A new generalised two-component Camassa–Holm type system with waltzing peakons and wave breaking

Abstract

This paper mainly investigate the Cauchy problem for the generalised two-component Camassa–Holm type system, which includes the celebrated Camassa–Holm equation, Degasperis equation, Novikov equation, and the two-component cross-coupled Camassa–Holm system, Novikov system as special cases. Firstly, the local well-posedness of the system in nonhomogeneous Besov spaces \(B^{s}_{l,r}(\mathbb {R})\times B^{s}_{l,r}(\mathbb {R})\) with \(l,r\in [1,\infty ]\), \(s>\max \{2+1/l,5/2\}\) is established by using the Littlewood–Paley theory and transport equations theory. Moreover, we verify the blow-up occurs for this system only in the form of breaking waves. Finally, the waltzing peakons for the system and some numerical experiments to illustrate our results are performed.