非线性演化方程达布变换的构造方法研究

Study on Construction Method of Nonlinear Evolution Equations of Darboux Transformation

非线性演化方程有许多可积性质,如多孤子解、达布变换、贝克隆变换、Lax表示和非线性叠加公式等等,达布变换是求非线性演化方程显式精确解的十分有效的方法之一,从非线性方程的Lax表示出发,可以构造达布变换,从而证明非线性方程的可积性。本文以KdV方程为例,详细介绍了两种情况下达布变换的构造方法,同时也给出了相应的贝克隆变换,并在计算机上利用Maple软件得到了实现。

Nonlinear evolution equation has many integrable properties, such as multi- soliton solutions,Darboux transformation,Baicklund transformation, Lax representation and nonlinear superposition formula and so on, Darboux transformation is one of the very effective methods for solving nonlinear evolution equation which has explicit and exact solutions, from Lax representation of the nonlinear equation, it may construct Darboux transformation, thus proving the integrability of the nonlinear equations. The KdV equation as an example, detailing the construction method of Darboux transformation of the two cases, and gives the corresponding Backlund transformation, and it has been realized on the computer using Maple software.