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A Mel'nikov approach to soliton-like solutions of systems of discretized nonlinear Schrödinger equations
PHYSICA D ELSEVIER
Physica D
I I3 ( 199X) 397406
A Mel’nikov approach to soliton-like solutions of systems of discretized nonlinear SchrGdinger equations
Abstract We investigate a class of N coupled discretized nonlinear
Schriidinger
equations of interacting chains in ;I nonlinear
lattice. which, in the limit of zero coupling. become integrable Ablowitz-Ladik the existence of stationary localized excitations, in the form of soliton-like perturbed ‘N-dimensional
differential-difference
time-periodic
equation\.
We \tudl
states. by reducing the system to a
symplectic map, whose homoclinic orbits are obtained by a recently developed Mcl’nikov
analysis.
from the simple /eroh oi‘a Mel‘niko\ vector and illustrate our results in the cases N = 2 and 3. Copyright 0 1998 Elsevier Science B.V.
We find that, depending on the perturbation, homoclinic orbits can be accurately located
1. Introduction There has recently been an impressive
activity in the study of the nonlinear
dynamics
of lattice systems, with
special attention focused on the existence of time-periodic excitations. commonly called breathers 1l-7 1. One 01 the most popular examples of such a system, in one dimension (ID), is described by various types of discretized nonlinear SchrBdinger equations (DNLS) whose physical significance has been discussed in several papers in the literature [8-131. Among these DNLS equations. introduced
the one that has been particularly
signiticant
is the discretization
by Ablowitz and Ladik (AL) [ 141.
i +,, -2~,l+[~,,+l+~,I__I][l+I*(~,,12]=0. which is known to be a completely
integrable
12=0.*1.12
.....
(I)
system. One then observes that the AL chain supports time-periodic.
spatially localized solutions of the form qf17(t) = c#J,! exp(iu).
-”Corresponding author. 0167.2789/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights rcaerved P/I so 167.2789(97)00295-9
(2)
M. Kollmann, T. Bounti.s/Physica
398
D 113 (1998) 397406
whose time independent
variable I$,, satisfies a mapping equation
which is also integrable
[ 15,161. In fact, for w > 0 one can use that for these solutions
an angular momentum
integral of (3) vanishes identically (corresponding to zero flux at the boundaries) in order to restrict oneself to the case of real @,, [ 171. It is then easy to solve analytically the map (3) in terms of Jacobi elliptic functions and find explicit expressions
for the two separatices
(closed homoclinic
orbits) emanating
from the unstable saddle point at
the origin of the @,,, & _ I plane. The next step is to consider non-integrable perturbations of (I), which support time-periodic solutions of the form (2) and ask under what conditions such solutions still exist, having a soliton-like, spatially localized profile, similar to the one provided by the separatrix solutions of the integrable case (3). This question has been recently answered for a single DNLS chain by Hennig et al. [ 171, who used the approach of Mel’nikov functions to locate homoclinic orbits of a 2D non-integrable
perturbation
of the integrable
map (3) with real &.
In this paper, we use a recently developed higher-dimensional Mel’nikov analysis for maps [ 191 to establish the existence of soliton-like solutions for systems of N coupled DNLS equations of the form
a iz$!” _ 2+;j) j=l,2
,,..,
+ [+(j) n+, + $;j’,l]l
N,with$~“‘=+~N+”
+ ElI$!j)12] = Ef.(#j) I, J
= 0, for all n. In particular,
) @i*‘ II ))
.
(4)
we study time-periodic
localized excitations
of
(4): $ljj’(t)
= f$Aj’eXp(hjt),
(5)
assuming that the coupling terms ,fi are such that the time dependence in (4) to yield a system of N 2D mappings of the form 0)
x,,+l =
--LIJ;/‘+ )
can be eliminated
upon substitution
(j)
(j)
u,,+j = xn ?
of (5)
(6)
(1) (1) (N) (N) = .x::), & = (x,1 ).gj(x,) = (1 +(x,(j))')-'fj(x,,) andhave ,u,, 3 . . ..X.l .un scaled all variables so that ,u = 1 with kj E 1 + wj/2. j = 1. . . . , N. In Section 2 we briefly recall the results of [ 191 and derive a Mel’nikov vector for (6) in the case N = 2 with the coupling terms taken to be of the form where we have defined 4:”
RI (X,,) = Ylx:*) + &IQ. where the yj and Sj terms correspond
g*(x,,)
=
y2p
to conservative
+ s2.42),
and “dissipative”
(7) perturbations,
respectively.
(Note that the
6i terms are dissipative in the sense that they destroy the area preserving property of the mapping (3). This has nothing to do with damping in time, usually connected with an if $ term in the NLS equation, r E R). Plotting the two components of the Mel’nikov vector M =(A41 , M2) for this problem, we demonstrate that they vanish along distinct nodelines t(j) = (t:“. $‘), i.e. Mj(t(j))
= 0,
j = 1,2.
(8)
which need not intersect and may even altogether disappear if the parameters Sj are sufficiently do intersect,
however. this happens at an infinite number of points t:” = tf)
large. If the nodelines
= t;, k = 0. fl,
f2,
. . ., through
which one can establish the existence of homoclinic
orbits of (6) and hence soliton-like
solutions (5) of the original
DNLS system (3). In Section 3 we present numerical evidence of the accuracy of our Mel’nikov predictions with regard to homoclinic tangency. i.e. points t: at which the nodelines (8) intersect tmpt~fio/!\.. We find satisfactory agreement between analytical and numerical results over a significant range of parameter values. at least in the symmetric case j/l = ~2 (81 = S? ). Furthermore, we outline in Section 3 our results for N = 3. using conservative linear coupling ,f; between the chains. cf. (3). This shows that once the homoclinic one can construct the three soliton-like
orbit has been accurately
solutions which represent a time-periodic.
determined
from the points t;.
localized state
of all three chains.
It is straightforward to generalize these results to the arbitrary N case. Of course. if the localized solutions are dynamically stable it does not matter whether the initial conditions
are
chosen with great accuracy. In the case of N = 2 dnd N = 3 chains. integrating the system of ODES (3) in time. with linear coupling. we have found that our soliton-like solutions are indeed stable. Thus. we expect our Mcl’nikob analysis
to be more useful for other types of couplin, (7 where the question of stability of localized solutions
respect to initial conditions
2. Mel’nikov
analysis of two chains
Consider a system of difference equations x,,+I
of the form
Rx,,) + tG(x,,).
=
with
becomes more delicate
where F. G : R?
-
(9)
LR’~’are sufficiently
mapping (9) be symplectic l; (x,, ) = c; = const.
and completely
smooth functions integrable.
possessing
of the coordinates
x,,. Furthermore.
Icr the c = 0
N analytic constants of the motion (IO)
j = I.....N.
We also suppose that x = 0 is a saddle fixed point of the unperturbed system with N-dimensional (N-D) stable manifold WC;and N-D unstable manifold Wi. joining smoothly along a single homoclinic orbit x,,(‘j’(t). known explicitly as a function of the variable t = (tl. . . . tv). Then. as has been shown in [ 191. a Mel’niko\, vector function can be evaluated for (9). whose components
M;(t) =
-f
$,~,G(xj;“(t)).
j = I.....
are given by the summation
N.
formula
(II)
I!=-% where $,” IS the jth bounded solution of the adjoint variational problem associated with the homoclinic solution of (9). i.e. <,Z+1 = ij,, [ DF(~j,"'(t))l~' . DF being the Jacobian matrix of F. For integrable mappings F. such as the ones considered here. it is known that these bounded solutions can be written in the form [ I9 1 4,) 91r = Vlj(Xjr()‘(t))
(12)
and thus the components of the Mel’nikov vector in (1 1) can be immediately obtained from the knowledge ot the integrals (IO) and x/,“‘(t). Once the M,(t) are known one examines their zeros t”’ where M / (t”‘) = 0. If these zeros occur at (componentwise) identical points t(i) = t*. j = I. . N. and are .cimp/e. in the sense that n(t*)
s
det DM(t*) # 0
(13)
hf. Kollmann,
400
7: Bountis/Physica
D 113 (I 998) 397406
and an additional non-degeneracy condition [ 191 holds, then the invariant manifolds W: and WF intersect transversely at x, (t*). For the corresponding theory for ODES see [20]. Since the Mel’nikov vector is periodic in all its arguments
one can use this periodicity
to define a unit cell in t-space. In particular, with
M(t*) = M(t* + 7) = 0,
(14)
a periodic sequence of zeros of the Mel’nikov clinic orbit of the perturbed for perturbations
vector represents a necessary condition
map. It is interesting
of symplectic
integrable
that this condition
for the existence of a homo-
has also been shown recently to be suficienr
maps [21].
Let us turn now to the example of two coupled AL chains with weak linear coupling and dissipation according to Eqs. (4)-(7) for N = 2. With the phase space vector defined as x, = (x,, u,, , .v,, u,) this system can be written in the form (9) with
The conserved
quantities
(10) are given by the expressions
zl(x,)=X,2+U,2+X,211~-22k]x,u,,
12(x,)
=
[ 16,191
y,z + u,f + .v,u;
-
2kzy,v,,.
(15)
We now derive the first-order Mel’nikov vector for this system according to Eq. (11). In particular, for kj > 1 and yj = Sj = 0 the origin is a saddle point with a two-parameter (f], t2) family of homoclinic solutions x:“)w2)
=
~~l~n~~I~~~I~n-l~~l~~.~2~n~~2~~~2~n-ll(~2~~~
(16)
where we introduce the abbreviations S,(tj) = sech(nwj + tj) and T,(rj) = tanh(nzuj + rj), *Fj = sinh Wj, and wj = cash-’ kj, For wj > 0, this solution lies entirely in the positve quadrant of R4, as all of its coordinates are positive for all n. The adjoint variational solutions of the unperturbed problem are generated way from the integrals (15) through (12). Thus we obtain the adjoint variational row vectors G(” =
(-.S-I(~I)G-IOI),
Gy2) = n
(O,O,
&(o)Tn(o),
-Sn-l(t2)Tn-l(r2),
These are the bounded variational vector (11) as follows: Mj(t)
= Mj(tl,
Aj(tl,
r2) =
in a straightforward
O,O),
Si,(t2)Tn(t2)).
solutions, each yielding after a simple calculation
12) = -I/jsls2Aj(rl,
r2) - 6jSyB(tj),
j = 1, 2
a component
of the Mel’nikov
(17)
with cc c G(tl)T,(rj)&(r2), fl=-CC
B(x) =
5 &(x)T,(x)S,,-I(X). n=-CC
The determination of homoclinic points becomes now an easy task since, by means of Eqs. (14) and (17), the zeros t* of the Mel’nikov vector appear at the intersections of the constant level curves Sj
- -
ST
YjSlS2
= cj(t*)
Aj (t;, $1 zz
Bj(tT)
’
j=
1,2
(18)
M. Kollrnunn,
Fig. I. Components of the Mel’nikov MI (tl . t?): right: M?(tt . t?).
7: Bountis/Physica
D 1 I3 (1998)
397406
vector for N = 2, WI = w? = w = 2, y, = 0.07, y2 = 0.05. 8, z 0.001.
6, = 0.002.
Left:
The functions Cj (t) contain all the information of the Mel’nikov vector. Since they are invariant under translations tj + tj + wj, they demonstrate the periodicity of the Mel’nikov vector over the unit cell T = (WI, UQ). In Fig. I we plot the components
of the Mel’nikov
vector for a typical parameter setting.
The tangency condition (13) is likewise easily evaluated. Calculating the derivatives of Mj w.r.t. tt and tz from (17) and evaluating them on the zero curves we find as a necessary condition for homoclinic tangency
n(t)
= [(II(t) - b(tl)CI (t)l[Q(t)
- b(t2)C2(t)l
- c(t)? = 0.
(19)
where
c!j(t)=uj(rl7
t2) =
g
sn(tl)Sn(t2)(S,2(tj)
-
T,,2([j)),
,1=-m
la)
=
5
Mx)S,,-,(.&(x) - q(x) -
T,(x)T,,-I(X)).
tl=--cyj
cm
c(t) = dtl? t2) =
c
S,*(tl)Tn(tl)Sn(t2)~n(t:!).
n=-cc
In Fig. 2 we display the nodelines of Mt and M2 for three different sets of parameters (vi, Sj). Whereas in Fig. 2(a) no homoclinic points are present, a common zero can be induced varying only one of the dissipation parameters, as is seen subsequently. In Fig. 2(b) the tangency condition (13) is satisfied since the nodelines intersect tangentially, i.e. if,, Ml &,MZ = i_+> Ml a,, M2 at some t = t*. On the other hand, in Fig. 2(c) the transversality condition is true.
402
2.5
1 j
i
i
,..-
.i”‘.
..’
:. ;;
i
5
,:.’ ,.:.. ..::.. ,.:..‘..
~~ 2
.;:::
,_,._...” ,,..._
‘..
.i
._ i’ i
,...::.‘.’ ;_,:::
,:.
::.
I
;:::.. ..:::> _:::::: ..::y
.: ,..::::
1
,,::y ..:::y
_.,.S
I 7 :
_.,.::.:..
,i_.i. ,:i 1
a)
..y:::..
..‘..
..::.:::
-.
,...::v ,.$y
-2.5 IG’ -2.5
TL-
-:
2.5
tl
_2
.5
./
. . .._.
/
_____..._.,________,.........
r._............_.._ ..........._..... T..................
-2.5 Fig. 2. Nodelines 6 = 0.01800;
3. Accuracy
of the Mel’nikov
(b) 6 = 0.01531:
components MI (t) black. M:(t)
t1
2.5
gray for U’ = 2, y, = y2 = y = 0.01. 8, = 62 = S decreasing: (a)
(c)S = 0.
of the Mel’nikov
analysis and the case of three chains
To test the accuracy of the parameter values for homoclinic tangency predicted by our Mel’nikov analysis. we apply the following numerical method: We place 2500 initial conditions on the positive quadrant WY of the 2D linearized unstable manifold, very close to 0 and map them forward a few iterations until the bulk of points have returned close to 0 following the stable manifold WT . If some phase points leave the upper positive quadrant of R4 this serves as a criterion for the intersection of the invariant manifolds in the symmetric case of equal kj , vj, and Sj, j = I, 2, see Figs. 3(a) and (b). Although the stable manifold typically does not coincide with any of the zero planes that divide the upper positive quadrant of R4 from the rest of phase space, a subset of WJ that intersects the stable manifold transversely at time II, in the symmetric case, will intersect a zero plane at some time n’ > II. Therefore, the occurrence of transversal intersection is equivalent to some points on WY attaining negative coordinates.
M. Kdmann,
Fig. 3. Projections of the unstable manifold the Mel’nikov
WY computed numerically
prediction for tangential intersection
td) asymmetric case yz/y~
= S:/Sl
=
7: Bountis/Ply.sica
ye = 0.005,
D I13 (/09X) 397406
using the method of Section 3. at parameter values obtained from
61 = 0.00122.
(a) and (b) symmetric case yz/yl
= &/Al
=
I: Cc) and
I.1 Cw = I).
In Table I, we list for several parameter values (~1 = y2 =)y the (6 1 = 62 =)S melat which homoclinic tangency is predicted by the Mel’nikov analysis, as well as the corresponding a,,, identified by our numerical criterion. Clearly, at least in this symmetric setting, the agreement is quite satisfactory, even for relatively large values 0fy.
However, even though the above numerical
criterion
works well in the symmetric
case, the agreement
breaks
down as soon as the parameter values depart from symmetry. The reason for this is that, in the asymmetric case, our assumption that transversal intersections between WY and W$ necessarily imply that some points of W;l will eventually exit from R” is no longer valid. Indeed, looking at projections in (x, u) and (_v. v) spaces demonstrates the usefulness of our numerical criterion in the symmetric case (Figs. 3(a) and (b)), while for yi # ~2. 61 # 62, points xn E WT can escape from the positive quadrant of R4 (by attaining e.g. x,, < 0, u,, < 0 coordinates, as shown in Figs. 3(c) and (d), without any intersections between W;l and Wt). Thus. it is clear that in order to study homoclinic tangency in the asymmetric case, one needs a more precise numerical criterion. We are currently investigating different approaches (including one by which numbers of points on WT and W_tare iterated, respectively, backward and forward, while their mutual distances are being monitored)
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Fig. 4. Three Ablowitz-Ladik chains with nonlinear coupling as Eq. (20) with ~11 = 0.08. ~‘2 = 0. I, uq = 0. IS. (a) Nodelines Mel’nikov vector (MI black, M? dark gray. M3 light gray) as functions of rl and Q (ti = 0.15). One homoclinic orbit is indicated intersection of three nodelines. (h) Stationary solution of three coupled AL chains computed from the homoclinic orbit in (a).
localized solutions are indeed dynamically
stable by integrating
(20). c = 0.0 I, for three chains of length 150, using a symmetric
numerically
of the by the
the original system of ODES (4) with
Euler scheme with a time step of 0.00 I.
Thus. our theory generalizes the corresponding result of Hennig et al. [ 171 obtained for one chain to the case of N > 2 coupled DNLS chains, for which one can now analyze in detail the occurrence of soliton-like solutions. under different kinds of coupling
[22].
Acknowledgements One of us (TB) wishes to thank the Institute for Theoretical Physics of the University of Amsterdam for its hospitality during his stay in December 1996 and January 1997 when this work was completed. He also acknowledges partial support of a PENED grant from the GSRT of the Greek Ministry of Development. provided under the HCM Program of the EC under contract no. CHRX-CT94 0480.
MK is grateful for support
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Physica D
I I3 ( 199X) 397406
A Mel’nikov approach to soliton-like solutions of systems of discretized nonlinear SchrGdinger equations
Abstract We investigate a class of N coupled discretized nonlinear
Schriidinger
equations of interacting chains in ;I nonlinear
lattice. which, in the limit of zero coupling. become integrable Ablowitz-Ladik the existence of stationary localized excitations, in the form of soliton-like perturbed ‘N-dimensional
differential-difference
time-periodic
equation\.
We \tudl
states. by reducing the system to a
symplectic map, whose homoclinic orbits are obtained by a recently developed Mcl’nikov
analysis.
from the simple /eroh oi‘a Mel‘niko\ vector and illustrate our results in the cases N = 2 and 3. Copyright 0 1998 Elsevier Science B.V.
We find that, depending on the perturbation, homoclinic orbits can be accurately located
1. Introduction There has recently been an impressive
activity in the study of the nonlinear
dynamics
of lattice systems, with
special attention focused on the existence of time-periodic excitations. commonly called breathers 1l-7 1. One 01 the most popular examples of such a system, in one dimension (ID), is described by various types of discretized nonlinear SchrBdinger equations (DNLS) whose physical significance has been discussed in several papers in the literature [8-131. Among these DNLS equations. introduced
the one that has been particularly
signiticant
is the discretization
by Ablowitz and Ladik (AL) [ 141.
i +,, -2~,l+[~,,+l+~,I__I][l+I*(~,,12]=0. which is known to be a completely
integrable
12=0.*1.12
.....
(I)
system. One then observes that the AL chain supports time-periodic.
spatially localized solutions of the form qf17(t) = c#J,! exp(iu).
-”Corresponding author. 0167.2789/98/$19.00 Copyright 0 1998 Elsevier Science B.V. All rights rcaerved P/I so 167.2789(97)00295-9
(2)
M. Kollmann, T. Bounti.s/Physica
398
D 113 (1998) 397406
whose time independent
variable I$,, satisfies a mapping equation
which is also integrable
[ 15,161. In fact, for w > 0 one can use that for these solutions
an angular momentum
integral of (3) vanishes identically (corresponding to zero flux at the boundaries) in order to restrict oneself to the case of real @,, [ 171. It is then easy to solve analytically the map (3) in terms of Jacobi elliptic functions and find explicit expressions
for the two separatices
(closed homoclinic
orbits) emanating
from the unstable saddle point at
the origin of the @,,, & _ I plane. The next step is to consider non-integrable perturbations of (I), which support time-periodic solutions of the form (2) and ask under what conditions such solutions still exist, having a soliton-like, spatially localized profile, similar to the one provided by the separatrix solutions of the integrable case (3). This question has been recently answered for a single DNLS chain by Hennig et al. [ 171, who used the approach of Mel’nikov functions to locate homoclinic orbits of a 2D non-integrable
perturbation
of the integrable
map (3) with real &.
In this paper, we use a recently developed higher-dimensional Mel’nikov analysis for maps [ 191 to establish the existence of soliton-like solutions for systems of N coupled DNLS equations of the form
a iz$!” _ 2+;j) j=l,2
,,..,
+ [+(j) n+, + $;j’,l]l
N,with$~“‘=+~N+”
+ ElI$!j)12] = Ef.(#j) I, J
= 0, for all n. In particular,
) @i*‘ II ))
.
(4)
we study time-periodic
localized excitations
of
(4): $ljj’(t)
= f$Aj’eXp(hjt),
(5)
assuming that the coupling terms ,fi are such that the time dependence in (4) to yield a system of N 2D mappings of the form 0)
x,,+l =
--LIJ;/‘+ )
can be eliminated
upon substitution
(j)
(j)
u,,+j = xn ?
of (5)
(6)
(1) (1) (N) (N) = .x::), & = (x,1 ).gj(x,) = (1 +(x,(j))')-'fj(x,,) andhave ,u,, 3 . . ..X.l .un scaled all variables so that ,u = 1 with kj E 1 + wj/2. j = 1. . . . , N. In Section 2 we briefly recall the results of [ 191 and derive a Mel’nikov vector for (6) in the case N = 2 with the coupling terms taken to be of the form where we have defined 4:”
RI (X,,) = Ylx:*) + &IQ. where the yj and Sj terms correspond
g*(x,,)
=
y2p
to conservative
+ s2.42),
and “dissipative”
(7) perturbations,
respectively.
(Note that the
6i terms are dissipative in the sense that they destroy the area preserving property of the mapping (3). This has nothing to do with damping in time, usually connected with an if $ term in the NLS equation, r E R). Plotting the two components of the Mel’nikov vector M =(A41 , M2) for this problem, we demonstrate that they vanish along distinct nodelines t(j) = (t:“. $‘), i.e. Mj(t(j))
= 0,
j = 1,2.
(8)
which need not intersect and may even altogether disappear if the parameters Sj are sufficiently do intersect,
however. this happens at an infinite number of points t:” = tf)
large. If the nodelines
= t;, k = 0. fl,
f2,
. . ., through
which one can establish the existence of homoclinic
orbits of (6) and hence soliton-like
solutions (5) of the original
DNLS system (3). In Section 3 we present numerical evidence of the accuracy of our Mel’nikov predictions with regard to homoclinic tangency. i.e. points t: at which the nodelines (8) intersect tmpt~fio/!\.. We find satisfactory agreement between analytical and numerical results over a significant range of parameter values. at least in the symmetric case j/l = ~2 (81 = S? ). Furthermore, we outline in Section 3 our results for N = 3. using conservative linear coupling ,f; between the chains. cf. (3). This shows that once the homoclinic one can construct the three soliton-like
orbit has been accurately
solutions which represent a time-periodic.
determined
from the points t;.
localized state
of all three chains.
It is straightforward to generalize these results to the arbitrary N case. Of course. if the localized solutions are dynamically stable it does not matter whether the initial conditions
are
chosen with great accuracy. In the case of N = 2 dnd N = 3 chains. integrating the system of ODES (3) in time. with linear coupling. we have found that our soliton-like solutions are indeed stable. Thus. we expect our Mcl’nikob analysis
to be more useful for other types of couplin, (7 where the question of stability of localized solutions
respect to initial conditions
2. Mel’nikov
analysis of two chains
Consider a system of difference equations x,,+I
of the form
Rx,,) + tG(x,,).
=
with
becomes more delicate
where F. G : R?
-
(9)
LR’~’are sufficiently
mapping (9) be symplectic l; (x,, ) = c; = const.
and completely
smooth functions integrable.
possessing
of the coordinates
x,,. Furthermore.
Icr the c = 0
N analytic constants of the motion (IO)
j = I.....N.
We also suppose that x = 0 is a saddle fixed point of the unperturbed system with N-dimensional (N-D) stable manifold WC;and N-D unstable manifold Wi. joining smoothly along a single homoclinic orbit x,,(‘j’(t). known explicitly as a function of the variable t = (tl. . . . tv). Then. as has been shown in [ 191. a Mel’niko\, vector function can be evaluated for (9). whose components
M;(t) =
-f
$,~,G(xj;“(t)).
j = I.....
are given by the summation
N.
formula
(II)
I!=-% where $,” IS the jth bounded solution of the adjoint variational problem associated with the homoclinic solution of (9). i.e. <,Z+1 = ij,, [ DF(~j,"'(t))l~' . DF being the Jacobian matrix of F. For integrable mappings F. such as the ones considered here. it is known that these bounded solutions can be written in the form [ I9 1 4,) 91r = Vlj(Xjr()‘(t))
(12)
and thus the components of the Mel’nikov vector in (1 1) can be immediately obtained from the knowledge ot the integrals (IO) and x/,“‘(t). Once the M,(t) are known one examines their zeros t”’ where M / (t”‘) = 0. If these zeros occur at (componentwise) identical points t(i) = t*. j = I. . N. and are .cimp/e. in the sense that n(t*)
s
det DM(t*) # 0
(13)
hf. Kollmann,
400
7: Bountis/Physica
D 113 (I 998) 397406
and an additional non-degeneracy condition [ 191 holds, then the invariant manifolds W: and WF intersect transversely at x, (t*). For the corresponding theory for ODES see [20]. Since the Mel’nikov vector is periodic in all its arguments
one can use this periodicity
to define a unit cell in t-space. In particular, with
M(t*) = M(t* + 7) = 0,
(14)
a periodic sequence of zeros of the Mel’nikov clinic orbit of the perturbed for perturbations
vector represents a necessary condition
map. It is interesting
of symplectic
integrable
that this condition
for the existence of a homo-
has also been shown recently to be suficienr
maps [21].
Let us turn now to the example of two coupled AL chains with weak linear coupling and dissipation according to Eqs. (4)-(7) for N = 2. With the phase space vector defined as x, = (x,, u,, , .v,, u,) this system can be written in the form (9) with
The conserved
quantities
(10) are given by the expressions
zl(x,)=X,2+U,2+X,211~-22k]x,u,,
12(x,)
=
[ 16,191
y,z + u,f + .v,u;
-
2kzy,v,,.
(15)
We now derive the first-order Mel’nikov vector for this system according to Eq. (11). In particular, for kj > 1 and yj = Sj = 0 the origin is a saddle point with a two-parameter (f], t2) family of homoclinic solutions x:“)w2)
=
~~l~n~~I~~~I~n-l~~l~~.~2~n~~2~~~2~n-ll(~2~~~
(16)
where we introduce the abbreviations S,(tj) = sech(nwj + tj) and T,(rj) = tanh(nzuj + rj), *Fj = sinh Wj, and wj = cash-’ kj, For wj > 0, this solution lies entirely in the positve quadrant of R4, as all of its coordinates are positive for all n. The adjoint variational solutions of the unperturbed problem are generated way from the integrals (15) through (12). Thus we obtain the adjoint variational row vectors G(” =
(-.S-I(~I)G-IOI),
Gy2) = n
(O,O,
&(o)Tn(o),
-Sn-l(t2)Tn-l(r2),
These are the bounded variational vector (11) as follows: Mj(t)
= Mj(tl,
Aj(tl,
r2) =
in a straightforward
O,O),
Si,(t2)Tn(t2)).
solutions, each yielding after a simple calculation
12) = -I/jsls2Aj(rl,
r2) - 6jSyB(tj),
j = 1, 2
a component
of the Mel’nikov
(17)
with cc c G(tl)T,(rj)&(r2), fl=-CC
B(x) =
5 &(x)T,(x)S,,-I(X). n=-CC
The determination of homoclinic points becomes now an easy task since, by means of Eqs. (14) and (17), the zeros t* of the Mel’nikov vector appear at the intersections of the constant level curves Sj
- -
ST
YjSlS2
= cj(t*)
Aj (t;, $1 zz
Bj(tT)
’
j=
1,2
(18)
M. Kollrnunn,
Fig. I. Components of the Mel’nikov MI (tl . t?): right: M?(tt . t?).
7: Bountis/Physica
D 1 I3 (1998)
397406
vector for N = 2, WI = w? = w = 2, y, = 0.07, y2 = 0.05. 8, z 0.001.
6, = 0.002.
Left:
The functions Cj (t) contain all the information of the Mel’nikov vector. Since they are invariant under translations tj + tj + wj, they demonstrate the periodicity of the Mel’nikov vector over the unit cell T = (WI, UQ). In Fig. I we plot the components
of the Mel’nikov
vector for a typical parameter setting.
The tangency condition (13) is likewise easily evaluated. Calculating the derivatives of Mj w.r.t. tt and tz from (17) and evaluating them on the zero curves we find as a necessary condition for homoclinic tangency
n(t)
= [(II(t) - b(tl)CI (t)l[Q(t)
- b(t2)C2(t)l
- c(t)? = 0.
(19)
where
c!j(t)=uj(rl7
t2) =
g
sn(tl)Sn(t2)(S,2(tj)
-
T,,2([j)),
,1=-m
la)
=
5
Mx)S,,-,(.&(x) - q(x) -
T,(x)T,,-I(X)).
tl=--cyj
cm
c(t) = dtl? t2) =
c
S,*(tl)Tn(tl)Sn(t2)~n(t:!).
n=-cc
In Fig. 2 we display the nodelines of Mt and M2 for three different sets of parameters (vi, Sj). Whereas in Fig. 2(a) no homoclinic points are present, a common zero can be induced varying only one of the dissipation parameters, as is seen subsequently. In Fig. 2(b) the tangency condition (13) is satisfied since the nodelines intersect tangentially, i.e. if,, Ml &,MZ = i_+> Ml a,, M2 at some t = t*. On the other hand, in Fig. 2(c) the transversality condition is true.
402
2.5
1 j
i
i
,..-
.i”‘.
..’
:. ;;
i
5
,:.’ ,.:.. ..::.. ,.:..‘..
~~ 2
.;:::
,_,._...” ,,..._
‘..
.i
._ i’ i
,...::.‘.’ ;_,:::
,:.
::.
I
;:::.. ..:::> _:::::: ..::y
.: ,..::::
1
,,::y ..:::y
_.,.S
I 7 :
_.,.::.:..
,i_.i. ,:i 1
a)
..y:::..
..‘..
..::.:::
-.
,...::v ,.$y
-2.5 IG’ -2.5
TL-
-:
2.5
tl
_2
.5
./
. . .._.
/
_____..._.,________,.........
r._............_.._ ..........._..... T..................
-2.5 Fig. 2. Nodelines 6 = 0.01800;
3. Accuracy
of the Mel’nikov
(b) 6 = 0.01531:
components MI (t) black. M:(t)
t1
2.5
gray for U’ = 2, y, = y2 = y = 0.01. 8, = 62 = S decreasing: (a)
(c)S = 0.
of the Mel’nikov
analysis and the case of three chains
To test the accuracy of the parameter values for homoclinic tangency predicted by our Mel’nikov analysis. we apply the following numerical method: We place 2500 initial conditions on the positive quadrant WY of the 2D linearized unstable manifold, very close to 0 and map them forward a few iterations until the bulk of points have returned close to 0 following the stable manifold WT . If some phase points leave the upper positive quadrant of R4 this serves as a criterion for the intersection of the invariant manifolds in the symmetric case of equal kj , vj, and Sj, j = I, 2, see Figs. 3(a) and (b). Although the stable manifold typically does not coincide with any of the zero planes that divide the upper positive quadrant of R4 from the rest of phase space, a subset of WJ that intersects the stable manifold transversely at time II, in the symmetric case, will intersect a zero plane at some time n’ > II. Therefore, the occurrence of transversal intersection is equivalent to some points on WY attaining negative coordinates.
M. Kdmann,
Fig. 3. Projections of the unstable manifold the Mel’nikov
WY computed numerically
prediction for tangential intersection
td) asymmetric case yz/y~
= S:/Sl
=
7: Bountis/Ply.sica
ye = 0.005,
D I13 (/09X) 397406
using the method of Section 3. at parameter values obtained from
61 = 0.00122.
(a) and (b) symmetric case yz/yl
= &/Al
=
I: Cc) and
I.1 Cw = I).
In Table I, we list for several parameter values (~1 = y2 =)y the (6 1 = 62 =)S melat which homoclinic tangency is predicted by the Mel’nikov analysis, as well as the corresponding a,,, identified by our numerical criterion. Clearly, at least in this symmetric setting, the agreement is quite satisfactory, even for relatively large values 0fy.
However, even though the above numerical
criterion
works well in the symmetric
case, the agreement
breaks
down as soon as the parameter values depart from symmetry. The reason for this is that, in the asymmetric case, our assumption that transversal intersections between WY and W$ necessarily imply that some points of W;l will eventually exit from R” is no longer valid. Indeed, looking at projections in (x, u) and (_v. v) spaces demonstrates the usefulness of our numerical criterion in the symmetric case (Figs. 3(a) and (b)), while for yi # ~2. 61 # 62, points xn E WT can escape from the positive quadrant of R4 (by attaining e.g. x,, < 0, u,, < 0 coordinates, as shown in Figs. 3(c) and (d), without any intersections between W;l and Wt). Thus. it is clear that in order to study homoclinic tangency in the asymmetric case, one needs a more precise numerical criterion. We are currently investigating different approaches (including one by which numbers of points on WT and W_tare iterated, respectively, backward and forward, while their mutual distances are being monitored)
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t1
Fig. 4. Three Ablowitz-Ladik chains with nonlinear coupling as Eq. (20) with ~11 = 0.08. ~‘2 = 0. I, uq = 0. IS. (a) Nodelines Mel’nikov vector (MI black, M? dark gray. M3 light gray) as functions of rl and Q (ti = 0.15). One homoclinic orbit is indicated intersection of three nodelines. (h) Stationary solution of three coupled AL chains computed from the homoclinic orbit in (a).
localized solutions are indeed dynamically
stable by integrating
(20). c = 0.0 I, for three chains of length 150, using a symmetric
numerically
of the by the
the original system of ODES (4) with
Euler scheme with a time step of 0.00 I.
Thus. our theory generalizes the corresponding result of Hennig et al. [ 171 obtained for one chain to the case of N > 2 coupled DNLS chains, for which one can now analyze in detail the occurrence of soliton-like solutions. under different kinds of coupling
[22].
Acknowledgements One of us (TB) wishes to thank the Institute for Theoretical Physics of the University of Amsterdam for its hospitality during his stay in December 1996 and January 1997 when this work was completed. He also acknowledges partial support of a PENED grant from the GSRT of the Greek Ministry of Development. provided under the HCM Program of the EC under contract no. CHRX-CT94 0480.
MK is grateful for support
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