A Two-component Generalization of the Camassa-Holm Equation and its Solutions

Abstract

An explicit reciprocal transformation between a 2-component generalization of the CamassaHolm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain solutions of the 2-CH system from those of the first negative flow of the AKNS hierarchy. Interesting examples of peakon and multi-kink solutions of the 2-CH system are presented. Mathematics Subject Classifications(2000). 35Q53, 37K35

Two-component generalizations of the Camassa–Holm equation

“…by Chen, Liu and Zhang [5], and related to an alternative system of the form (3) presented by Falqui [9], namely ( ) ( )…”

“…After taking c 1 = 0, c 3 = 0 and rescaling suitably, the system (48) becomes the known two-component CH equation (5) from [5]. With a change of notation, the transformation (7) presented in the introduction is…”

“…As explained in [5], this change of independent variables transforms (5) to the first negative flow of the AKNS hierarchy (which, at the level of the Lax pair, is equivalent to the classical Boussinesq hierarchy, up to a gauge transformation). Under the reciprocal transformation, we have the following system of four equations relating u, v, m, n:…”

A Continuous Interpolation Between Conservative and Dissipative Solutions for the Two-Component Camassa–holm System

“…The 2CH system was introduced by Olver and Rosenau [35, Equation (43)] (see also [2,8,32]), and derived in the context of water waves by Constantin and Ivanov [11]. In this paper, the question of wave breaking is also analyzed.…”

“…The scalar CH equation, which corresponds to the case where ρ(t, x) = ρ 0 (x) = 0, was introduced by Camassa and Holm in the fundamental paper [7], and its analysis has been pervasive. Other generalizations of the Camassa-Holm equation exist; see, for example, [8,9,13,21,33]. The 2CH system experiences wave breaking in the sense that the spatial derivative of u becomes unbounded while keeping its H 1 (R) norm finite.…”