Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| author | Karpenko, I. Shepelsky, D. Teschl, G. |
| author_facet | Karpenko, I. Shepelsky, D. Teschl, G. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions Karpenko, I. Shepelsky, D. Teschl, G. Analysis of PDEs 35Q53 (Primary) 37K15, 35Q15, 35B40, 35Q51, 37K40 (Secondary) 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. |
| title | Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions |
| topic | Analysis of PDEs 35Q53 (Primary) 37K15, 35Q15, 35B40, 35Q51, 37K40 (Secondary) |
| url | https://arxiv.org/abs/2507.02100 |