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arXiv · 2507.02100

Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions

I. Karpenko·D. Shepelsky·G. Teschl
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Abstract

We consider the Cauchy problem for the modified Camassa-Holm equation \[ u_t+\left((u^2-u_x^2)m\right)_x=0,\quad m\coloneqq u-u_{xx}, \quad t>0,\ \ -\infty<x<+\infty \] subject to the step-like initial data: $u(x,0)\to A_1$ as $x\to-\infty$ and $u(x,0)\to A_2$ as $x\to+\infty$, where $0<A_1<A_2$. The goal is to establish the global existence of the solution of this problem.

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I. Karpenko, D. Shepelsky, G. Teschl. 2025-07-02. Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions. https://arxiv.org/abs/2507.02100

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