非线性演化方程的非局域对称研究及其相互作用解的构造

With the further research of nonlinear systems, it is found that so much real physical phenomena is too complicated that the pure soliton patterns and the soliton-soliton interaction patterns are not enough to describe them. Thus it is necessary to build more possible excited patterns for nonlinear evolution equations. Based on providing some nonlocal symmetries of nonlinear evolution equations, the main target of this project is to construct ‘soliton + nonlinear wave’ solutions for nonlinear evolution equations and coupled nonlinear evolution equations by using the tools of symmetry reduction theory and CTE/CRE method. It will be realized through the following specific goals: (1) Introduce suitable nonlocal variables to build nonlocal symmetries for more nonlinear evolution equations, by localizing these nonlocal symmetries to reveal a possible link between the finite transforms of nonlocal symmetries. (2) Combining the nonlocal symmetries and usual point Lie symmetries, the interactions between soliton and different nonlinear waves of several important nonlinear systems will be constructed with the help of the Lie point symmetry reduction theory, (3) CTE/CRE method will be applied to more coupled nonlinear evolution equations or nonlinear evolution equations with variable coefficients to directly construct more interaction solutions. Finally, the new research results will be applied to some actual physical problems to provide them reasonable explanations. The results of this project will not only enrich the symmetry theory to provide new ways to solve and reveal analytic properties of nonlinear evolution equations, and will also give valuable theory foundation for the practical applications of nonlinear evolution equations.

随着非线性研究的不断发展，很多真实的自然现象非常复杂以至于无法用纯粹的孤立子或多孤立子模式描述，于是构造非线性演化方程的更多可能的激发模式成为必要。本项目从研究非局域对称出发，以延拓非线性系统为对象，以对称约化方法和CTE/CRE方法为工具，构造非线性演化方程（组）的孤立子和非线性波相互作用解。主要研究内容包括：（1）引入非局域变量，构造非线性演化方程的非局域对称，并研究对应的有限变换问题；（2）结合非局域对称和通常的局域点李对称对一些重要非线性演化方程进行对称性约化，求解群不变解满足的低维约化方程并构造孤立子和非线性波相互作用解；（3）应用CTE/CRE方法直接构造非线性演化方程（组）或变系数方程的相互作用解，最后将研究成果应用于实际物理问题。本课题的研究成果不但能丰富对称理论，为求解非线性演化方程和揭示方程的解析性质提供新途径，而且为非线性模型的实际应用提供理论基础。

目前，科学家们已经找到了很多研究非线性波的模型，同时也利用各种数学工具找到了这些模型的非线性激发。但随着非线性研究的不断深入，人们发现很多真实的自然现象非常复杂以至于无法用纯粹的孤立子或多孤立子模式描述，构造非线性演化方程的更多可能的激发模式成为必要。本项目从寻找非线性演化方程的非局域对称出发，借助对称约化方法和相容tanh函数展开法，构造了多个非线性方程的孤立子--非线性波相互作用解，并进一步分析了解的动力学行为和可能应用。研究结果包括：(1)通过截断Painleve展开法构造(2+1)维KD方程、(2+1)维CDGKS方程等的非局域留数对称；(2)将非局域对称局域化为延拓系统的李点对称，运用对称约化构造(2+1)维CDGKS方程、修正KdV方程等的孤立子-非线性波相互作用解；(3)应用相容tanh函数展开法构造修正Boussinesq方程等的孤立子-椭圆周期波相互作用解；(4)基于相互作用解，详细分析它们的动力学性质和在实际物理问题中的可能应用；(5)将研究对象推广到非局域非线性薛定谔方程和高阶KdV方程，构造非局域非线性薛定谔方程的有理函数解并分析高阶KdV方程的解析性质和稳定性。本课题的研究成果不但丰富了对称理论，为求解非线性发展方程和揭示方程的解析性质提供新途径，而且为非线性模型的实际应用提供理论基础。