Scattering of regularized-long-wave solitary waves. (English) Zbl 0599.76028

The Lagrangian density for the regularized-long-wave equation (also known as the BBM equation) is presented. Using the trial function technique, ordinary differential equations that describe the time dependence of the position of the peaks, amplitudes, and widths for the collision of two solitary waves are obtained. These equations are analyzed in the Born and equal-width approximations and compared with numerical results obtained by direct integration utilizing the split-step fast Fourier-transform method. The computations show that collisions are inelastic and that production of solitary waves may occur.


76B25 Solitary waves for incompressible inviscid fluids
35Q99 Partial differential equations of mathematical physics and other areas of application
76M99 Basic methods in fluid mechanics
70Sxx Classical field theories
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[1] Peregrine, D. H., J. Fluid Mech., 25, 321 (1966)
[2] Meiss, J. D.; Horton, W., Phys. Fluids, 25, 1838 (1982) · Zbl 0497.76114
[3] Benjamin, T.; Bona, J.; Mahoney, J., Phil. Trans. R. Soc. London, A272, 47 (1972) · Zbl 0229.35013
[4] Kruskal, M. D., (Moser, J., Dynamical Systems, Theory and Applications (1976), Springer: Springer Berlin)
[5] Bona, J. L.; Pritchard, W. G.; Scott, L. R., Phys. Fluids, 23, 438 (1980) · Zbl 0425.76019
[6] Olver, P. J., Math. Proc. Camb. Phil. Soc., 85, 143 (1979) · Zbl 0387.35050
[7] Tsujishia, T., Conservative Laws of the BBM Equation (1979), preprint
[8] Hill, E. L., Rev. Mod. Phys., 23, 253 (1951) · Zbl 0044.38509
[9] Courant, R.; Hilbert, D., (Methods of Mathematical Physics, vol. I (1953), Wiley: Wiley New York), chap. IV · Zbl 0729.00007
[10] Tappert, F., Lect. Appl. Math., 15, 215 (1974)
[11] Makino, M.; Kamimura, T.; Taniuti, T., J. Phys. Soc. Jap., 50, 980 (1981), Dr. T. Kamimura has generously provided us with his program. This algorithm was also used in the 2-D case by
[12] Eilbeck, J. C.; McGuire, G. R., J. Comp. Phys., 23, 63 (1977) · Zbl 0361.65100
[13] Santarelli, A. R., Nuovo Cimento, 46, 179 (1978)
[14] Courtenay Lewis, J.; Tjon, J. A., Phys. Lett., 73A, 275 (1979)
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