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arXiv 2101.02489math.APnlin.SI
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修正的Camassa-Holm方程在不同时空孤子区域的渐近逼近

On asymptotic approximation of the modified Camassa-Holm equation in different space-time solitonic regions

Yiling Yang, Engui Fan

备注：
90 pages. arXiv admin note: substantial text overlap with arXiv:2012.15496
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摘要：

本文研究了修正的Camassa-Holm（mCH）方程初值问题在孤子区域\begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+κu_{x}=0, \quad m=u-u_{x x}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align}中的长时间渐近性态，其中$κ$为正常数.基于对mCH方程和散射矩阵的Lax对的谱分析，利用Riemann-Hilbert（RH）问题的解刻画了Cauchy问题的解.进一步利用Deift-Zhou最速下降法的$\overline\partial$推广，得到了解$u(x,t)$在$x/t$不同时空孤子区域的不同长时间渐近展开式.这些渐近近似可以用一个$N(Λ)$ -孤子来表征，当一个人从$\overline\partial$方程通过具有不同残差阶的区域时，该$N(Λ)$ -孤子的参数被局域孤子-孤子相互作用的总和所调制：对于$ξ=\frac{y}{t}\in(-\infty,-0.25)\cup(2,+\infty)$是$\mathcal{O}(|t|^{-1+2ρ})$，对于$ξ=\frac{y}{t}\in(-0.25,2)$是$\mathcal{O}(|t|^{-3/4})$。我们的结果也证实了mCH方程的孤子分辨猜想和N-孤子解的渐近稳定性。

英文摘要：

In this paper, we study the long time asymptotic behavior for the initial value problem of the modified Camassa-Holm (mCH) equation in the solitonic region \begin{align} &m_{t}+\left(m\left(u^{2}-u_{x}^{2}\right)\right)_{x}+κu_{x}=0, \quad m=u-u_{x x}, \nonumber &u(x, 0)=u_{0}(x),\nonumber \end{align} where $κ$ is a positive constant. Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a Riemann-Hilbert (RH) problem. Further using the $\overline\partial$ generalization of Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution $u(x,t)$ in different space-time solitonic region of $x/t$. These asymptotic approximations can be characterized with an $N(Λ)$-soliton whose parameters are modulated by a sum of localized soliton-soliton interactions as one moves through the region with diverse residual error order from $\overline\partial$ equation: $\mathcal{O}(|t|^{-1+2ρ})$ for $ξ=\frac{y}{t}\in(-\infty,-0.25)\cup(2,+\infty)$ and $\mathcal{O}(|t|^{-3/4})$ for $ξ=\frac{y}{t}\in(-0.25,2)$. Our results also confirm the soliton resolution conjecture and asymptotically stability of N-soliton solutions for the mCH equation.

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