Construction of 2-Peakon Solutions and Nonuniqueness for a Generalized mCH Equation

Hao Yu,Kelei Zhang,Aiyong Chen,Manuel De Le N

doi:10.1155/2021/9363673

Abstract

For the generalized mCH equation, we construct a 2-peakon solution on both the line and the circle, and we can control the size of the initial data. The two peaks at different speeds move in the same direction and eventually collide. This phenomenon is that the solution at the collision time is consistent with another solitary peakon solution. By reversing the time, we get two new solutions with the same initial value and different values at the rest of the time, which means the nonuniqueness for the equation in Sobolev spaces H s is proved for s < 3 / 2 .

Highlights

The Camassa-Holm (CH) equation [1,2,3] is an integrable system with a bi-Hamiltonian structure, which is derived by Camassa and Holm using the asymptotic expansion in the Hamiltonian for Euler’s equation
A special kind of weak solution for this equation describes the solitary wave at the peak, called peakons [4, 5], whose wave slope is discontinuous at the peak
The interactions between any number of peakons were described by the multipeakon solutions [6, 7], in the form of a linear superposition of peakons whose amplitude and velocity change with time

Summary

Introduction

The Camassa-Holm (CH) equation [1,2,3] is an integrable system with a bi-Hamiltonian structure, which is derived by Camassa and Holm using the asymptotic expansion in the Hamiltonian for Euler’s equation. The nonuniqueness results of Himonas and Holliman [33] show that solutions to the Cauchy problem for the FORQ equation are not unique in Hs when s < 3/2. We consider the Cauchy problem for a generalized mCH (gm- CH) equation which has the following form mt This equation is obtained by Anco and Recio [37], by extending a Hamiltonian structure of the CH equation. The results of Anco and Recio [37] show that the gmCH equation admits peakon traveling wave solutions and multipeakon solutions. Based on the conservation laws in [38], the Cauchy problem and nonuniqueness of the peakon solutions in this paper are studied Under this premise, we obtain our main result, and its proof is closely related to the conservation of norms.

Nonuniqueness on the Line

Nonuniqueness on the Circle