Smooth multisoliton solutions of the Geng-Xue equation

Abstract

We present a reciprocal transformation which links the Geng-Xue equation to a particular reduction of the first negative flow of the Boussinesq hierarchy. We discuss two reductions of the reciprocal transformation for the Degasperis-Procesi and Novikov equations, respectively. With the aid of the Darboux transformation and the reciprocal transformation, we obtain a compact parametric representation for the smooth soliton solutions such as multi-kink solutions of the Geng-Xue equation.

A view of the peakon world through the lens of approximation theory

“…Further research is needed to clarify this, but clearly this requirement of peakons being non-overlapping is incompatible with letting 𝑣 = 𝑢, so at present it is not clear to us whether it is justifiable to use Novikov peakons as a source of counterexamples for the GX equation, as has been done in the literature [151]. Speaking of literature, we are aware of a few analytic studies [241,288,151,10,56,298] as well as some papers about integrability aspects [210,212,204,211]. There is a also a bewildering array of other multi-component peakon equations generalizing the Novikov and/or GX equations, which we will not attempt to survey here, although we may mention the work by Zhao and Qu [325] who have classified all two-component Novikov-type cubic equations which admit peakons in the standard weak sense and in addition conserve the integral ∫ (𝑢 2 + 𝑢 2 𝑥 + 𝑣 2 + 𝑣 2 𝑥 ) 𝑑𝑥 (the GX equation is not one of them).…”