2012/01/31 by Vaycheslav M. Boyko, Michael Kunzinger, Roman O. Popovych
Physics and Astronomy · Mathematics · #math-ph #math.AP #math.MP #msc:35B06 #msc:35A30 #msc:35C05
paper · pdf · doi:10.1063/1.4965227
published as J. Math. Phys. 57 (2016), 101503, 34 pages · 38 pages, advanced version. Extension of results of arXiv:0808.3577 to the case of a greater number of independent variables
arxiv created 2017/12/04 · arxiv updated 2017/12/05
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.