Soliton and antisoliton resonant interactions
- 21 December 1987
- journal article
- Published by IOP Publishing in Journal of Physics A: General Physics
- Vol. 20 (18), 6223-6235
- https://doi.org/10.1088/0305-4470/20/18/022
Abstract
Using the Hirota formalism, Gibbon et al (1976), have shown that the evolution equation ut+ux-uxxt+(4u2+2yxzt)x=0 with u=yt=zx, has the same solitary wave as the regularised long wave (RLW) equation ut+ux-uxxt+6(u2)x=0 and an exact two-soliton solution describing the elastic collision of two sech2 profile solitary waves. Performing a more detailed analysis the authors show that the two-soliton solution can also represent other processes like the resonant or the singular collision of two RLW-type solitary waves. The interaction type depends on the values of a characteristic parameter of the solution. They also prove that with the bilinear form associated with the evolution equation, a three-soliton solution of the Hirota type cannot exist. They then study the equation ut+ux-uxxt+3(u2)x+6utzx=0 with u=zt, associated with another bilinear form, which has the same solitary wave as the evolution equation. They prove the existence of N-soliton solutions, for arbitrary N, and analyse the behaviour of the solitonic solutions. As in the first case, the two-soliton solution can describe elastic, resonant or singular interaction of two RLW-type solitary waves. A remarkable feature of the resonant triad is that it always involves one positive and two negative waves. This triad corresponds to a fundamental vertex for the analysis of the elastic soliton-antisoliton interaction.Keywords
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