非线性系统的非局域对称及其应用

非局域对称作为对称理论重要组成部分, 近年来逐渐引起人们关注. 本文以势Korteweg-de Vries (KdV)方程、修正Korteweg-de Vries (mKdV)方程和Kadomtsev-Petviashvili (KP)方程为例, 分别介绍了对应非线性系统与B?cklund变换相关的非局域对称、非局域留数对称与Darboux变换相关的非局域对称. 通过引入3个辅助变量, 将KP方程与Darboux变换相关的非局域对称局域化为Lie点对称. 运用对称约化方法简单概述了KP方程的相似约化解, 其中包括孤立子和Boussinesq波相互作用解、孤立子和KdV型波相互作用解以及非均匀背景下的单孤立波解.

Nonlocal symmetries of nonlinear system and corresponding applications

The nonlocal symmetry, as an important part of symmetry theory, has attracted more and more attention in recent years. Taking the potential KdV equation, the mKdV equation and the KP equation as examples, we introduce the B?cklund transformation related nonlocal symmetry, the nonlocal residual symmetry and the Darboux transformation related nonlocal symmetry of nonlinear systems. After inducing three auxiliary variables, the Darboux transformation related nonlocal symmetry of KP equation is localized into the Lie point symmetry of an extended nonlinear system. Based on this system, we briefly summarize the similarity reduction solutions of the KP equation by applying the symmetry reduction method, which includes the interaction solution between soliton and Boussinesq wave, the interaction solution between soliton and KdV-type wave, and the single solitary wave with a nonhomogeneous background.