The direct method for multisolitons and two-hump solitons in the Hirota-Satsuma system

The coupled Korteweg-de Vries (cKdV) system, originally proposed by Hirota and Satsuma [1], modeling the interaction of two long waves evolving with different dispersion relations, is considered. Multisoliton solutions are explicitly found in the Hirota-Satsuma system by means of an extension of the direct method due to Hirota [2]. Such multisolitons are found to be a complete family of interacting classical solitons consisting of the trivial ones corresponding to the decoupled N-soliton solution of the KdV equation and the ones coupled to the M-cKdV solitons, here simply referred to as the classical (N, M)-soliton solutions which are N+M interacting solitons indeed. Explicit analytical solutions, expressed in terms of hyperbolic functions, are found for the (0, 1), (1, 1) and (0, 2) cases while the process to generate the general (N, M)-soliton solutions is systematically described in detail. In the particular scenario of the (1, 1)-soliton, a novel two-hump one-soliton solution is obtained when a negative prestressing is prescribed for one of the waves in the merging of the 1-KdV and 1-cKdV solitons. Special attention is emphasized to the phase shift for the (1, 1) and (0, 2) interaction cases to asymptotically get a particular remarkable long range interaction phenomenon. Finally, some numerical evolutions are also considered to illustrate the interaction dynamics of the multisolitons and the new two-hump soliton in the different situations considered.