Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation

Aiyong Chen; Xinhui Lu

doi:10.3934/dcds.2020090

Article Contents

2020, Volume 40, Issue 3: 1703-1735. Doi: 10.3934/dcds.2020090

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Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation

Aiyong Chen^1,2, and
Xinhui Lu^2,

1.
Department of Mathematics, Hunan First Normal University, Changsha, Hunan 410205, China
2.
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Received: April 30, 2019

Revised: August 31, 2019

Published: December 2019

This work are supported by the National Natural Science Foundation of China (No. 11671107 and No. 11971163), Guangxi Natural Science Foundation of China (No. 2015GXNSFGA139004) and Program for Innovation Project of GUET Graduate Education (No. 2019YCXS080)

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Abstract

The orbital stability of peakons and hyperbolic periodic peakons for the Camassa-Holm equation has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Comm. Pure. Appl. Math. 53 (2000) 603-610] and Lenells in [J. Lenells, Int. Math. Res. Not. 10 (2004) 485-499], respectively. In this paper, we prove the orbital stability of the elliptic periodic peakons for the modified Camassa-Holm equation. By using the invariants of the equation and controlling the extrema of the solution, it is demonstrated that the shapes of these elliptic periodic peakons are stable under small perturbations in the energy space. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation.

Keywords:

Camassa-Holm equation,
peakon,
periodic peakon,
orbital stability,
elliptic functions and elliptic integrals.

Mathematics Subject Classification: Primary: 37K05, 37K45; Secondary: 70H14, 70H33.

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Figure 1. (a) Phase portrait of the system (2.5) (b) The algebraic curve defined by H(φ, y) = 0

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Figure 2. The profile of elliptic periodic peakon for c = 1

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