基于近似对称的扰动方程的若干研究

Approximate generalized variable separation (AGVS) of nonlinear perturbed equations are studied based on approximate symmetry group theory and perturbation theory.. Firstly, AGVS of nonlinear mixed type of equations with perturbation are investigated, and definition of approximate generalized variable separation solutions (AGVSS) is given. In the meanwhile, a sufficient and necessary condition is obtianed with respect to approximate generalized conditional symmetry. AGVSSs.are constructed by above obtained condition. Subsequently, approximate conservation.laws、approximate symmetries、approximate Lie algebra structures and approximate.adjiont representation of the resulting equations are discussed.. Secondly, approxiamte derivative-dependent generalized variable separation equations with perturbation are studies, and approximate derivative-dependent generalized variable separation solutions (ADDGVSSs) are construced, which some properties are discussed. At the same time, approximate conservation.laws、approximate symmetries、approximate Lie algebra structures and approximate.adjiont representation of the resulting equations are investigated.

本项目以近似对称群理论和扰动理论为基础，研究非线性扰动系统的近似广义分离变量及其性质。. 首先，研究(1+1)维非线性混合型扰动方程的近似广义分离变量问题，给出该类型非线性扰动方程的近似广义分离变量解的定义，并寻求该类型扰动方程容许近似广义条件对称的充要条件. 构建所得分类方程的近似广义分离变量解.探讨分类非线性扰动方程的近似守恒律、近似对称、近似李代数结构和近似伴随表示。. 其次，带有扰动的非线性波动型方程的近似导数相关的广义分离变量问题，构造所得分类方程的近似导数相关的广义分离变量解；研究近似导数相关的广义分离变量解的性质. 同时，研究分类中所致非线性扰动方程的近似守恒律、近似对称、近似李代数结构和近似伴随表示等。

本项目以近似对称群理论和扰动理论为基础，研究非线性扰动系统的近似广义分离变量及.其性质。首先，研究(1+1)维非线性混合型扰动方程的近似广义分离变量问题，给出该类型非.线性扰动方程的近似广义分离变量解的定义，并寻求该类型扰动方程容许近似广义条件对.称的充要条件. 构建所得分类方程的近似广义分离变量解.探讨分类非线性扰动方程的近.似守恒律、近似对称、近似李代数结构和近似伴随表示。其次，带有扰动的非线性波动型方程的近似导数相关的广义分离变量问题，构造所得.分类方程的近似导数相关的广义分离变量解；研究近似导数相关的广义分离变量解的性质.. 同时，研究分类中所致非线性扰动方程的近似守恒律、近似对称、近似李代数结构和近.似伴随表示等。本项目的特色是以对称群理论、分离变量理论和扰动理论基础，研究求取非.线性扰动方程的近似解方法和其它若干问题。这是理论上和方法上创新。这既能.完善对称群理论，拓宽非线性扰动系统近似解的研究范围，又能为其它数学领域.提供新的研究方法。