Multi-soliton and multi-cuspon solutions of a Camassa–Holm hierarchy and their interactions

Abstract

In this paper, we study an integrable Camassa–Holm hierarchy whose high-frequency limit is the Camassa–Holm equation. By a method associated with the Darboux transform, we construct the explicit multi-soliton and multi-cuspon solutions. Then, we study in detail the interactions of soliton–cuspon and soliton–soliton. Further, an interesting phenomenon is found: the soliton with smaller amplitude can travel faster than the one with larger amplitude when they interact. At last, we investigate in detail the head-on collision of one soliton and one cuspon.

Bäcklund Transformations for the Camassa–Holm Equation

“…Similar results can be obtained for HS and the full family of equations (19). The existence of an infinite number of conservation laws for CH follows from the bihamiltonian structure for CH discovered in [9].…”

“…This is also the characteristic for a symmetry of the full family of equations (19). The next thing to do is to find a (formal) asymptotic series solution of (11)-(12) for small |α|.…”

“…Equation (19) includes the KdV, CH, and Hunter-Saxton (HS) [29] equations. The KdV equation can be obtained by putting c 1 = c 4 = 0.…”

“…The first investigations of two wave solutions were [60] and [22], both of which required some element of numerical computation. However, since then, a substantial literature [31,39,49,18,38,43,53,44,52,19,63] has developed on multisoliton, multicuspon and soliton-cuspon solutions. The known methods for analytic construction of solutions include a determinantal formula based on the inverse scattering approach, a Hirota bilinear form for CH and a reciprocal transformation relating the CH hierarchy to the KdV hierarchy.…”

“…The integrability of the CH equation was already firmly established in [9], where a Lax pair and a bihamiltonian structure were given, and much further evidence for this has accumulated since then. There is an inverse scattering formalism [13,14], explicit formulas can be found for multipeakon,multisoliton, multicuspon and soliton-cuspon solutions [60,4,5,6,22,31,39,49,18,38,43,53,44,51,52,19,63] there are an infinite number of local conservation laws [23,56,57,28,25,27,35,11,30,24], and there is a rich algebra of symmetries [56,57,28,25,27,24]. Other significant works on CH include studies of the stability of peakon and other exact solutions [15,16,17,36] and interesting numerical studies [32,45,21,12].…”