| 特木尔朝鲁,额尔敦布和,夏铁成.具源项的波动方程的非古典对称[J].数学年刊A辑,2012,33(2):193~204 |
| 具源项的波动方程的非古典对称 |
| Nonclassical Symmetry of the Wave Equation with Source Term |
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| DOI： |
| 中文关键词: 非古典对称 分类 精确解 公开问题 |
| 英文关键词:Nonclassical symmetry, Classification, Exact solution, Open problem |
| 基金项目:国家自然科学基金，上海海事大学自然科学基金 |
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| 中文摘要: |
| 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巳知源项外,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. |
| 英文摘要: |
| 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. Clark-
son 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. |
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