Camassa–Holm Cuspons, Solitons and Their Interactions via the Dressing Method

Abstract

A dressing method is applied to a matrix Lax pair for the Camassa-Holm equation, thereby allowing for the construction of several global solutions of the system. In particular solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa-Holm equation ar re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.

Integrable negative flows of the Heisenberg ferromagnet equation hierarchy

“…The projector P again is given in terms of the components of F as in (34). The eigenfunction Ψ is defined as in (40), the expression (42) is the same, in addition…”

“…A review on the subject could be found in [32,34]. The peakon, soliton and cuspon solutions are derived for example in [2,3,35,43,44,45,12,40]. Analytic and numerical aspects are studied e.g.…”

“…It has been also realised that both equation can be considered as "negative" flows of an AKNS or ZS hierarchy [1,52,53], e.g. [13,14,40]. The Fokas-Lenells equation [24,42] is another example of a non-evolutionary equation from the DNLS-hierarchy (the hierarchy of the Derivative Nonlinear Schrödinger Equation).…”

A New 2 + 1-Dimensional Integrable Variable Coefficient Toda Equation

“…Recently, the dressing method is extended to a matrix Lax pair for Camassa-Holm equation in ref. [13], in which interactions between soliton and cuspon solutions of the system are studied. The dressing method as nonlinear superposition in Sigma models has been researched by Dimitrios Katsinis et al in ref.…”