Numerical Solution of coupled-KdV systems of Boussinesq equations: II. Generation, interactions and stability of generalized solitary waves J. L. Bona
a,1
V. A. Dougalis b,c and D. E. Mitsotakis b,c
a Department
of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA b Department of Mathematics, University of Athens, 15784 Zographou, Greece c Institute of Applied and Computational Mathematics, FO.R.T.H., P.O. Box 1527, 71110 Heraklion, Greece
Abstract In this paper we investigate numerically generalized solitary wave solutions of two coupled KdV systems of Boussinesq type. We present numerical experiments describing the generation and evolution of such waves, their interactions, the resolution of general initial profiles into sequences of such waves, and their stability under small perturbations. Key words: Generalized solitary waves, coupled KdV systems, Boussinesq systems, one- and two-way propagation, resolution into solitary waves, stability of solitary waves.
1. Introduction In the preceding paper we considered coupled KdV systems of Boussinesq type, mainly the coupled KdV system, [5] ηt + ux + (ηu)x + 16 uxxx = 0, ut + ηx + uux + 16 ηxxx = 0,
(1)
and the symmetric coupled KdV system, [4] ηt + ux + 12 (ηu)x + 16 uxxx = 0, ut + ηx + 12 ηηx + 32 uux + 16 ηxxx = 0, 1
(2)
Email addresses:
[email protected] (J. L. Bona ),
[email protected] (D. E. Mitsotakis). Corresponding author
Preprint submitted to Elsevier Science
11 October 2005
We discussed briefly the known existence and uniqueness results for both systems and showed, by applying the theory in [9], that they possess generalized solitary wave (GSW) solutions consisting of main ‘solitary’ pulses that decay to oscillatory profiles of small amplitude. We constructed such generalized solitary waves by solving the systems of nonlinear o.d.e.’s that we obtain from (1) and (2) when we assume that they possess travelling wave solutions and impose periodic boundary conditions. We also constructed and tested a high order, unconditionally stable fully discrete scheme for solving the initial- and periodic boundaryvalue problem for (1) and (2). The scheme uses the Galerkin method with smooth periodic splines for the spatial discretization and the fourth-order accurate two-stage Gauss Legendre implicit Runge-Kutta scheme (coupled with Newton’s method and appropriate efficient linear systems solvers) for time-stepping. In the paper at hand, a sequel to Part I, we use the above fully discrete scheme to study numerically several interesting properties of these systems related to the GSW’s. In Section 2 we generate such waves starting from general initial profiles leading to two-way propagation and, in Section 3 starting from special initial conditions that give rise to solutions that practically propagate towards one direction. We try to isolate structures resembling GSW’s by means of iterative cleaning. In Section 4 we let these cleaned wave profiles collide with each other and study the details of their interaction. We also study the analogous interaction of ‘exact’ GSW’s, generated by the o.d.e. systems referred to earlier. In Section 5 we study the stability of these waves under small perturbations and comment on the related property of resolution, as t grows, of general initial profiles into sequences of such waves. As pointed out in Part I, the GSW’s of (1) and (2) are quite similar. We therefore present in this note mainly numerical solutions of (1). The phenomena exhibited by the solution of the symmetric system (2) are entirely analogous, but we point out some interesting differences. It should be noted that system (1) is more challenging to integrate numerically in time compared to the L2 -conservative symmetric system (2).
2. Generation of GSW’s by evolution If we consider the initial-value problem for (1) or (2) with a given general initial profile (η0 (x), u0 (x)) and integrate in time using the evolution code described in Part I, we observe that one or more solitarywave like pulses are generated, followed by smaller amplitude dispersive oscillatory tails, and accompanied by very small amplitude oscillations possibly indicating that the main pulses are GSW solutions of (1). This is illustrated in Figure 4(a), (b), which shows the right-travelling wavetrain (there is also a symmetric one travelling to the left but not shown in the figure) produced at t = 160 by the evolution of (1) generated by a Gaussian initial condition In order to study the apparent GSW’s generated in this manner (‘by evolution’) we need to separate a main pulse from its dispersive tail. This may be done by iterative ‘cleaning’, [3], in which a solitary wave, after being allowed to distance itself from its trailing tail, is cut from the rest of the solution and used as initial condition for a new evolution. It develops again an oscillatory dispersive tail of smaller amplitude, and the process is repeated until a ‘clean’ solitary wave emerges that travels shedding dispersive oscillations of extremely small amplitude that remains under a specified tolerance, e.g. 10−10 . For Boussinesq systems known to possess solitary waves and having dissipative terms like −uxxt , −ηxxt this process converges fast, cf. e.g. [3], [2], [8]. For systems like (1) and (2) two issues complicate matters: The absence of dissipative terms and the fact that generalized solitary waves emerge consisting of main pulses and ripples. The absence of dissipative terms slows down the process considerably. As an illustration, we consider the KdV equation 2
1 ut + ux + uux + uxxx = 0, (3) 6 which we solve in [−150, 150] with periodic boundary conditions starting from a Gaussian u(x, 0) = 2 0.5e−(x+100) /25 and using cubic splines with h = 0.1 and the two-stage Gauss-Legendre Runge-Kutta scheme with k = 0.02. In the first cleaning iteration we let the solution evolve up to t = 160 (cf. Figure 1), cut the dispersive tail by setting uh = 0 in two intervals [−150, 90.2], [113.2, 150] and translate back 0.8
0.1
0.0001
0.7
8e-005 0.08
0.6 6e-005 0.5
0.06 4e-005
0.4 0.04 2e-005
0.3
0.02
0.2
0
0.1
-2e-005 0
0 -4e-005 -0.1 -150
-100
-50
0
50
100
150
-0.02 -150
-100
-50
(a)
0
50
100
150
88
89
90
(b)
91
92
93
94
(c)
Fig. 1. Evolution of a Gaussian, KdV equation (3), t = 70; (b) is a magnification of (a), and (c) shows the smooth truncation in the vicinity of x = −90.2.
the remaining pulse so that its peak is again at x = −100. (It should be noted that “cutting”, i.e. setting the numerical solution uh = 0 in a set [−150, α] ∪ [β, 150], is done smoothly by setting the coefficients 2e-007
5e-008 4e-008
1.5e-007
4e-010
3e-008 1e-007 2e-008 5e-008
2e-010
1e-008
0
0
0
-1e-008
-5e-008
-2e-008
-2e-010
-1e-007 -3e-008 -1.5e-007
-2e-007 -150
-4e-008
-100
-50
0
(a)
50
100
150
-5e-008 -150
-4e-010
-100
-50
0
(b)
50
100
150
-150
-100
-50
0
50
100
150
(c)
Fig. 2. Cleaning of the cut solitary wave of Figure 1. (a): After the 4th iteration, (b): After the 7th iteration, (c): After the 15th iteration. (All at t = 70).
of the B-spline basis functions in the representation of uh equal to zero before and after some index. Thus the truncated solution is in C 2 [−150, 150]). During the evolution with the new initial profile small amplitude oscillations are still produced behind the main wave due to the cutting, so the process has to be repeated. The amplitude of the oscillatory noise is decreasing rather slowly. For example, Figure 2 shows three stages of cleaning of the solitary wave. The ‘clean’ solitary wave that is produced after 15 iterations propagates retaining for long time (up to t = 200 at least) six constant digits in its speed (1.24912) and its amplitude (0.747367), and an L2 -shape error (relative to its shape at t = 0) less that 3.7 · 10−6 . We proceed now to integrate the system (1) in time with the evolution code, using cubic splines with h = 0.02, k = 0.004 on [−150, 150], starting from Gaussian like initial conditions and aiming to produce a ‘clean’ solitary wave-like solution. The initial profile is resolved into two wavetrains moving in opposite 3
directions; Figure 3(a)–(b) shows the η component of the solution at t = 30. Note that in addition to 0.18
0.0001
0.0001
5e-005
5e-005
0
0
-5e-005
-5e-005
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -150
-100
-50
0
50
100
150
-0.0001 -150
-100
-50
(a)
0
50
100
150
-0.0001 -150
-100
-50
(b)
0
50
100
150
(c)
Fig. 3. Evolution of a Gaussian, system (1), η at t = 30; (b) is a magnification of (a), and (c) is the new initial profile after the first cleaning.
the dispersive tails following the two main pulses, small amplitude ripples have developed in front or the main pulses. We clean the solution, keeping only the rightward moving wavetrain by zeroing the solution in [−150, −100.6] ∪ [50.44, 150] and using the resulting profile as new initial condition setting t = 0 again. At t = 160 this new profile has evolved into the one shown in Figure 5 (a)–(b) (Note the ripples that have appeared). We translate it back so that its maximum is at x = −50 and we clean the solution in 0.2
0.0001
0.0001
5e-005
5e-005
0
0
-5e-005
-5e-005
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -150
-100
-50
0
(a)
50
100
150
-0.0001 -150
-100
-50
0
(b)
50
100
150
-0.0001 -150
-100
-50
0
50
100
150
(c)
Fig. 4. Evolution of profile (c) of Figure 3, t = 160; (b) is a magnification of (a), and (c) is the initial profile after translation and the second cleaning.
[−150, −117.26] ∪ [−36.3, 150]. The result is shown in Figure 4(c). We repeat this procedure two more times. After the last (fourth) cleaning we let the solution evolve up to t = 70; the result is shown in Figure 5(a)–(b). At this point, the leading pulse has distanced itself sufficiently from the trailing dispersive tail. Hence, after translating back so that the maximum is at x = −100, we isolate the main pulse by cutting out dispersive tails and ripples i.e. zeroing the solution in [−150, −114.4] ∪ [−86.44, 150] (fifth cleaning). The resulting pulse is shown in Figure 5(c). If we let this profile evolve, we will observe again ripples appearing in front of the main pulse and wrapping around due to the periodicity. If we clean the solution two more times and start from the profile shown in Figure 6(a) we obtain the evolution shown in Figure 6(b)–(f). Ripples develop in front of the wave, wrap around the boundary due to periodicity, ‘climb’ over the main pulse and eventually seem to be of approximately uniform amplitude. These ripples are not an artifact of the numerical integration. When we repeated the computation with smaller h and k, and also when we used quintic splines for the spatial discretization and even another time 4
0.2
0.0001
0.0001
5e-005
5e-005
0
0
-5e-005
-5e-005
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -150
-100
-50
0
50
100
150
-0.0001 -150
-100
-50
(a)
0
50
100
150
-0.0001 -150
-100
-50
(b)
0
50
100
150
(c)
Fig. 5. Evolution after fourth cleaning, t = 70; (b) is a magnification of (a), and (c) is the new initial profile after translation and the fifth cleaning.
0.0001
0.0001
0.0001
5e-005
5e-005
5e-005
0
0
0
-5e-005
-5e-005
-5e-005
-0.0001 -150
-100
-50
0
50
100
150
-0.0001 -150
-100
(a) t = 0
-50
0
50
100
150
-0.0001 -150
0.0001
0.0001
5e-005
5e-005
5e-005
0
0
0
-5e-005
-5e-005
-5e-005
-100
-50
0
(d) t = 120
50
100
150
-0.0001 -150
-100
-50
0
-50
0
50
100
150
50
100
150
(c) t = 80
0.0001
-0.0001 -150
-100
(b) t = 40
50
100
150
-0.0001 -150
-100
(e) t = 160
-50
0
(f) t = 200
Fig. 6. Evolution after the seventh cleaning at t = 0(40)200.
stepping method (a third order Gauss-Lobatto implicit Runge-Kutta scheme), we obtained, at the same cleaning level, exactly the same profiles. The profile of the ripples changes with the number of cleanings of course but it is convergent and by the seventh cleaning it has stabilized. Figure 7, for example, gives some idea of this convergence: It shows the ripples after the sixth cleaning (dotted line) and after the seventh cleaning (solid line) at t = 120. The profile after the eighth cleaning is identical within line thickness with that after the seventh cleaning. Thus, it is reasonable to conclude that after the 7th cleaning, the noise due to the cleaning process has fallen to at least two orders of magnitude below the ripples. The slow increase of the ripple amplitude with t could be a linear growth due to the numerical integration. The properties of the main pulse are not easy to measure due to the superposition of ripples and main pulse. The amplitude of its η-component is shown in Figure 8. The speed of the pulse was found to be 5
4e-005
4e-005
4e-005
2e-005
2e-005
2e-005
0
0
0
-2e-005
-2e-005
-2e-005
-4e-005
-4e-005
-150
-140
-130
-120
-110
-100
-4e-005
-100
-90
(a)
-80
-70
-60
-50
-50
(b)
4e-005
4e-005
2e-005
2e-005
2e-005
0
0
0
-2e-005
-2e-005
-2e-005
-4e-005
-4e-005
10
20
30
40
50
-30
-20
-10
0
130
140
150
(c)
4e-005
0
-40
-4e-005
50
60
(d)
70
80
90
100
100
110
(e)
120
(f)
Fig. 7. Magnification of the ripples, t = 120, —:7th cleaning, - - -:6th cleaning
0.18932
0.18931
0.1893
0.18929
0.18928
0.18927
0.18926
0.18925
0.18924 0
50
100
150
200
250
300
Fig. 8. Amplitude of the main pulse of Figure 6 as a function of t.
1.090±0.005. Overall, the computation was of generally good quality. The invariants udx, ηdx, ηudx were conserved up to 10 digits up to t = 300. The Hamiltonian preserved 5 digits up to t = 150 and 4 digits up to t = 300. The ripples at t = 120, cf. Figure 7, have an amplitude (7th cleaning) that does does not exceed 4 · 10−5 . We measured their propagation speed and period at different subintervals of [−150, 150] and found that the ripples approximate well solutions of the linearized system. For example, in the case of the ripples of figure 7(e), the measured average spatial period was equal to about 1.77 (hence k ∼ = 3.55) and the average speed about 1.11. (The wave number computed for this speed from the linearized dispersion relation for (1), k = 6(c + 1), is equal to about 3.56.) It is safe then to conclude, that by t = 120, say, this process generates a good approximation to a generalized solitary wave of (1). 6
3. One-way propagation It is well-known, cf. e.g. [1], that if the initial values η0 (x), u0 (x) of the solution of a Boussinesq system are suitably chosen, then the solution that emerges moves in one direction. In the case of (1) we took such special initial data, of the form 1 (4) η(x, 0) = φ(x), and u(x, 0) = φ(x) − φ2 (x), 4 √ 3A 3 1 where φ(x) = Asech2 2 x is a solitary wave of the KdV equation ut + ux + 2 uux + 6 uxxx = 0, and integrated in time using the evolution code with cubic splines in [−150, 150] with h = 0.1, k = 0.02. The resulting evolution of the η component of the solution is shown in Figure 9 in the case A = 0.6. 0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 -0.1 -150
0 -100
-50
0
50
100
150
-0.1 -150
-100
-50
t=0
50
100
150
50
100
150
t = 10
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 -0.1 -150
0
0 -100
-50
0
50
100
150
t = 30
-0.1 -150
-100
-50
0
t = 40
Fig. 9. One-way propagation for (1) with initial condition (4), A = 0.6.
The waveform that emerges moves indeed to the right shedding a dispersive tail behind it; small ripples appear in front (as e.g. in [7], where another one-way model p.d.e. is being considered) and by t = 60, have turned around the boundary and are starting to interact with the main pulse. Figure 10(a) shows the amplitude of the main pulse as a function of t for the evolution of Figure 9, while Figure 10(b) shows the amplitude when the computation is repeated on [−500, 500]. The main pule initially loses height as the dispersive tail and the ripples are formed. The further drop of the amplitude in (a) right after t = 60 occurs when the ripples overtake the main pulse. In (b) this has not occurred up to t = 100 due to the larger spatial interval. The apparently slow linear growth of the amplitude in (b) is probably due to the numerical integration. We observed a similar general behavior for initial amplitudes A up to about 1.2. For A 1.2 the solution blew up; we are not certain, given the present, uniform mesh evolution code that we are using, whether this is a numerical artifact or a property of the solution of the system. However, when the computations were repeated with smaller h and k and quintic splines blow-up occurred at about the same t. We observed no blow-up in the case of the symmetric KdV system (2) for the same ranges of A and final T . We also obtained similar results for the symmetric coupled KdV system (2). For example, taking now one-way data appropriate for (2), i.e. of the form 7
0.605
0.605
0.6 0.6
0.595 0.595 0.59 0.59 0.585
0.585 0.58
0.575
0.58 0
10
20
30
40
50
60
70
80
90
100
0
20
40
(a)
60
80
100
(b)
Fig. 10. Amplitude of the main pulse as a function of t; (a): Evolution of Figure 9, (b): Same example but in [−500, 500].
1 η(x, 0) = φ(x), u(x, 0) = φ(x) + φ2 (x), (5) 4 with φ(x) as before, we obtain, for A = 0.6, the evolution shown in Figure 11. The size of the emerging ripples is larger now, so by t = 120 the main pulse is still losing mass (cf. Figure 12) which is apparently feeding new ripples. 0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 -0.1 -500
0 -400
-300
-200
-100
0
100
200
300
400
-0.1 -500
500
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0 -0.1 -500
-400
-300
-200
-100
0
100
200
300
400
500
-400
-300
-200
-100
0
100
200
300
400
500
0 -400
-300
-200
-100
0
100
200
300
400
-0.1 -500
500
Fig. 11. One-way propagation for (2) with initial conditions (5), A = 0.6, t = 0, 40, 80, 120.
0.6
0.59
0.58
0.57
0.56
0.55
0.54 0
20
40
60
80
100
120
Fig. 12. Evolution of the amplitude of the main pulse up to t = 120, evolution of Figure 11.
8
It should be noted that ‘cleaning’ the main pulse that is produced from this type of ‘one-way’ initial data is sometimes very difficult. For example, after four cleaning iterations, performed in the usual way, the ‘cleaned’ main pulse that is produced by the evolution of Figure 9 of system (1) is shown in magnification in Figure 13. Subsequent cleanings yielded exactly the same picture. The ripples in this example are quite
0.04
0.02
0
-0.02
-0.04
-150
-100
-50
0
50
100
150
Fig. 13. ‘Cleaned’ main pulse, evolution of Figure 9, after 4 cleaning iterations
large, in comparison to those emerging from the Gaussian initial condition of Section 2. As a result, it seems that the dispersive tail cannot be entirely cleaned away as it is apparently reinforced by interaction with the ripples. The amplitude of this ‘cleaned’ pulse as a function of t oscillates with small variation about a mean value of 0.587.
4. Head-on collisions of generalized solitary waves In this section we briefly address the issue of interaction during a head-on collision of two generalized solitary waves by giving a report of two numerical experiments that we performed. We consider the system (1) and, starting with initial values of the form η(x + 100) + η(x − 100) and u(x + 100) − u(x − 100), where η and u are cleaned main pulses obtained from a Gaussian as described in Section 2, we integrate numerically in time. We observe the evolution shown in Figure 14. Figure 15 shows magnifications of some frames of Figure 14 that enable us to see some details of the interaction. As t grows, before the interaction, ripples appear in front and after the main pulses. 9
0.2
0.2
0.2
t=0
t=40
t=80
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
0
-150
-100
-50
0
50
100
0
-150
150
0.2
-100
-50
0
50
100
150
-150
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
50
100
150
-150
0.2
-100
-50
0
50
100
-150
150
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
-50
0
50
100
-50
0
50
100
150
150
t=140
0.15
-100
-100
t=130
0
150
0.2
0.2 t=120
-150
100
0
0
0
50
t=110
0.15
-50
0
t=100
0.15
-100
-50
0.2
0.2 t=90
-150
-100
0
-150
-100
-50
0
50
100
-150
150
-100
-50
0
50
100
150
Fig. 14. Collision of ‘cleaned’ main pulses of system (1)
After the interaction (t = 110) we observe an interaction of the dispersive tails following the two main pulses as they separate and an enlargement of the ripples in front of them. 0.003
0.003
0.003
0.002
0.002
0.002
0.001
0.001
0.001
0
0
0
-0.001
-0.001
-0.001
-0.002
-0.002
-0.002
-0.003 -150
-100
-50
0
50
100
150
-0.003 -150
-100
-50
0
50
100
150
-0.003 -150
-100
-50
0
50
100
150
Fig. 15. Magnification of profiles in Figure 14 at t = 40, 80, 110.
Figure 16(a) shows the (total) amplitude of the solution as a function of t. During the interaction the amplitude rises to about 0.39833, which is approximately 5.21% above the sum of the amplitudes of the two initial pulses, and returns to its previous value modulo small oscillations due to ripples. Figure 16(b) 10
is a phase-shift diagram showing the paths in the t, x plane of the two main pulses. As expected both 0.4
102 100
0.35
98 96
0.3
94 t
0.25
92 90 88
0.2
86 84 0
50
100
150
82 −10
200
−5
0 x
(a)
5
10
(b)
Fig. 16. Total amplitude and phase shift diagram for the head-on collision of Figure 14.
main pulses are delayed after the interaction. In a similar manner we may study interactions of GSW’s that are obtained as solutions of the periodic two-point boundary value problem for the system of o.d.e.’s analyzed in Section 4 of paper I, and then used as initial conditions in the evolution code. For example, Figure 17 shows the collision of two GSW’s for the system (1) on [−50, 50]. The initial conditions are obtained as outlined above, are centered at x = ±20 and are given opposite velocities. (The GSW’s have speed 1.2 i.e. c = 0.2 in the notation of Section 4 of Paper I, and were constructed with L = 50). The simulation of their temporal evolution was done with cubic splines and h = 0.05, k = 0.01. The features of the collision are basically the same as before, but the oscillations produced after the interactions are larger. Figure 18 shows the evolution of the total amplitude of the solution and the phase shift diagram near the collision point. 1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
-50
0
-40
-30
-20
-10
0
10
20
30
40
50
-50
0
-40
-30
-20
-10
0
10
20
30
40
50
-50
1
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
-50
0
-40
-30
-20
-10
0
10
20
30
40
50
-50
-40
-30
-20
-10
0
10
20
30
40
50
-40
-30
-20
-10
0
10
20
30
40
50
0
-40
-30
-20
-10
0
10
20
30
40
50
-50
Fig. 17. Collision of two GSW’s for system (1)
11
1
30
0.9 25 0.8 20
0.7 t 0.6
15
0.5 10 0.4
0
5
10
15
20
25
30
5 −15
35
−10
−5
0 x
5
10
15
Fig. 18. Total amplitude vs. time and phase shift diagram for the collision of Figure 17
5. Stability of GSW’s and resolution into GSW’s We close this note with a brief report on some numerical experiment that we performed aimed at producing computational evidence of the stability of the GSW’s of the coupled KdV systems under small perturbations and of the related property of the resolution of general initial conditions into sequences of such waves. To illustrate stability we considered as η0 (x) and u0 (x), a GSW for the system (1) generated as solution of the periodic two-point boundary value problem for the o.d.e. system. (These were GSW’s with speed cs = 1.2 (c = 0.2), constructed with L = −150) We perturbed η0 (x) by a factor r and took as initial conditions for the evolution code on [−150, 150] the functions η0 (x) = rη0 (x) and u0 (x). Figure 19 shows the evolution in the case r = 1.05. As a result of the small perturbation, a dispersive tail appears and the whole profile is superimposed with ripples. Figure 20(a) shows the evolution of the amplitude of the main pulse. When the perturbation factor r grows to 1.5 (Figure 21, evolution shown in perspective), the left-travelling wave train has a front pulse that will probably evolve into a smaller main pulse. Note that the ripples now are larger in amplitude. Figure 20(b) shows the associated evolution of the amplitude of the larger main pulse as a function of time.
0.5
0.45
0.45
0.4
0.4
0.35
0.35
0.4 0.35
0.45
0.3
0.3
0.25
0.25
0.3 0.25 0.2
0.2
0.15
0.15
0.2 0.15 0.1 0.05
0.1 0.05
0
0 -0.05 -150
0.1 0.05
-100
-50
0
50
100
150
-0.05 -150
0
-100
-50
0
50
100
150
-0.05 -150
-100
-50
0
50
100
150
Fig. 19. Evolution of a perturbed GSW for (1). Perturbation of η0 (x), r = 1.05,t = 0, 40, 100.
These numerical experiments (and many other similar ones that we performed for the systems (1) and (2) as well) suggest that the GSW’s are “asymptotically stable” under small perturbations. A detailed numerical study of asymptotic stability properties of the solution of other Boussinesq systems is made in [8]. A manifestation of the stability of the GSW’s is the resolution property of arbitrary initial profiles into sequences of GSW’s. (We have already seen an example of the resolution of a Gaussian initial condition 12
0.458
0.66
0.456
0.64
0.454 0.62 0.452 0.6 0.45 0.58 0.448
0.56
0.446
0.444
0.54 0
20
40
60
80
100
20
0
40
60
80
100
Fig. 20. Evolution of the amplitude of the main pulse, Figures 19 and 21. (a):r = 1.05, (b):r = 1.5.
0.8 0.6
η
0.4 0.2 0 −0.2 100 200 100
50
0
t
−100 0
x
−200
Fig. 21. Perspective view of the evolution of the perturbed GSW of (1) with r = 1.5. 2
in Section 2). If we consider the system (1) and take initial data of the form η0 (x) = ae−λx , u0 (x) = 0 in [−150, 150], we observe that the initial profile is resolved into leftward- and rightward-travelling sequences of GSW’s whose number depends on the parameters a and λ of the initial Gaussian, cf. Figure 22. Of course, ripples appear as well. 0.6
0.5
0.7
0.8
0.6
0.7
0.6
0.5 0.4
0.5 0.4 0.3
0.4 0.3 0.3
0.2 0.2
0.2 0.1 0.1
0
-0.1 -150
0.1
0
-100
-50
0
(a)
50
100
150
-0.1 -150
0
-100
-50
0
(b)
50
100
150
-0.1 -150
-100
-50
0
50
100
150
(c)
Fig. 22. Resolution of a Gaussian; system (1), a = 1, (a): λ = 1/5, (b ): λ = 1/15, (c): λ = 1/35
A similar picture is obtained for the system (2) albeit with larger ripples, cf. Figure 23. When we took larger amplitude a in the Gaussian (for example for a = 2) we observed that the numerical solution of (1) blew up in finite time. We cannot be certain of course whether this is a numerical or a real phenomenon; however, when we repeated the experiment with different mesh parameters h and k we observed blow-up at the same time. It should be noted that no blow-up was observed in the case of 13
0.6
0.5
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.4
0.3
0.2
0.1
0
-0.1 -150
0
-100
-50
0
(a)
50
100
150
-0.1 -150
0
-100
-50
0
(b)
50
100
150
-0.1 -150
-100
-50
0
50
100
150
(c)
Fig. 23. Resolution of a Gaussian; system (2), a = 1, (a): λ = 1/5, (b ): λ = 1/15, (c): λ = 1/35
the symmetric KdV system (2) as far as we can see. See [8] for a more detailed study of resolution into solitary waves in the case of other classes of Boussinesq systems. References [1] A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen and J. Wu, Comparisons between the BBM equation and a boussinesq system, Adv. Differential Equations, (to appear). [2] D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of the Bona-Smith system by the finite element method to appear. [3] J. L. Bona and M. Chen, A Boussinesq system for two-way propagation of nonlinear dispersive waves, Physica D 116(1998), 191–224. [4] J. L. Bona, T. Colin, and D. Lannes, Long wave approximations for water waves, Arch. Rational Mech. Anal., (to appear). [5] J. L. Bona, M. Chen, and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media: I. Derivation and Linear Theory, J. Nonlinear Sci., 12(2002), 283–318. [6] J. L. Bona, V. A. Dougalis, and D. E. Mitsotakis, Numerical solution of Coupled KdV systems of Boussinesq equations. I: The numerical scheme, This journal. [7] E. S. Belinov, R. Grimshaw and E. P. Kuznetsova, The generation of radiating waves in a singularly-perturbed Kortewegde Vries equation, Physica D, 96(1993), 270–278. [8] V. A. Dougalis, A. Duran, M. A. Lopez-Marcos and D. E. Mitsotakis, A numerical study of the stability of solitary waves of the Bona-Smith system to appear. [9] E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders, with applications to homoclinic orbits in reversible systems. Lecture Notes in Mathematics, 1741, Springer-Verlag, Berlin, (2000).
14
