Volume 108A, number 1 PHYSICS LETTERS 11 March 1985

GENERATION OF BOUSSINESQ SOLITONS VIA BOUNDARY CONDITIONS

Carlos CAMACHO and Fernando LUND

Departamento de Eisica, Facultad de Ciencias Fisicas y Matemriticas, Universidad de Chile, Casilla 5487, Santiago, Chile

Received 6 December 1984

The generation of solitons is studied solving the Boussinesq equation on a half-line with box-type boundary conditions. A train of decreasing-amplitude solitons is generated. The width of the box controls the number of solitons while its height determines the amplitude of the leading one.

One of the most interesting challenges of soliton physics is to understand how these nonlinear lumps are coupled to an external source: what is their excita- tion mechanism? The problem is difficult, of course, because of its nonlinear nature. Any wave-like phenom- enon described by a linear evolution equation can be

decomposed into normal modes and if there is an ex- ternal source, Fourier analysis says what is to be done:

decompose the source into the corresponding normal modes, each driving one Fourier component separate- ly and the resulting time evolution is found by super- position.

No linearity means no superposition and things have to be rethought. In a sense made precise by the

inverse scattering method [l] solitons behave as normal modes of systems bearing them, and it is natural to try to decompose the driving term in a soliton equation along those modes. Steps along this difficult road have been taken recently by Kaup [2] for the driven Toda lattice, as well as by Boiti, Leon and Pempinelli [3] for the sine-Gordon equation

with a peculiar coupling to an external field. A more pedestrian approach was started by Chu,

Xiang and Baransky [4] who solved numerically the Korteweg-de Vries equation on a half-line subject to prescribed boundary conditions chosen to mimic the situation in a shallow water channel which is initially quiescent and whose surface is pushed up at one end and then allowed to come back to its initial position. The remarkable finding of Chu et al.

[5] was that a number of identical solitons peel off this boundary condition, their amplitude being twice the height of the boundary perturbation, and their number depending on the duration of the perturba-

tion. In order to understand the generality of these

results we have solved numerically the Boussinesq equation

%t = P2% + rcu2xx + axxxx (1)

on the half-line with boundary conditions similar to those considered by Chu et al. [4], which are also relevant to the generation of Love wave solitons

studied by one of us [5]. We take eq. (1) whose

initial value problem is ill-posed rather than the one with the opposite sign for the dispersive term, be- cause it is the one appearing in physical applications. This is no great difficulty, it only means that the space discretization has to be taken large enough.

ukt) t

‘B tlno. of grid points)

Fig. 1. The three different boundary conditions used to solve eq. (1) on the half-tine.

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Volume 108A, number 1 PHYSICS LETTERS 11 March 1985

The boundary condition u(t, 0) we take is shown in fig. 1. Initial conditions are ~(0, x) = u,(O, x) = 0. The numerical scheme we use away from x = 0 is straightforward. Eq. (1) is approximated by

(At)-2 (u;+’ - 2~7 + ui”- ‘)

= (Ax)-2 p2(~/+1 - 2~7 + ui”_ l)

+ (Ax)-2y[(~i”+ I)2 - 2(~7)~ + (ui”_ 1)2]

+ (Ax)-4 e(ui”,, - 4~7+~ + 6~7 - 4ui”_ 1 + pi”_ 2) , (2)

where we have defined ~7 z u(jAx, nat). This has to be modified for j = 1, where we use a scheme similar to the one of Chu et al. [4], approximating values at the point x = 3Ax/2, t = (N + 1/2)At by

uXX Ly (2Ax)-2 (Go - E1 - ii2 t US) ,

(u2xx - (~Ax)-~ [(G;) - (UT) - (G;) t (+)] ,

uXXXX - (Ax)-~ (.GO - 4til+ 6a2 - 4@3 + ti4) ,

utt ‘CI (2At)-2 ($u”O+~ - .;+l t $u”o t ;u;+~

- 2u;+l - ;u; t uy+l -2U7 t.q-1 tL@),

(3)

where Uj = i&jr+’ + $), and imposing that (1) holds. We have chosen f12 = 30.25, y = 4 and E = 100;

At = 0.68 and Ax = 4. For numerical stability of the wave equation one must have Ax > @At. Moreover, short wavelengths have to be cut off and we found

A

/- 550 600 x

1.. t=300

Fig. 2. The time evolution of boundary condition A. Three solitons are formed. The amplitude of the leading one is about twice the height of the boundary perturbation. Time and space are given in number of grid points.

Fig. 3. The time evolution of boundary condition B. Four solitons are formed. The amplitude of the leading one is about twice the height of the boundary perturbation. Time and space are given in number of grid points.

that for 0,~ and E as above good results were ob- tained for Ax 2 3.6.

The evolution of the three different boundary conditions of fig. 1 is given in figs. 2-4. Clearly, for small trmes the utt = p2uX, portion of the equa- tion dominates as it should. After a while the non- linear and dispersive terms become dominant and the original perturbation breaks up in a number of soli- tons. As for KdV [4] their number increases with height and width of the boundary condition. As op-

Fig. 4. The time evolution of boundary condition C. Seven solitons are formed. Here as well the leading soliton grows to twice the amplitude of the boundary perturbation. Time and space are given in number of grid points.

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Volume 108A, number 1 PHYSICS LETTERS 11 March 1985

posed to what happens in the KdV case, however, they are not all of the same height; rather, their amplitudes increase roughly linearly. The amplitude of the leading soliton is quite sensitive to the height of the boundary perturbation but not to its width, and is very roughly twice the amplitude of the bound-

ary perturbation, as for the KdV solitons. Notice that it would not do to eliminate the

P2%X term from (1) by a change of variables as one

does with the ux term in

2.Q + UX + uux + UXXX = 0

to get KdV in order not to wait a transient time to have solitons appear. The reason is that in the equa-

tion

all wavelengths blow up in time, not just short ones. As a check on the numerics we verified that

J au %dx=O

0

after the boundary perturbation had died out. Also,

the bumps in figs. 2-4 were fitted with the analytical solution

u = 3(c2 - P2)

2Y sech21;(v)l’2(x-ct)l ,

and excellent agreement was found, well within the numerical accuracy. This accuracy is limited by the size of the space grid, which cannot be taken too small in order to cut off unstable short wavelengths. Finally, the measured velocity is related to the ampli- tude as it should according to (4).

Comparison with the work of Kaup [2] on the forced Toda lattice is not so straightforward due to the different boundary conditions imposed. In any case it is also true there that solitons are generated via boundary conditions. Also, the numerical aver- aging used in eq. (3) for points near the boundary is reminiscent of the averaging used in Kaup’s analytical work where the difficulty is also to determine the motion of the point right next to the boundary.

This work was supported by the Departamento de Investigation y Bibliotecas, Universidad de Chile, and Fondo National de Ciencia y Tecnologia, Chile.

References

[l] M.J. Ablowitz and H. Segur, Solitons and the inverse scattering transform (SIAM, Philadelphia, 1981) and references therein.

[2] D.J. Kaup, J. Math. Phys. 25 (1984) 277, 282. [3] M. Boiti, J.P. Leon and F. Pempinelli, Lecce preprint

UL-DF-26-83/84 (June, 1984). [4] C.K. Chu, L.W. Xiang and Y. Baransky, Comm. Pure

Appt. Math. 36 (1983) 495. [5] F. Lund, Pure AppL Geophys. 121 (1983) 17.