Cauchy problem for a generalized cross-coupled Camassa-Holm system with waltzing peakons and higher-order nonlinearities

Abstract

In this paper, we study the Cauchy problem for a generalized cross-coupled Camassa-Holm system with peakons and higher-order nonlinearities. By the transport equation theory and the classical Friedrichs regularization method, we obtain the local well-posedness of solutions for the system in nonhomogeneous Besov spaces $B_{p,r}^s\times B_{p,r}^s$ with $1\le p,r\le +\infty$ and $s>\max \{2+\frac{1}{p},\frac{5}{2}\}$. Moreover, we construct the local well-posedness in the critical Besov space $B_{2,1}^{5/2}\times B_{2,1}^{5/2}$ and the blow-up criteria. In the paper, we also consider the well-posedness problem in the sense of Hadamard, non-uniform dependence, and Hölder continuity of the data-to-solution map for the system on both the periodic and the non-periodic case. In light of a Galerkin-type approximation scheme, the system is shown well-posed in the Sobolev spaces $H^s\times H^s,s>5/2$ in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous. Furthermore, we prove the Hölder continuity in the $H^r\times H^r$ topology when $0\le r<s$ with Hölder exponent $\alpha$ depending on both $s$ and $r$.

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• On an integrable two-component Camassa-Holm shallow water system
• The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations
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