P;)(o,
If (4.2)
then by the implicit function theorem there exists a smooth function c = C(E), with c(0) = 0, such that F(c(E), E) = 0 for E sufficiently small. Thus, assuming that (4.2) holds, the manifolds intersect transversely.
As a side remark,
c’(0) = -FJO,
O)/F,.(O, 0) .
Theorem 4.1. Suppose Proof.
showing
(4.3)
that (2H - B)H 2 0. Then
This is an immediate
Before
note that
consequence
that (4.2) holds,
of Lemmas
it is necessary
{r = O}. This can be accomplished
near
(4.2)
holds,
with F,.(O, 0) > 0.
4.8-4.10.
to derive
0
approximations
for Wg near A, and for IV>
by using the fact that each of these manifolds
the flow generated by (2.4),,,. Before making any statements is necessary. For notational ease, for the rest of the paper
about set
Wz, however,
is invariant
under
the following
lemma
b=2H-B.
(4.4)
Lemma 4.2. The manifold
& satisfies
(a) Mii(O, 0, 0) = 6
b 0
(b) M:(O,O,O)=
-&bH
(c) M~(O,O,O)=&b(W(r,,)-ir). 0
Proof.
When
g(R)
c = E = 0 the critical
= f12 + b(M)*
- h(M)
manifold
&,, exists,
so that the function
Mr(fl,
0,O) satisfies
= 0.
Differentiating g and setting R = 0 yields part (a) of the lemma. Set 77= (a, c, E). Since M”(0, 0,O) = 0, the functions comprising M’(R,
c, E) = r&
+ r,c + r,E + 0( 1~1~),
M”(R,
c, 6) = u,.c + U,E + O( 1#)
)
J& can be written
T. Kapitula
where
the coefficients
I Physica
are to be determined.
invariant under the how generated ut determines (c).
This will be accomplished
by (2.4), .~. Note that U, determines
The coefficient
yn is already given in (a). To determine so that (M’)’ = M’M”,
implies
47
D 82 (199-f) 36-59
by using
the fact that
part (b) of the lemma,
the other coefficients,
note that, on J&, Y’= TU
rJ2’ + f7(/~1’) = r,,(u,.c + U,E) + 0(1q12) .
Similarly,
Substituting
+ (O(r(,)
(4.6)
proves
4.3. The
proof
satisfies,
for r near r,,,
b(r - r,)) + O((r - r,,)‘)
bH + 0(r - r,,)
b( W(r,,) - a) + U(r - r,,) .
of (b) and (c) follows
immediately
all that is left is to prove (a). set u = %!“(r, 0, O,O), so that when
= [r, u(r, fl),
0,0,0]
and u = 0 for points
rh’(r)
the fact that
c = E = 0, WF
J&!C WY
of
has the graph
theorem
,
about
each point p E A0 satisfy
= 0,
with the associated eigenvector being (-r, -y, 0). Here r represents the coordinate critical manifold A,,. Let y+ denote the positive eigenvalue, so that y&(O) = +rh’(r)
Then
and the results
p E A(,, it is clear that uI(r,,, 0) = 0. By Taylor’s
0) = Urrl(ro, O)(r - r,,) + C((r - rJ)
R’+
from
.
so all that is left is to calculate u,.fj(rg, 0). The eigenvalues for the linearization of (2.4) y’+
yields
(_r,~c~i,,3f2
0,0,O)= &
Since A,, C WT Gr,
WG
manifold
Lemma 4.2. Thus. For convenience, Wz
then
0
.R:l(r,O,O.O)=-$&
The
the coefficients
the lemma.
(C) 92:I@, Proof.
and equating
(4.6)
(W(rd - a>r, = r,,q ,
(4 %Xr, (),O,0) = (b)
- CT)C+ 0(17~/‘) .
into (4.5)
= rouC ,
-Hr,?,r,,
Lemma
(45)
on A
R’ = -Hrf,c
which
J& is
and that
u, = y+ lr, which
- 0’.
yields
M’(f2,
0,O) of the
T. Kapitula I Physica D 82 (1995) 36-59
48
In this calculation leads
the result given in Lemma
to the conclusion
4.2(a) is explicitly
used.
Further
manipulation
of the lemma.
Remark 4.4. If H = 0, then
in [9] that 3 fY(r,0, 0,O) > 0 for 1 % r,, - r > 0.
it is shown
Remark 4.5. If b = 0, then %~j(r, 0, 0,O) = 0, as in this case (2.4) is invariant R+ -0. For the next lemma,
recall
Lemma 4.6. The manifold (a) Z:‘(r,
Y;p:'(O,0,O)
=
-
Recall
which describes DC@+,
for r sufficiently
small,
O(r)
+ +
bcr r’ + Q(r’) .
u2+cu+~-f12 i
(c+2u)Q+Ea
the critical
0
1
point
=
(u, , 0,)
(b) and (c) will only entail 2u+), and
II 0
)
of (2.4) when r = 0. This critical
showing
that u+(c, E) and n+(c,
point is contained
E) behave
in W.>, so
appropriately.
Since
0, 0,O) = diag(2u+,
G,(u+,O,O,O)=[u+,O]‘, the result
c = E = 0, U, = -fi.
Eq. (2.6),
G(u,fl*c,E)= that proving
the transformation
+ 0(r)
+
(d) 2’z(r, 0,O) = 2a+%$Proof.
under
r2 + 0(r’)
+
(b) iYf(r, 0, 0) = - $ (C)
that when
IV> satisfies,
0,O) = - $
of the above 0
now follows
G,(u+,O,0,0)=[0,(~]~-, via an application
of the implicit
function
theorem.
Note
also that
(R+)(
=
(u+), = 0. For the rest of this proof define 77= (r, c, E). The proof of (a) and (d) is a bit more complicated, and the ideas of Lemma 4.2 need to be used. Part (a) will be proven first; therefore, set E = 0 in (2.4). Note that for (2.4),. the manifold W> is tangent to the subspace spanned by the vectors [l, 0, 0, 01, [0,0, 0, l] as r-0. Thus, the functions comprising W’$ have the expansion Y7”(r, c, 0) = u, -+c
+ S(lql’)
,
Y”(Y, c, 0) = .C2,,r’+ Q,.rc + f&c’+ In this expansion
the results
CyIql”) .
of the previous
paragraph
are being
used.
49
T. Kapitula I Physica D 82 (1995) 36-59
It will now be shown order.
that Q, = R,,. = Q,. = 0, so that the expansion
for 3”
will necessarily
be third
of W,b,,
By the invariance (3?“)’ = -(c + 2Y’)3?
- cHr’ ,
(4.7)
so that 2&Y
+ L?,,r’c + a( ln13) = -2U+ (0,,Y2 + Q,YC + @)
r’ = r9”
Since
2f&u+r’ Equating
on WL, this further + &u+rc
coefficients
The function LP(r, where found.
implies
that
+ O(lql’) = -2u+(f&r’ now proves
+ C( InI’)
+ QCrc + i&c’)
+ 6((q13) .
the claim.
3”’ now has the expansion
c, 0) = R,r3 + &r’c
+ fiyrc2 + R,c’ + fT(I~I”) ,
the coefficients 0, need to be determined. Again using (4.7), it can be seen that
u+(IfI,r’
+ 2R,r2c + &rc’)
To prove
+ tY([ql”) = -Hr’c
part
(a) of the lemma,
only
(2.4),.
As r-0,
Y’(r,
to the subspace
the manifold
0, E) = u, + u,,r2 + up
?? )= -
L!?“(r, 0, The result
W,k, is tangent
comprising
of part
T$
+
spanned
and r’ = rY,
U II
+ u,,e2 + 0(171’) ,
E + 0( 1771’).
(b) is used here. that u,, = 0. The invariance - h(r) + br’9”
+
+ 0(1~1’) = -2u+u,,re
+
+ (3”)’
u,p)
the coefficients =
_-
[l, O,O, 01, [0,0,O. 11, so that
of WL implies
?? z+r’5’f”
(4.8)
so that
u+(2u,,r’+ Equating
by the vectors
then
SU,
yields
CT2
h”(0)
so that the claim
’
u,,
=jg
1
U,,
= 0,
is proved.
To prove (d), the 0(/7~]‘) t erms of .Y’ must be determined. calculated. Furthering the expansion for 9’ and .L@’yields
Specifically,
the coefficient
iY?(r, 0, E) = u, + u,,r2 + u,,~~ + u,r3 + uzr2E + u3rE2 + u4rfz3 + fT(l~l’) , LP(r,
and consider
have the expansion
It will first be shown (3”)’ = -(Z’)’
be
- 2u+(L?,r” + Rzr’c + R,rc’ + fl.,c3) + 0(1~/“) .
Thus, 0, = fi, = 0, = 0 and 4u+R, = -H, which yields the result. To finish the proof of the lemma, all that is left to show is part (d). Set c = 0 in (2.4), the functions
a2 need
0, E) = - 7j$- E + f&re + .Le,e2 + 0(l71’) I
u,,, must be
T. Kapitula I Physica D 82 (1995) 36-59
50
Recall
that it has already
the results
of the previous
u+(3u,?
+ 2u2A
4u+u2 =a+u+ finishes
Remark
4.7.
that Q2,, = 0. The coefficient
paragraph,
then
(4.8) and
that
-&,
the proof.
0
If H = 0, then .Zp(r, 0,O) = 0, as in this case (0 = 0} is invariant
Now that the behavior follow
it is seen after some manipulation
Using
gives
of Wz
the tangent
intersect
hyperplanes
shall be done via the use of 2-forms. Kopell [lo], among others. For each coordinate satisfy the variational
transversely,
as they are carried This approach
for all c in Eq. (2.4).
near ,&,, and {r = 0}, respectively,
and W$ have been characterized
must be shown that these manifolds must
u2 will now be calculated.
+ $YE?) + O(l#)
the r’e coefficients
Equating
which
shown
i.e., that (4.2) holds. To accomplish along
by the flow generated
has been
used in Jones
it
this, one
by (2.4),.,,.
et al. [9] and Jones
This and
l-form 6x. These l-forms x of (2.4),.,, (x = r, u, etc.) there exists an associated equations associated with (2.4),.,,. When linearizing about the heteroclinic orbit
that exists for c = E = 0, the variational
equations
take the form
6r’ = uSr + r6u , 6~’ = -A’(r)Gr - 2uSu + br%R - USC + a+r’u& Ml’ = -2uSR
,
- Hr26c + (W(r) - (T)& ,
&‘=O, &‘=O. From
these
(4.9) l-forms
exterior
products
of forms
of any degree
can be constructed;
however,
as already
to a 2-plane T a stated, the interest here is in 2-forms P,, = 6x A Sy. Each such 2-form associates number that is the area of the projection of a unit square of T onto the coordinate planes of the two coordinates specified by P. In order to evaluate P, consider the following. Let N represent a k-dimensional manifold, k 2 2, and let T,,N represent the tangent space to N at a point p E N. Furthermore, let {a,(p)} represent a basis for TpN. Now suppose that S(p) C TpN represents a two dimensional subspace spanned by the vectors {a,(p), a,(p)}, 1 5 i, i 5 k. These vectors may be thought of as rows to a 2 x m matrix, A(p). If x and y represent two of the m coordinates of TpN, then Pxy is the determinant of the 2 x 2 submatrix obtained by considering the xth and yth columns of A(p). Note that this implies that PxY = -PYx and that Px, = 0. Finally, the evolution equation for Px4. is given by the product rule, i.e., P:, = 6x’ A sy + sx A Sy’ . The proof
of Theorem
4.1 will be established
in a series
of lemmas.
In the proofs
of these
lemmas
51
T. Kapitula I Physica D 82 (1995) 36-59
T(Z) and U(Z) will represent
the wave at a point
z. Furthermore,
Eq. (4.9)
calculating the appropriate 2-form equations. A key observation vector field is in the tangent space of both WY and W>>. It will be assumed behavior
in the following
of the manifolds
Proof.
as they hit the section
in the
so that r(0) = r,, i.e., the
&r(O, 0,O) > 0. was already
proved
in [9].
52 = [O, g’:‘, 0, 1901 3
and note that 5, E TWF. The satisfies the evolution equation Pi, = -UP,,
- ru2 )
2-form
PJO)
P,, , when
applied
to the subspace
sgn(P&))
spanned
by these
vectors,
= r’(0) 82; .
It is clear that if P,,(z) < 0 for z + 0, then Pr,(0) < 0. But Lemma 0, sgn(% r) = sgn(bH),
and since the vector
field is tangent
to [-1,
4.3(b) -3
ensures
that for 1 P r,, - r >
r, 0, 0, 01, it follows
that
+ -sgn(W
for z <<0. Thus, the wave.
if bH > 0, then
Consider
5, = [r’, u’,
the conclusion
as r’ < 0 along
of the lemma,
0,O)) = sgn(6).
the vectors
0, 0, 01,
s~=[o>~~,~,o,ol~
Pru, when applied
P:, = -UP,,
Pr,(0) < 0, which implies
0
Lemma 4.9. sgn(&F)(O,
The 2-form equation
is that
the vectors
5, = [r’, U’, O,O,O] ,
Proof.
proofs
I (see Eq. (4.1)).
here that bH > 0, for if H = 0, the result
It will be assumed
Consider
proofs that the wave has been translated
will be observed
Lemma 4.8. If bH 2 0, then
will be used extensively
in the following
+ br’u ,
of TWY spanned
to the subspace
by these vectors,
satisfies
the evolution
Pr,(0) = r’(0) &!y2
If b > 0, then P,,(z) < 0 for z G 0 implies implies Pr,(0) > 0.
that
PrL1(0)< 0. Similarly,
b < 0 and
P,,(z) > 0 for z 40
By Lemma 4.3, sgn (%yJ) = sgn(b) for 1% r. - r > 0. Since the vector field is tangent to [ -1, %!r, O,O, 0] for r near rO, for z < 0 it is true that sgn(P,,(z)) = -sgn(b). Since r’ < 0, this concludes the proof. 0 From the remark following Lemma 4.6, if H = 0 then pf(O, 0,O) = 0. Furthermore, in this case it has been shown in [9] that &‘r(O, 0,O) < 0. Thus, in the proof of the next lemma it will be assumed that H ZO. Lemma 4.10. If bH 2 0, then
sgn(W.
&y(O, 0) < 0. Furthermore,
the manifold
pfi
satisfies
sgn(pF(O,
0)) =
52
T. Kapitula I Physica D 82 (1995) 36-59
Proof.
Consider
the vectors
5, =[o,~:,~~,o,l], and note vectors,
that
s,=[o,~;,9~,1,0],
5, E TW”,.
The 2-forms
P,,, and Pfl,, when
applied
to the subspace
spanned
by these
satisfy
PI, = -2uP,,
+ br’P,,,
PJO)
)
= -Lq
By Lemma
P& = -~uP,,~ + Hr2 P&(O) = -9; and Z;p:’< 0 for 0 < r G 1. Thus,
4.6, sgn(_!G!?:‘)= sgn(H)
> 0 1 wR&N
P,,(z)
+ u ,
for z % 0
= -w(H)
First consider Pfj,. It is clear that H > 0 implies that Pn,(z) < 0, and that H < 0 implies z 2 0, so that sgn(P,,,(z)) = -sgn(H) for positive z and sgn( 9:)
= sgn(H)
and bH > 0, P,,, = 0 implies Now consider P,,,. Since sgn(P(,,) = -sgn(H) for large z implies that Pl,e(0) > 0, which concludes the proof. Now that
Pn,(z) > 0 for
it has been
characterized in order discussion surrounding
determined to prove equations
that the manifolds or disprove (2.12) and
intersect
F,(O, 0) = (&;1:’- 9Yir:’ + &;$?;)(o,
,
the flow on A,., persists. to show
of Theorem
P,,,(z) > 0 0 must
be
Specifically, by the that c’(0) does not
4.1. To show that c’(0) is
0)
does not depend on this parameter. For the following lemmas, in order sgn( W(r) - a) = -sgn(a)
transversely,
that the heteroclinic orbit (2.13) 1 ‘t will be necessary
depend on the parameter a_. Recall Eq. (4.3) and the statement independent of a ~, it is sufficient to show that
that Pit, < 0. Thus,
to make
a definitive
statement
it is required
that
0 5 r 5 r, ,
W(r,,) - (T # 0 ,
(4.10)
which leaves open the possibility (4.10) is strengthened to sgn( W(r) - 0) = -sgn(a)
,
for O(r) - (T to have zeros between 0 I r 5 r(, ,
r, and r,,. In the special
case that (4.11)
the statement of the next theorem is considerably simplified. As such, (4.11) will be assumed to be restrictions on d, , d,, throughout the rest of this paper. Both (4.10) and (4.11) can be considered and (T in parameter space, with the admissible domain for (4.11) being a subset of that for (4.10). It should be noted here that (4.10) and (4.11) are not necessary conditions to show that F, is independent of a_. These conditions are required only so that a more precise characterization of the wave speed c(e) can be achieved. Theorem
4.11.
sgn(a+) Then
Assume
= -sgn(ba)
sgn(F,(O,
that (4.11) .
0)) = sgn(ba).
holds,
and that
T. Kapirula I Physica D 82 (1995) 36.59
The proof of this theorem, lemmas. depends
As before, these on the parameters
that the behavior
Consider
that sgn(a+)
and
note
will be accomplished
in a series of
been
determined
= -sgn(bv).
Then
in Lemma
4.9.
sgn( &r(O, 0,O)) = sgn(ba).
the vectors
5,=[r’,u’,O,O,O],
satisfies
theorem,
lemmas will state how the behavior of each of the functions comprising P, associated with (2.4), and they will be proved via the use of 2-forms. Recall
of &yj has already
Lemma 4.12. Suppose Proof.
as in the proof of the previous
53
Sz=[O,%~,O,O,l],
5, E TW’I,“. The
that
the evolution
Pi,, = -UP,,
P,,, , when
2-form
applied
to the subspace
spanned
by these
vectors,
equation
+ u+r3u2 ,
Pr,,(0) = r’(0) &I .
Furthermore, by Lemma 4.3(c) and (4.11), sgn(% z) = sgn(ba) for 0 < r. - r 6 1, so that sgn(P,,(z)) sgn(ba) for 2 GO. Since r3u2 2 0 for 2 5 0, P,,, = 0 implies that sgn(Pi,) = sgn(a+). The fact sgn( & z) = -sgn(P,,,(O)) finishes the proof. Remark
4.13. If (4.10)
sgn(b(w(r,,)
holds,
As in Lemma
hypothesis
of this
lemma
must
be changed
to sgn(a+)
=
0)) = sgn(a).
4.10, consider
~,=[O,~p:‘,,ip~,O,l], and note satisfies
the
that 0
- a)).
Lemma 4.14. sgn(&:)(O, Proof.
then
= -
that
the vectors
~,=[O,=Y~,~~,l,O],
5, E TWF.
The 2-form
Pilc = -2uP,,,. + W(r) -
(T
P,,(O)
,
Pclr, when
applied
to the subspace
spanned
by these
vectors,
= 9fl .
of the lemma will It will be shown that sgn(P,,,.(z)) = sgn(a) f or z 2 0, from which the conclusion follow. Using (4.11), it is clear that Pflc = 0 implies that sgn(Pi,,.) = -sgn(a). Thus, if sgn(P,,,(z)) = sgn(a) for z +O, the claim is proved. However, by Lemma 4.6(b), this is necessarily true. 0 Lemma 4.15. Suppose Proof.
As in Lemma
that sgn(a+)
applied
to the subspace
+ br’P,,, + u,r’u
,
0)) = -sgn(bc).
= -sgn(2u+X
+ ba)
spanned
by these
vectors,
satisfies
P[,,(O) = 9:
Recall that in the previous lemma it was shown In the statement of Lemma 4.6(d) it is stated sgn(Zz)
sgn(pr(O,
52=[O,~:,~:.I,Ol.
P,,,-, when
P:, = -2uP,,
Then
4.14, let
5,=[O,~:,~:,O,I], The 2-form
= -sgn(bu).
that sgn(P&)) that
= sgn(cT) for z 2 0.
54
T. Kapitula I Physica D 82 (1995) 36-59
for 0 < r + 1. Thus,
for z 9 0, P,,,.(z) exhibits
as x < 0 and the hypotheses under
these
same
sgn(P:,,)
hypotheses,
= sgn(br”P,,, = sgn(ba
This
when
the same behavior.
implies
The proof
of the lemma
that for z + 0, sgn(Pi,,(z))
= -sgn(ba).
is now clear, Furthermore,
PUC= 0
+ a+?~)
- a +)
= sgn(ba) This yields
of the lemma
. 0
the result.
section
will be closed
previous
two theorems
Theorem
4.16.
intersects
W>, transversely;
sgn(c’(0))
with the following
of this section
Suppose
theorem,
which
is just a simple
consequence
of the
and Eq. (4.3).
that bH ~0,
that
furthermore,
Eq.
(4.11)
the resulting
holds,
and that
wave speed
sgn(a+)
= -sgn(ba).
Then
WY;
C(E) satisfies
= -sgn(ba)
Remark 4.17.
Note
that a_ has no influence
on c’(O).
It might be of some interest to consider the effect of perturbing Then the statement of Lemma 4.6(d) becomes
and the equations Pi,, = -UP,,
in Lemma
4.12 and Lemma
+ (a+r’u - x*)ru
,
4.15 become,
PI’,<= -2uP,,,
+ br’P,,,
x. Suppose
that x = -3116Q
+ EX*.
respectively, + a+r2u - ,y* .
Since ru < 0 along the wave, it can be easily checked that if sgn(x*) = sgn(a+), then the conclusion each of these lemmas is unchanged. This yields the proof for the remark following Theorem 1.1.
of
5. Flow on A, W>> transversely. The In the previous section conditions were derived under which Wp intersects task is to now determine the flow on the slow manifold A,,, so that the existence of a heteroclinic orbit for E nonzero can be proved or disproved. Since only the wavespeed guaranteed by Theorem 4.16 will be used, the manifold will hereafter be referred to as A,. For the rest of this section, the hypotheses of Theorem 4.16 will be assumed. The strategy in this section is twofold. First, it must be shown that in a suitable domain of parameter space critical points exist in A,. Following the discussion in Section 2, this is tantamount to showing that the curves given by (2.12) and (2.13), i.e., by OTC,+ br’@, R,=
- h(r) = 0 ,
W(r) - (+ - c’(0)Hr2 c’(O)( 1 - /?r’)
+ O(E) >
(5.1)
55
T. Kapitula I Physica D 82 (1995) 36-59
intersect
for (r, L?) sufficiently
on the critical
manifold
close to (r,,, 0). In the above
expression,
0,, represents
the IZ coordinate
JY~,. and
p = a_ /c’(O) . It is important arbitrarily.
to note
Also,
W(ro) - (T - c’(0)
Finally,
here
solutions Hri
4.16 and
Remark 4.17 guarantee (r,,, 0) unless
that
p can be chosen
will not be D(E) from
E = 0, (5.1)
has a solution
and
dfiR,
then by the implicit assumed
Theorem
= 0 .
note that if, when
dR,. --F#O, dr
that
of (5.1)
without
function
theorem
loss of generality
the intersection
persists
for E sufficiently
small.
As such, it can be
that E = 0 in (5.1).
of that point must be Once a critical point has been shown to exist on A,, the stability characteristics determined. Let (r,, 0, 0,) represent the critical point on Jtl, closest to (r,, 0,O). Linearizing (2.4) about this point manipulation, D =~c’(O)(l
and
taking
-@rz)(bri
the
determinant,
+20.+)(d$%)
D, of the
resulting
3 x 3 matrix
yields,
after
some
+ 0(~‘).
In the above. dR,. dr
W’(r)(I
- pr*)
+ 2pr( W(r) - CT)- 2c’(O)Hr
c’(O)( 1 - pr’)’
+ Q(E) 3
(5.2)
dfl, ---_ dr Note
i’(r) - 2brfl,il br2 + 20,
that at (r, 0) = (r,,, 0) E Al,,,
d@ti Qro> -= dr
br(?l ’
(5.3)
which is nonzero and finite, as b f 0 and h’(ro) < 0. Recall a basic fact of linear algebra, which states that D is the product of the eigenvalues associated with the linearization of (2.4) about (r*, 0,fi.J. Since it is already known that one of these eigenvalues is positive and one is negative, it is clear that D < 0 implies that (r*, 0, 0,) is a repeller on &,, and that D > 0 implies the critical point is an attractor on the manifold. Thus, if D < 0, the (r*, 0, 0,) possesses must coincide with Wz in a a two dimensional unstable manifold, W”“, which by uniqueness neighborhood of the critical point. Therefore, since this critical point is the closest to (r,, 0,O) on .M,, by invoking Theorem 4.3 one can conclude that the singular heteroclinic orbit persists for E nonzero. If D < 0, then the question remains unanswered, as in this case the one dimensional unstable manifold, does not W”“, of (r,, 0,0 ,) is a submanifold of W>, and the fact that WY intersects W> transversely W>. imply that W u’ intersects Now, in showing that one, and possibly more, critical points exist on AE, two cases will be considered. It will first be assumed that H is arbitrary, modulo the fact that bH 2 0 be satisfied. Under
T. Kapitula
56
this assumption,
it will be necessary
0 5 IHI G 1, then the second W(r) -
0,. = Under
(T
c’(O)( 1 - @Y?)
this scenario
Jtlt. Before
stating
sgn(%
Theorem
5.1.
the appropriate
any results,
sgn(bc) then
condition
some notation
to
on p yields
the existence
of multiple
reduces
to the requirement
for all rE [r_, r,]. f&.(r*) and
E A,.
of Theorem
Furthermore,
orbit
p = a_ /c’(O)
to (2.4)
and
Thus,
sgn(c’(0))
if IpI is sufficiently
dR - +J(r*))
4.16 hold.
Then
for lb1 sufficiently
= -sgn(ba),
= -sgn(2
of (5.1)
large
the hypothesis
D (0,
by
4.16 implies
which concludes
actually
and (5.2)
there
exists
shows that by making
an r, E (r_, r,)
1~ I sufficiently
such that fl,r,(rs) =
(r*)) ,
that sgn(c’(0))
sgn(P)
orbit.
large.
w(D) = -w(c’(O)) w(P) w(b) w(b) = -sgn(c’(O)) sgn( /?) .
Thus,
on
large there
in the theorem
which proves the first part of the theorem. To prove the second part of the theorem, it must now be shown that the hypotheses For ]p] sufficiently large, sgn(1 - prz) = -sgn(p). Thus, for 0~ E + 1,
= sgn(ba)
on
if
exists which is 0(e) close to the singular
that a_ > 0 be sufficiently
= sgn(b)
sgn(D)
points
(5.4)
For each given 6 > 0, an examination
But Theorem
critical
,
5.2.
Since
that
must be set. By (5.3)
that the hypotheses (r,, 0, 0,)
point
= -sgn(P)
dR. 2
If it is assumed
function theorem yields the existence of r_ < r(, < r, with Q,, existing S’mce a,,,(O) = 0, it is clear that $2, have opposite signs; furthermore,
Remark
sgn
result.
,
for 0 < E G 1 a heteroclinic
Proof. large,
reduces
a definitive
of the implicit
Assume
a critical
in (5.1)
36-59
+a(lHI+E).
[r_, y,]. Let 0, = fi,(r,). (5.4) sgn(0,) = -sgn(b).
exists
D 82 (1995)
for I/3/ +O to achieve
equation
(r,,)) = -sgn(b)
so an invocation
I Physica
= -sgn(ba),
imply that D < 0.
(5.5) so that
. the proof.
0
T. Kapitula
Now
that
the
existence
question
has
I Physica
been
settled,
heteroclinic orbits will be addressed. Specifically, not only have an orbit close to the singular orbit, be given, and suppose that the parameters can clearly be done, as W(r) is a quartic.
D 82 (1995) 36-59
the
question
57
of the
existence
of multiple
it will be shown that if IHI G 1, then it is possible to but also an orbit existing on &,. Let r. < Y, < r2 < r+
d, , dr, (T have been manipulated Since p can be chosen arbitrarily,
so that W(r,) - v = 0. This r, E (r,, r,) can be given,
1 - prf = 0. Now, for IHI < 1,
with
W(r) - fl
f12,.=
+ 0( pf
c’(O)( 1 - pr’)
so by making
r2 - r, sufficiently
Furthermore,
at these
Recall
Eq. (4.1). G’(r,))
sgn(
+ E) )
small 0,. will intersect
lj,, transversely
at points
r(, < r’r < rC < rT < r+.
points
It is a trivial
= sgn(cT) .
consequence
sgn( i’(r?))
of this condition
= -sgn(o)
that
.
(5.6)
Since dR,. dY(r,) the choice
w’(r,) =
of rC implies
all this together
sgn(c’(0))
then
sgn(b) sgn(a)
sgn(D) Thus,
=
1
-sgn(b)
+ Q(lHI + c> ,
that
(r: )I = sgn(a)
sgn/% Putting
c’(O)( 1 - /3rf)
sgn(a)
yields that for 0 < E 6 1, ,
r = rT
,
r = rt
if bu < 0. the point (ry ,O, KIT) is a repeller has th e reverse properties. The following
(rz, 0, C) Theorem
on A,;
otherwise,
theorem
it is an attractor.
has now been
The
point
proved.
5.3. Assume
r,, < r,
that the hypothesis of Theorem 4.16 hold, and that IHI =G1. Suppose that are chosen so that r2 - r, is sufficiently small, and that the parameters d,, dz, (T have
been chosen so that W(r,) - cr = 0. Finally, let rC E (r,, r7), and let p > 0 be chosen so that 1 - pr’ = 0. Then if ba < 0 and 0 < E G 1, there exists two heteroclinic orbits to (2.4), one of which is close to the singular orbit, and the other of which lies on the slow manifold .M,. Remark
5.4. Since
bcr < 0 implies
that c’(0) > 0, the requirement
that p > 0 implies
that u_ > 0.
As a final remark, it is possible to have two, or even three, heteroclinic orbits existing on Ju,. This can be achieved by making A,, sufficiently large, so that the curve 0,. intersects .&,, at more than two points. Since A,, is just an arc contained in an ellipse in (r’, CI)-space, this can be achieved by making b sufficiently small. For example, it can be checked that if IHI @ 1 and b’
T. Kapitula I Physica D 82 (1995) 36-59
58
transversely in at least 3 places, slow manifold A,.
which will yield the existence
of at least two heteroclinic
orbits
on the
Acknowledgements I would
like to thank
application suggestions,
Jim Keener
for his assistance
in helping
me to understand
of formal asymptotics. I would also like to thank the referees which helped to markedly improve the quality of this paper.
for their
the theory
and
comments
and
Appendix In this appendix wave
is singular
an expression in nature,
for the wave speed
the machinery
of matched
will be derived asymptotics
via formal must
asymptotics.
be used.
Since the
However,
since
interest here is only in finding the wave speed, this issue will not be pursued further. Let p(z) denote the wave that exists when E = 0, and recall that in this case 0 = 0. Writing perturbations as
the the
Y = p + EY, + 2Y* + . . . , n = dl,
+ E2L$ + . . . ,
c=c,E+C2E2+-., it is found
that at O(E) the equations
r; + (h(p) + pA’(p))r, = -clp’ must be satisfied. Since p+O as z*
(P2G)(4 Note
= c,H
that unless
expression
1
p’(s) ds -
the parameters
1P’(S)(P'(S))' 08
a+p’p’+ bp(p’0,) , (p%,) = -c,Hp4 + p’(W(p) -
x, the second equation T z
into the equation
a+
+
I z
can be integrated,
$(.Y) (O(p(s))
are chosen
ds - b
1P(S) P’(S) 1 R
so that
- a) ds
correctly,
for Y, and invoking
a)
0, is unbounded
the Fredholm
P’@>
(4p(t>>
as z*
alternative
- (7) dt ds
--x.
then
Substitution
yields
of this
that
.
5
It might be of some interest to compare this expression to the note that the condition bH 2 0 is not unreasonable, as the fact multiplying c, to be inverted. Furthermore, the conditions on a+ Thus, there is good agreement between the asymptotics and the remark, note that the theorem does not yield, as the asymptotics
conditions given in Theorem 1.1. First that p’ < 0 then allows the expression and W(Y) - u guarantee the sign of c, conditions of the theorem. As a final do, the potentially useful information
T. Kapitula I Physica D 82 (1995) 36-59
that for fixed b and H the derivative
of the wave speed depends
linearly
59
on the parameters
a,,
d, , d2,
IT.
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