Non-interlacing peakon solutions of the Geng–Xue equation

The aim of the present article is to derive explicit formulas for arbitrary non-overlapping pure peakon solutions of the Geng–Xue (GX) equation, a two-component generalization of Novikov’s cubically non-linear Camassa–Holm type equation. By performing limiting procedures on the previously known formulas for so-called interlacing peakon solutions, where the peakons in the two component occur alternatingly, we turn some of the peakons into zero-amplitude ‘ghostpeakons’, in such a way that the remaining ordinary peakons occur in any desired configuration. A novel feature compared to the interlacing case is that the Lax pairs for the GX equation do not provide all the constants of motion necessary for the integration of the system. We also study the large-time asymptotics of the non-interlacing solutions. As in the interlacing case, the peakon amplitudes grow or decay exponentially, and their logarithms display phase shifts similar to those for the positions. Moreover, within a group of adjacent peakons in one component, all peakons but one have the same asymptotic velocity. A curious phenomenon occurs when the number of such peakon groups is odd, namely that the sets of incoming and outgoing velocities are unequal.