Enhanced group classification of nonlinear diffusion-reaction equations with gradient-dependent diffusion

Abstract

We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion-reaction equations with gradient-dependent diffusivity using the two-step version of the method of furcate splitting. For simultaneously finding the equivalence groups of an unnormalized class of differential equations and a collection of its subclasses, we suggest an optimized version of the direct method. The optimization includes the preliminary study of admissible transformations within the entire class and the successive splitting of the corresponding determining equations with respect to arbitrary elements and their derivatives depending on auxiliary constraints associated with each of required subclasses. In the course of applying the suggested technique to subclasses of the class under consideration, we construct, for the first time, a nontrivial example of finite-dimensional effective generalized equivalence group. Using the method of Lie reduction and the generalized separation of variables, exact solutions of some equations under consideration are found.

arXiv comment30 pages, 2 tables, minor corrections

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• Symmetries, conservation laws and difference schemes of the (1+2)-dimensional shallow water equations in Lagrangian coordinates
• Mapping method of group classification
• On the ineffectiveness of constant rotation in the primitive equations and their symmetry analysis
• Equivalence groupoid of a class of general Burgers-Korteweg-de Vries equations with space-dependent coefficients
• Group classification and exact solutions of a class of nonlinear waves
• Group classification of the two-dimensional Green-Naghdi equations with a time dependent bottom topography
• Exact Solutions and Reductions of Nonlinear Diffusion PDEs of Pantograph Type