Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity
Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity
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DOI：
10.1063/1.4807417
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发表时间：
2013-02
影响因子：
1.3
通讯作者：
Y. Matsuno
中科院分区：
文献类型：
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作者：
Y. Matsuno
给出了具有立方非线性的修正的Camassa-Holm(MCH)方程的光滑亮多孤子解的紧致参数表示。我们首先通过倒数变换将MCH方程变换为相伴的MCH方程，然后在相伴的MCH方程的解与Ablowitz等人提出的浅水波模型方程之间找到了一种新的Backlund变换。我们将这一结果与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、双孤子解以及一般多孤子解的性质。证明了只有当孤子的振幅参数满足一定条件时，才能保证解的光滑性。我们还发现，在一个参数的临界值，如果超过这个临界值，孤子解表现出不同的奇性。
We present a compact parametric representation of the smooth bright multisoliton 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 Backlund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz et 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 differe...