Soliton and antisoliton resonant interactions

Using the Hirota formalism, Gibbon et al (1976), have shown that the evolution equation u_t+u_x-u_xxt+(4u²+2y_xz_t)_x=0 with u=y_t=z_x, has the same solitary wave as the regularised long wave (RLW) equation u_t+u_x-u_xxt+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 u_t+u_x-u_xxt+3(u²)_x+6u_tz_x=0 with u=z_t, 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.