On solutions of the Camassa-Holm equation

Abstract

The techniques that have been developed for the application of the inverse-scattering transform method to the solution of the Camassa-Holm equation, principally by Constantin, are implemented. We use this approach, first, to represent the known solitary-wave solution in a simple parametric form, and then, second, to obtain the general two-and three-soliton solutions. (These latter two solutions require rather extensive use of mathematical packages, Mathematica and Maple, in order to complete the construction of solutions.) A number of examples are presented; the phase shifts, evident after an interaction, are found and the special limit that recovers the peakon solutions is discussed.

Long time asymptotics of the Camassa–Holm equation on the half-line

“…Particularly, being considered as a model of unidirectional wave propagation [13,14,18,20,35], it accounts for both waves existing globally for all t and waves blowing up at a certain finite critical time t = t cr , depending on the form of the initial data. For ω > 0, it possesses analytic solitary waves -solitons [16,36,39]; moreover, any initial data from a large class of decreasing, as |x| → ∞, functions evolves, for large time, into a train of solitons [10]. For ω = 0, the equation supports weak, non-analytic solutions, called "peakons", which are stable patterns interacting like solitons [13,21,1,2].…”

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

“…Recall that the Camassa-Holm equation (1.1) has a similar relation with the first negative flow of the KdV hierarchy, the corresponding reciprocal transformation (also called a hodograph transformation) was found by Fuchssteiner in [15]. Several attempts have been made to obtain solutions of the Camassa-Holm equation (1.1) from that of the first negative flow of the KdV hierarchy by using this reciprocal transformation in, for example, [9,18,20,21,22,28]. However, since the inverse of this reciprocal transformation involves the solving of a nonlinear ODE of second order, only particular solutions like the multi-soliton solutions were obtained in explicit forms by using this approach.…”

“…When κ = 0, the Camassa-Holm equation (1.1) has a peculiar property that its soliton solutions become piecewise smooth and have corners at their crests, such solutions are weak solutions of (1.1) and are called "peakons". Since the works of Camassa and Holm, this equation has become a well known example of integrable systems and has been studied from various point views in, for example, [1,3,9,10,14,15,18,20,26,27,28] and references therein.…”

Symbolic Software for the Painlevé Test of Nonlinear Ordinary and Partial Differential Equations

“…Recently, progress has been made using Mathematica and Maple in applying the ISTmethod to difficult equations, including the Camassa-Holm equation [21]. While there is as yet no systematic way to determine if a differential equation is solvable using the IST-method [27], having the Painlevé property is a strong indicator that it will be.…”