# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020245

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

 School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China

* Corresponding author

Received  June 2020 Revised  July 2020 Published  September 2020

Fund Project: This work is supported by the NSF-China grant-11631007 and grant-11971251

In this paper, we provide a classification to the general two-component Novikov-type systems with cubic nonlinearities which admit multi-peaked solutions and $H^1$-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 $H^1$-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 $H^s({\mathbb R})$ with $s>3/2$, 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.

Citation: Min Zhao, Changzheng Qu. The two-component Novikov-type systems with peaked solutions and $H^1$-conservation law. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020245
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