Singular reduction modules of differential equations

Details Singular reduction modules of differential equations Singular reduction modules of differential equations

2012-01-16

arxiv.org/abs/1201.3223v1

Physics

Mathematical Physics

30 pages. Extension of results of arXiv:0808.3577 to the case of a greater number of independent variables

Scientific paper

The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be favourably enhanced by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and second-order quasi-linear equations are studied. Reductions of differential equations to algebraic equations and first-order ordinary differential equations are considered in detail within the framework proposed.

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