Nonlocal symmetry and generation of solutions for reaction-diffusion equations
Although this paper is brief, it contains much of interest to those who use Lie symmetries in the solution of partial differential equations. The essential feature is the ability to linearise certain partial differential equations by means of nonlocal transformations. If the linear equation possesses a reasonable number of nontrivial (i.e. apart from \(u\partial_u \) and \(f (t,x)\partial_u \), where \(u \) is the dependent variable and \(f \) is a solution of the linear equation) Lie point symmetries, similarity solutions can be constructed and the mapping of solutions into solutions through the Lie bracket of a nontrivial symmetry with the solution symmetry obtained with another nontrivial symmetry. The solutions can then map to solutions of the original nonlinear equation. The linear superposition principle of the linear equation is also mapped to a nonlinear superposition principle for the nonlinear equation. The presentation of the material is quite clear and this makes the paper a good starting point for a student to gain experience in the construction of solutions to nonlinear partial differential equations using symmetry methods.
scientific article; zbMATH DE number 5015617