摘要
给出了具源项的波动方程的非古典对称的完全分类和相应源项的所有可能的具体表达式.除了古典对称对应的已知源项外,获得了允许非古典对称的新源项,其中包括著名的演化方程,如线性(齐次和非齐次)波动方程,双曲Liouville方程和Klein-Gordon方程等.这些结果解答了Clarkson在2001年中提出的关于波方程非古典对称的公开问题.同时,用分类中得到的对称,通过求不变解构造了以上演化方程的一些新的精确解.
The complete nonclassical symmetry classification of wave equation with a source term is given.Mathematical forms are obtained for the source terms which permit a non-classical symmetry.In addition to the known source terms obtained from classical symmetry methods,new source terms which admit nonclassical symmetry are found.Some important evolution equations,such as the linear（homogeneous and inhomogeneous） wave equations, the hyperbolic Liouville equation and the Klein-Gordon equation,are included.As a result, the open problem about the nonclassical symmetry classification mentioned by P.A.Clarkson in 2001 for the wave equation is clarified.Several new exact solutions to these evolution equations are constructed through solving their invariant solutions by using the obtained symmetries in the classification.
出处
《数学年刊（A辑）》
CSCD
北大核心
2012年第2期193-204,共12页
Chinese Annals of Mathematics
基金
国家自然科学基金(No.11071159
No.61072147)
上海海事大学自然科学基金(No.20110008)资助的项目
关键词
非古典对称
分类
精确解
公开问题
Nonclassical symmetry
Classification
Exact solution
Open problem
