An isospectral problem for global conservative multi-peakon solutions of the Camassa-Holm equation (Q2249770)

The paper presents a proof of true integrability of the initial-value problem for the Camassa-Holm equation, \[ u_t - u_{xxt} = 2u_xu_{xx} - 3uu_x +uu_{xxx}, \] whose integrability in the general case is formal, as the equation admits collapsing solutions. In particular, this happens with multi-peakon solutions, which may be constructed in the form of \[ u(x,t)=\sum_{n=1}^Np_n(t)\exp(-|x - q_n(t)|). \] These solutions develop a singularity in the case when amplitudes \(p_n\) have different signs. Then some peakons collide, which gives rise to the blowup. The present work produces a proof of the true integrability of class of a \textit{conservative solutions} to the Camassa-Holm equations, i.e., those conserving the dynamical invariant \[ \int_{-\infty}^{+\infty}[u^2(x) + (u_x)^2]dx. \] The proof is based on the Lax pair for the equation, which may be applied in a consistent form to the conservative solutions. In the appendix to the paper, an explicit exact conservative two-peakon solution is constructed.