Singular reduction modules of differential equations

Abstract

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 improved 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 to first-order ordinary differential equations are considered in detail within the framework proposed and are related to previous no-go results on nonclassical symmetries.

38 pages, advanced version. Extension of results of arXiv:0808.3577 to the case of a greater number of independent variables

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Cited by in corpus (8)

• Point and generalized symmetries of the heat equation revisited
• Extended symmetry analysis of generalized Burgers equations
• Extended symmetry analysis of two-dimensional degenerate Burgers equation
• Extended symmetry analysis of remarkable (1+2)-dimensional Fokker-Planck equation
• Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein-Gordon equations
• Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system
• Classification of reduction operators and exact solutions of variable coefficient Newell-Whitehead-Segel equations
• Surprising symmetry properties and exact solutions of Kolmogorov backward equations with power diffusivity