On the Cauchy problem and peakons of a two-component Novikov system. (English) Zbl 1462.35134
Summary: We study a two-component Novikov system, which is integrable and can be viewed as a two-component generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the Cauchy problem in Besov spaces \(B_{p,r}^s\) with \(1\leq p,r\leq+\infty, s > \max\{1+1/p,3/2\}\) and Sobolev spaces \(H^s(\mathbb{R})\) with \(s > 3/2\), and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.
MSC:
| 35C08 | Soliton solutions |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35G55 | Initial value problems for systems of nonlinear higher-order PDEs |
Keywords:
two-component Novikov system; Hamiltonian structure; Camassa-Holm type equation; well-posedness; peaked solitonReferences:
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