English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider multipeakon solutions, and to some extent also multishockpeakonsolutions, of a coupled two-component integrable PDE found by Geng and Xue as ageneralization of Novikov's cubically nonlinear Camassa-Holm type equation. Inorder to make sense of such solutions, we find it necessary to assume thatthere are no overlaps, meaning that a peakon or shockpeakon in one component isnot allowed to occupy the same position as a peakon or shockpeakon in the othercomponent. Therefore one can distinguish many inequivalent configurations,depending on the order in which the peakons or shockpeakons in the twocomponents appear relative to each other. Here we are in particular interestedin the case of interlacing peakon solutions, where the peakons alternatinglyoccur in one component and in the other. Based on explicit expressions forthese solutions in terms of elementary functions, we describe the generalfeatures of the dynamics, and in particular the asymptotic large-timebehaviour. As far as the positions are concerned, interlacing Geng-Xue peakonsdisplay the usual scattering phenomenon where the peakons asymptotically travelwith constant velocities, which are all distinct, except that the two fastestpeakons will have the same velocity. However, in contrast to many other peakonequations, the amplitudes of the peakons will not in general tend to constantvalues; instead they grow or decay exponentially. Thus the logarithms of theamplitudes (as functions of time) will asymptotically behave like straightlines, and comparing these lines for large positive and negative times, oneobserves phase shifts similar to those seen for the positions of the peakons.In addition to these K+K interlacing pure peakon solutions, we also investigate1+1 shockpeakon solutions, and collisions leading to shock formation in a 2+2peakon-antipeakon solution.

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