The two-component $ \mu $-Camassa–Holm system with peaked solutions

Yingying Li; Ying Fu; Changzheng Qu

doi:10.3934/dcds.2020253

Article Contents

2020, Volume 40, Issue 10: 5929-5954. Doi: 10.3934/dcds.2020253

This issue Previous Article A degenerate KAM theorem for partial differential equations with periodic boundary conditions Next Article Gradient regularity for a singular parabolic equation in non-divergence form

The two-component $ \mu $-Camassa–Holm system with peaked solutions

1.
School of Mathematics and Statistics and Center for Nonlinear Studies, Ningbo University, Ningbo 315211, China
2.
School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

Received: November 30, 2019

Revised: April 30, 2020

Published: June 2020

The work of Li is partially supported by the NSF-China grant-11971251. The work of Fu is partially supported by the NSF-China grant-11631007 and grant-11471259, and the National Science Basic Research Program of Shaanxi (Program No. 2019JM-007 and 2020JC-37). The work of Qu is partially supported by the NSF-China grant-11631007 and grant-11971251

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Abstract

This paper is mainly concerned with the classification of the general two-component $ \mu $-Camassa-Holm systems with quadratic nonlinearities. As a conclusion of such classification, a two-component $ \mu $-Camassa-Holm system admitting multi-peaked solutions and $ H^1 $-norm conservation law is found, which is a $ \mu $-version of the two-component modified Camassa-Holm system and can be derived from the semidirect-product Euler-Poincaré equations corresponding to a Lagrangian. The local well-posedness for solutions to the initial value problem associated with the two-component $ \mu $-Camassa-Holm system is established. And the precise blow-up scenario, wave breaking phenomena and blow-up rate for solutions of this problem are also investigated.

Keywords:

The Camassa-Holm equation,
the $ \mu $-Camassa-Holm equation,
two-component $ \mu $-Camassa-Holm system,
peaked solution,
conservation law,
blow-up.

Mathematics Subject Classification: 35Q51, 37K06.

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References

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