Abstract

We construct and study Darboux transformations for the \((2{+}1)\)-dimensional Camassa–Holm system. We apply a reciprocal transformation that relates the \((2{+}1)\)-dimensional Camassa–Holm system and the linear system associated with the modified Kadomtsev–Petviashvili hierarchy. Using three Darboux transformation operators, we obtain three types of solutions for the \((2{+}1)\)-dimensional Camassa–Holm system, of which one is a multisoliton solution. In addition, we briefly discuss rational solutions.

中文翻译：

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使用Darboux变换获得（2 + 1）维Camassa-Holm系统的多孤子解

摘要

我们构造和研究\（（2 {+} 1）\）维Camassa–Holm系统的Darboux变换。我们应用了倒数变换，该变换将\（（2 {+} 1）\）-维Camassa–Holm系统和与修改后的Kadomtsev–Petviashvili层次结构关联的线性系统相关。使用三个Darboux变换算子，我们获得了\（（2 {+} 1）\）维Camassa-Holm系统的三种类型的解决方案，其中一种是多孤子解决方案。此外，我们简要讨论了合理的解决方案。