On the well-posedness of the Cauchy problem for the two-component peakon system in $$C^k\cap W^{k,1}$$

Abstract

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class \((m,n)\in C^{k}(\mathbb {R}) \cap W^{k,1}(\mathbb {R})\) with \(k\in \mathbb {N}\cup \{0\}\). This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao:

$$\begin{aligned} \partial _t m(t,x)= \partial _x[m(t,x)(u(t,x)-\partial _xu(t,x)) (u(-t,-x)+\partial _x(u(-t,-x)))], \end{aligned}$$

where \(m(t,x)=\left( 1-\partial _{x}^2\right) u(t,x)\). Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class \(C^k\cap W^{k,1}\). Moreover, we derive criteria for blow-up of the local solution in this class.