Two-peakon dynamics in the Clifford algebra generalization of the Camassa-Holm equation

Abstract

We study the dynamics of a two-component perturbation of the Camassa--Holm equation arising from a reformulation of the Euler--Bernoulli beam problem, recently extended to a general Clifford algebra setting. We focus on the original case associated with a Clifford algebra with two generators and Minkowski signature, for which the resulting equation admits nonsmooth soliton solutions (peakons) carrying internal degrees of freedom. We investigate analytically and numerically the dynamics of a two-peakon solution and establish the existence of a synchronized exchange of energy between spatially separated peaks, a phenomenon previously observed only numerically. We obtain a complete description of the long-time dynamics: the amplitudes approach a periodic orbit determined by the spectral invariants, and the resulting hidden periodicity governs the persistent exchange between the two peakons. The limiting orbit and its averaged dynamics are described explicitly in terms of elliptic functions and complete elliptic integrals. We also derive an exact identity relating the peak separation to the accumulated imbalance of the two amplitudes. This identity, combined with a frozen-parameter comparison argument, yields exponential decay of the interaction between the peakons and exponential convergence of the amplitude variables to the limiting periodic orbit. Moreover, once the asymptotic regime is reached, the separation increases from one period of the internal oscillation to the next, even though its instantaneous rate may continue to change sign. These results reveal a dynamical feature absent from the scalar Camassa--Holm equation: a persistent oscillatory transfer of energy between increasingly separated peakons, coupled with quantitatively controlled asymptotic decoupling.