Analytical instability of the Klein-Gordon equation

Journal of Computational North-Holland

and Applied

Mathematics

21 (1988) 17-26

Analytical instability of the Klein-Gordon

17

equation

A.H.J. CLOOT and B.M. HERBST Department Received

of Applied Mathematics,

30 September

University of the Orange Free State, Bloemfontein

9300, South Africa

1986

Abstract: In this work we investigate the stability of the Klein-Gordon equation by means analysis. Two conditions relating to instabilities on two different time-scales are obtained. numerically which also illustrates the behaviour of the instabilities on different time-scales.

Keywords:

Stability,

Klein-Gordon

of a multiple-scales These are verified

equations.

1. Introduction It is well-known (see e.g. Whitham [lo]) that the modulations of weakly nonlinear deep-water waves are unstable with respect to small perturbations. The modulations are described by a nonlinear Schriidinger (NLS) equation with a cubic nonlinearity. Fairly detailed analyses of the stability of the nonlinear Schriidinger equation in analytical and numerical contents are available, see e.g. [l], [4], [5], [7], [9] and [ll]. In these numerical studies special attention is given to the way the modulational instabilities are reflected in the numerical schemes. If a NLS is studied the instabilities show up as an exponential growth on the laboratory time-scale. However, in practice the modulational instabilities often show up as an exponential growth on a slow time-scale [3], i.e. a linear growth on the laboratory time-scale. It is therefore of interest to study situations where instabilities show up on different time-scales. For this purpose we study the Klein-Gordon equation. (%-%,>u=JXu) where F(U) is a well-behaved to be known and in addition maxI]u(x, X

t*)-u(x*,

(1.1) function of U. In addition satisfying t*)jl
we assume the solution

V’t*>O

u(x, t) of (1.1)

(1.2)

and x, x * E (- co, co), i.e. we assume the solution to be a slowly varying function of x and t. Several conditions regarding the stability of solutions satisfying these assumptions will be derived by means of a multiple scales analysis. In particular, it will be pointed out that different kind of instabilities arise on different time-scales; the instabilities occurring on the faster time-scales being the more severe. 0377-0427/87/$3.50

0 1987, Elsevier Science Publishers

B.V. (North-Holland)

A. H. J. Cloot, B. M. Herbst / Instability

18

For our numerical

experiments

of Klein-Gordon

equation

we choose

F(u)

=A24 + BU3

(1.3a)

F(u)

= -sin

(1.3b)

and u.

The stability conditions will be tested numerically which will confirm the analytical results. In the process the instablilities occurring on the different time-scales will be demonstrated. Although we believe the results of this paper to be of independent interest, we also use it as preparation for the more difficult question regarding the stability of the numerical scheme. These results are presented in an accompanying paper [2].

2. Multiple scales analysis Suppose that. at time t = t * one adds a small perturbation, equation describing the behaviour of the solution is: (a,, - a,& One may expand

I;(u)

(a,, - a&

around

(a,,-

the

t);

(2.1)

u( x, t *) and write

= F”‘[ u ( x, t*)]6+qF(2qU(X,

t*) =A(x),

Using the assumption

u(x,

+ 8) = F(u + 8).

which is nothing else but the equation the initial conditions: 8(x,

8, to the solution

describing

IV(x)

t*)]8+

the behaviour 8(x,

II * I;

(1.2), (2.2) can be rewritten

8,,)8=F”‘[u(x*,

t*)]s2++F’3’[U(X,

t*>]s’

of the perturbation

(2.2)

submitted

t*) = d(x).

to

(2.3)

as

:F(2)[~(~*,

t*)]8*+

$FC3+(x*,

t*)]s’

(2.4)

where F(‘)[u(x* t*)], i = 1, 2, 3, are now considered to be constant coefficients. In order to sobe (2.4) we use the method of multiple scales [lo] and put 6 = E 6”6,,

x, = CX,

t, = ct.

(2.5)

n=l

We will refer to t, = t as the laboratory We write in the usual way 3, = c cnaxn, n=O Substituting

these expressions

time-scale

and tj, j = 1, 2,. . . . as the slow time-scales.

a, = c &l,/ n=O

(2.6)

into (2.4) one finds to O(E)

or

P-7) where we define, Lj

Csk >= (at,t, - %,,- F(‘))‘/c.

(2.8)

A.H.J. Cloot, B.M. Herbst / Instability of Klein-Gordon equation

19

To O(e*) one finds: L,( S,) = - 2[a,0,, - &_)sl

+ :P’*‘q

(2.9)

and finally at 0( c3):

We now proceed to solve (2.7) (2.9) and (2.10). Assuming a L-periodic solution of (2.7) of the form 6, = E (A, eiK + c.c) n=O

(2.11)

0, = knxO - o, t,,

(2.12)

where k, = 2~n/L

and C.C. denotes the complex conjugate of the preceding terms, it follows that w, satisfies the linear dispersion relation: w* = k* _ J’(i) n n

(2.13)

From (2.11), (2.12) and (2.13) it is clear that a necessary condition time-scale is F(l)<

for stability on the to

min(k2). n

(2.14)

Since 6 is a small but otherwise arbitrary perturbation, (2.14) implies that F”‘
vx*,

(2.15)

t*.

If (2.11) is substituted into (2.9) we obtain L&92)

=

a

,,a,

[ 4j

+ ;P

c

+

&%A)

[ (A,A,q_&+,

h,

+

-

c.c.1

C.C.)

+

(A,AJEc#&,

+

C.C.)]

(2.16)

m,l

where we use the simplified notation +, = e%,

G~+[= ei(@,*&). _ In order to find a bounded solution we must remove the secular term: c (~,l~,w, + %l&kJ+~

= 0

(2.17)

or c.QtlAn + k,&A,

= 0

Vn

(2.18)

plus a similar expression for the complex conjugate A,*. The solution for 6, is easily found to be: ,Sj, =

_

SF(z)

c m,I

(2.19)

A.H.J.

20

Cloot, B.M. Herbst / Instability

of Klein-Gordon

equation

where Di( I, m) = (0, &(I,

+ WJ’ - (k, + kJ2 + F(i),

(2.20)

m) = (am - WJ’ - (k, - k,)* + F(l).

(2.21)

It is obvious that Di(m,

I, F(l)) =Di(l,

m; F(i)),

i= 1, 2.

(2.22)

Finally, replacing S, and 6, in (2.10) by their expressions (2.11) and (2.19) it follows, after rearranging, that L&)

= c n

c

((A&4,

+A,A;“A,)(+P-

+(F’2’)2/D2(k,

I))

(k,l,m)=n +LdkLd,~$$@(~‘-

+ a

(%4*4

+

GL247

+(F(2’)2/Dl(k,

> %I -

c

(4,r,4

I)))&

-

&%

j &I

n +R(m,

z,

k;

@(k&n)

> I(k,l,m)+n

*+

By the notation (k, 1, m) = n we mean all combinations +k+,+::

=

(2.23)

c.c.

of k, 1 and m such that

6, *

Thus, R in (2.23) contains all the nonsecular terms. Removing the secular terms from (2.23), one obtains 2ik$,;A. +

+ k&A, c

> - (%,r,4? - %,x,4)

(L&&d;”

+A,A,*A,)G,(k,

(m,l,k)=n

1) +

c

A;A,A,G,(k,

l) = 0,

(2.24)

(m,l,k)=n

where Gj(Z, m) = $F (3) - +(F’2’)2/Dj(l,

m),

j = 1, 2.

(2.25)

The condition (2.18) implies that the complex amplitude A,, remains constant in a coordinate system moving at the group velocity, VP:= da,,/ d k, = k,/w,

.

Let us therefore assume that the wave is quasi-monochromatic we transform to the moving coordinate system {:=x1-

V&=$2-v&2),

with dominant wave number k. If

7:=t2,

then (2.24) becomes 2i~~,A+(l-V~~~~~A+qlA~*A=O

(2.28a)

4= +(F’3’_

(2.28b)

where +(F(2))2/F(1)j.

A.H.J.

Cloot, B.M. Herbst / Instability

of Klein-Gordon

equation

21

From (2.15) it follows that sign(l-

Vg2)= sign( --F(i))

= +l,

(2.29)

7:= 7/2w.

(2.30)

which allows the change of variables f:= l/(1 - vp)1’2, Equation (2.28) now becomes (2.31)

i&4 + ZIgA + q 1A 12A = 0. Equation (2.31) is the cubic Schriidinger equation and is well known to be stable if

(2.32a)

q
and unstable if (2.32b)

q>o.

(see e.g. Whitham [l]). More specifically, if it is assumed that (2.33) where [a,“/-=~1 and

if p#O,

p,,,=mp

it has been demonstrated [ll] that all a,“( 7) will grow exponentially in 7”provided that (2.34)

PL2m<2q/%12. Thus, using (2.28b) it follows that the solution will be stable if

(F(3)- $( F(2))2/F(1)) 1a, 12 < ( pJ2.

(2.35)

Therefore, a sufficient condition for stability on the 7” (i.e. slow) time-scale with respect to arbitrary small perturbations, is given by J?(3)

<

(2.36)

4 ( p))2/p.

Before we illustrate the way in which these instabilities occur in practice, we first, in the next section, make a few remarks of a general nature.

4. Discussion of the theoretical results Our assumptions and the stability conditions (2.15), (2.36) deserve the following comments. (a) If one cannot assume that P(x,

t*) = P(x*,

t*)

for all x E ( - co, cc) one may need to subdivide the domain in such a way that (1.2) is satisfied for each part. In this case the stability conditions must be satisfied for all parts. (b) For a general operator F(u) in (1.1) we must, to ensure stability, satisfy the stability conditions for all values of x and t. Since 6 is an undampened wave, initial stability is not sufficient to ensure that it will not eventually reach a region where it may become unstable.

A.H.J. Cloot, B.M. Herbst / Instability of Klein-Gordon equation

22

(c) The instabilities of Section 3 occur on a slow time-scale and may only show up after long times. The linear instability which may be prevented by (2.15) occur on a fast time-scale, i.e. exponential growth on the laboratory time-scale, and is therefore much more severe. (d) Due to the recurrence associated with the cubic Schrbdinger equation [ll], the instability which occur when (2.36) is violated, will assume a periodic nature. After an initial growth of the perturbation, it will decline and this process is then repeated periodically. This kind of instability may in certain applications be associated with physical phenomena.

5. Numerical experiments In our difference scheme in for F(u). time-scales,

numerical experiments the spatial derivative in (1.1) was replaced by a central scheme and the resulting system of ODE’s solved by the Runge-Kutta-Merson the NAG-library. We illustrate the theory using the two expressions (1.3a) and (1.3b) Since our main interest lies in the way the instabilities show up on the different we do not include the rather boring pictures of the stable solutions.

5.1. Polynomial Consider

expressions for F(u)

the set of equations (a,,-k3,,)u=au+ibu3,

a, b= +l

(5.1)

and investigate the stability of their common trivial solution u = 0 (see also Table 1). It is in this case easy to calculate a priori the value of F(‘) and predict the results as follows: $‘(I) = a Considering can be written

F(*) = 0 > J’(3) = b a perturbation of the form 6 = A(1 + e cos px), (c < 1) the conditions

bA* - /_L*< 0.

a < 0, Choosing

(5 4

c = 0.1 we have tested the different

5.2. The sine-Gordon Now consider

possible

cases. The results are described

in Table 2.

equation

the sine-Gordon

(a,,-a,,)~=

of stability

-sin

equation (5 *3)

2.4,

Table 1 Various choices for the coefficients in (5.1) Case 1 a
Case 2 aO

Case 3 a
v = 0 always stable

v = 0 unstable for particular modes

v = 0 always unstable

A.H.J.

Cloot, B.M. Herbst / Instability

of Klein-Gordon

equation

23

Table 2 N

a

b

0.35 0.25 0.25 0.5 0.5 0.5

-1 -1 -1 -1

-1 1 1 1 1

1 1

-1

’ In this case the instability

observed

Theoretical prediction

Numerical result

0.4 0.2 0.4 0.4 0.2 0.2

stable stable unstable stable unstable unstable

stable stable unstable stable unstable unstable

is the linear instability

Fig. 1. Unstable

Fig. 2. Unstable

A

solution

solution

Fig. 1 a a

which occurs on the laboratory

in a slow time-scale

for F(u)

= au + Abu3.

on the fast time-scale

for F(u)

= au + &bu3.

Fig. 2(a) Fig. 2(b) scale.

A.H.J.

24

Cloot, B.M. Herbst / Instability

Fig. 3. Instability

of Klein-Gordon

on a slow time-scale

for F(u)

equation

= - sin(u).

together with the initial conditions U(X, 0) = 0.4,

U,(X, 0) = 0

(5.4)

and periodic boundary conditions. The solution consists of an oscillation in time with the maximum amplitude not exceeding the initial value. Since the solution remains constant in space, (1.2) is satisfied. The coefficients F(j) in (2.2) are F(l) = -cos

u,

FC2) = sin u>

FC3) = cos u.

(5.5)

The conditions (2.15) and (3.7) are given by -cos

UC0

(5.6)

and q1=i(f+2s

in2 u)/cos u < 0

(5 -7)

respectively. These conditions cannot be satisfied simultaneously and the solution will therefore never be unconditionally stable. Since (5.6) prevents the more severe linear instability we ensure that it is satisfied by using maxlu] < $rr. (5.8) X,f The choice (5.4) implies that u E [ - 0.4, 0.41 which in turn leads to values of q1 in (5.7) in the range [i, 0.5981. If the perturbation S=A(l+ecos/_Lx),

C=Cl

(5 -9)

A. H.J. Cloot, B.M. Herbst / Instability Table 3 Numerical

experiments

conducted

with the sine-Gordon

of Klein-Gordon

25

equation

equation

P

A

41

Theoretical prediction

Numerical

i

0.84 0.528

0.598

Unstable Unstable

Unstable Unstable

Fig. 3(a)

$ :

0.44

Unstable

: ;

0.26 0.24

0.598

Unstable Unstable

0.598

Unstable

Fig. 3(c) Fig. 4(a) Fig. 4(b)

+

0.23

0.598

Stable

:

0.096

7 4

0.3

0.598 0.598

Stable Stable

0.5 0.598

is added to the solution solution provided that P2>2G

result

Fig. 3(b)

Unstable Slightly unstable Stable Stable Stable

of (5.3) - (5.4) the condition

(3.5) ensures

the stability

(5.10)

IA12.

Because of the variation in qi mentioned above we need to consider (a) A E (0, p//G)-always stable. (b) A E [p/.,/G,

of the perturbed

p//‘-]-sometimes

the following

values of A:

unstable.

(c) A E [p//w, l]-always unstable. In case (b) the choice of A is such that (5.10) is violated only at certain times when qi is close to its maximum value. The The various possibilities offered by the various choices of A were tested numerically. numerical experiments are summarized in Table 3. For these experiments we used p = i in which case the following intervals for A need to be considered: (a) A E [0, 0.2291. (b) A E [0.229, 0.251. (c) A E [0.25, 11.

Fig. 4. Marginally

unstable

solution

for F(u)

= -sin(u),

26

A.H.J.

Cloot, B.M. Herbst / Instability

of Klein-Gordon

equation

The value of q1 in Table 3 is its value at the time when the perturbation (5.9) was added to the solution. Figure 4 show the results on the edge of the stability/instability regions and it is just possible in these cases to discern the instabilities. The table also shows that the numerical results are in excellent agreement with the theory.

6. Conclusions In this paper we conducted numerical experiments on the instabilities, appearing on the different time-scales in the Klein-Gordon equation. The numerical experiments clearly demonstrate the validity of the analysis and, more importantly, show the different time-scales involved. The instabilities on the slow time-scales take a long time to develop and is therefore prone to round-off errors. Since we are dealing with unstable, albeit weakly unstable, processes these errors may seriously affect our numerical solutions. This important question is addressed in [2].

References [l] T.B. Benjamin and J.E. Feir, The disintegration of wave trains on deep water. Part 1. Theory, J. Fluid Mech. 27 (1967) 417-430. [2] A. Cloot and B.M. Herbst, A model for the propagation of rounding error in a Klein-Gordon equation, in preparation. [3] H. Hasimoto and H. Ono, Nonlinear modulation of gravity waves, J. Phys. Sot. Jap. 33 (1972) 805-811. [4] B.M. Herbst, A.R. Mitchell and J.A.C. Weideman, On the stability of the nonlinear Schrodinger equation, J. Comp. Phys. 60 (1985) 263-281. [5] E. Infeld, Quantitative theory of the Fermi-Pasta-Ulam recurrence in the nonlinear Schrodinger equation, Phys. Reu. Lett. 47 (1981) 717-718. [6] A. Jeffrey and T. Kawahara, Asymptotic Methods in Nonlinear Wave Theory (Pitman, Boston, 1982). [7] M. Stiassni and U.I. Kroszynski, Long-time evolution of an unstable water-wave train, J. Fluid Mech. 116 (1982) 207-225. [8] J.T. Stuart and R.C. Di Prima, The Eckhaus and Benjamin-Feir resonance mechanisms, Proc. Roy. Sot. London 362 (1978) 27-41. [9] J.A.C. Weideman and B.M. Herbst, Recurrence in semi-discrete approximation of the nonlinear Schriidinger equation, SIAM J. Sci. Stat. Comput. (1986) submitted for publication. [lo] G.B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974). [ll] H.C. Yuen and W.E. Ferguson, Relationship between Benjamin-Feir instability and recurrence in the nonlinear Schrodinger equation, Phys. Fluids 21 (1978) 1275-1278.