具有立方非线性项的修正Camassa-Holm方程的贝克隆变换和光滑多孤子解
Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity
摘要：
本文给出了一类具有立方非线性项的修正Camassa-Holm（mCH）方程的光滑亮多解的一个参数表示.首先通过一个互逆变换将mCH方程变换为一个相应的mCH方程，然后在相应的mCH方程的解和由Ablowitz { \it at al}引入的浅水波模型方程之间找到一个新的贝克隆变换.我们将这个结果与SWW方程和修正的Korteweg-de Vries方程的多孤子解的表达式相结合，得到了mCH方程的多孤子解。随后，我们研究了单孤子解和双孤子解以及一般多孤子解的性质。我们证明了只有当孤子的振幅参数满足一定的条件时，解的光滑性才能得到保证。我们还发现，在一个临界值的参数，超过该值的解决方案变得奇异，孤子解表现出不同的功能，从CH方程的峰子解决方案。然后，通过大时间的渐近分析，得到了相移的计算公式，证实了多孤子解的孤子性质。最后，我们利用贝克隆变换得到了mCH方程的无穷多个守恒律。
英文摘要：
We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel Bäcklund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz {\it at al}. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Korteweg-de Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one- and two-soliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Last, we use the Bäcklund transformation to derive an infinite number of conservation laws of the mCH equation.
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