Dynamics of interlacing peakons (and shockpeakons) in the Geng-Xue equation

TL;DR

This paper analyzes interlacing peakon and shockpeakon solutions of the Geng-Xue equation, revealing their explicit dynamics, asymptotic behavior, and phase shifts, with a focus on their scattering and exponential amplitude growth or decay.

Contribution

It provides explicit formulas and detailed analysis of interlacing peakon and shockpeakon solutions, including their asymptotic and collision behaviors, for the Geng-Xue equation.

Findings

Peakons exhibit scattering with constant velocities, except the two fastest sharing the same velocity.

Amplitudes grow or decay exponentially, with logarithms behaving linearly over time.

Phase shifts occur in amplitudes similar to position phase shifts.

Abstract

We consider multipeakon solutions, and to some extent also multishockpeakon solutions, of a coupled two-component integrable PDE found by Geng and Xue as a generalization of Novikov's cubically nonlinear Camassa-Holm type equation. In order to make sense of such solutions, we find it necessary to assume that there are no overlaps, meaning that a peakon or shockpeakon in one component is not allowed to occupy the same position as a peakon or shockpeakon in the other component. Therefore one can distinguish many inequivalent configurations, depending on the order in which the peakons or shockpeakons in the two components appear relative to each other. Here we are in particular interested in the case of interlacing peakon solutions, where the peakons alternatingly occur in one component and in the other. Based on explicit expressions for these solutions in terms of elementary functions, we describe the general features of the dynamics, and in particular the asymptotic large-time behaviour. As far as the positions are concerned, interlacing Geng-Xue peakons display the usual scattering phenomenon where the peakons asymptotically travel with constant velocities, which are all distinct, except that the two fastest peakons will have the same velocity. However, in contrast to many other peakon equations, the amplitudes of the peakons will not in general tend to constant values; instead they grow or decay exponentially. Thus the logarithms of the amplitudes (as functions of time) will asymptotically behave like straight lines, and comparing these lines for large positive and negative times, one observes phase shifts similar to those seen for the positions of the peakons. In addition to these K+K interlacing pure peakon solutions, we also investigate 1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2 peakon-antipeakon solution.