The two-component Novikov-type systems with peaked solutions and $ H^1 $-conservation law

Abstract

<p style="text-indent:20px;">In this paper, we provide a classification to the general two-component Novikov-type systems with cubic nonlinearities which admit multi-peaked solutions and <inline-formula><tex-math id="M2">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-conservation law. Local well-posedness and wave breaking of solutions to the Cauchy problem of a resulting system from the classification are studied. First, we carry out the classification of the general two-component Novikov-type system based on the existence of two peaked solutions and <inline-formula><tex-math id="M3">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-conservation law. The resulting systems contain the two-component integrable Novikov-type systems. Next, we discuss the local well-posedness of Cauchy problem to the resulting systems in Sobolev spaces <inline-formula><tex-math id="M4">\begin{document}$ H^s({\mathbb R}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ s>3/2 $\end{document}</tex-math></inline-formula>, the approach is based on the new invariant properties, certain estimates for transport equations of the system. In addition, blow up and wave-breaking to the Cauchy problem of a system are studied.</p>