On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries
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2025
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| author | Weng, Weifang Yan, Zhenya |
| author_facet | Weng, Weifang Yan, Zhenya |
| contents | We explore the $N_{\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+κu_x=0,\quad m=u-u_{xx}$ with $\lim_{x\rightarrow \pm \infty }u(x,t)=0$. We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with $\ell=n=1$, the corresponding $N_{\infty}$-soliton is the one-soliton solution which the discrete spectral point is the center of the region; ii) when the region is a quadrature domain with $\ell=n$, the corresponding $N_{\infty}$-soliton is an $n$-soliton solution; iii) when the discrete spectra lie in the line region, we provide its corresponding Riemann-Hilbert problem,; and iv) when the discrete spectra lie in an elliptic region, it is equivalent to the case of the line region. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_03485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries Weng, Weifang Yan, Zhenya Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs We explore the $N_{\infty}$-soliton asymptotics for the modified Camassa-Holm (mCH) equation with linear dispersion and boundaries vanishing at infinity: $m_t+(m(u^2-u_x^2)^2)_x+κu_x=0,\quad m=u-u_{xx}$ with $\lim_{x\rightarrow \pm \infty }u(x,t)=0$. We mainly analyze the aggregation state of $N$-soliton solutions of the mCH equation expressed by the solution of the modified Riemann-Hilbert problem in the new $(y,t)$-space when the discrete spectra are located in different regions. Starting from the modified RH problem, we find that i) when the region is a quadrature domain with $\ell=n=1$, the corresponding $N_{\infty}$-soliton is the one-soliton solution which the discrete spectral point is the center of the region; ii) when the region is a quadrature domain with $\ell=n$, the corresponding $N_{\infty}$-soliton is an $n$-soliton solution; iii) when the discrete spectra lie in the line region, we provide its corresponding Riemann-Hilbert problem,; and iv) when the discrete spectra lie in an elliptic region, it is equivalent to the case of the line region. |
| title | On the $N_\infty$-soliton asymptotics for the modified Camassa-Holm equation with linear dispersion and vanishing boundaries |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2501.03485 |