Symmetry Operators of the Nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov Equation with a Quadratic Operator

Abstract

A class of nonlinear symmetry operators has been constructed for the many-dimensional nonlocal Fisher- Kolmogorov-Petrovskii-Piskunov equation quadratic in independent variables and derivatives. The construction of each symmetry operator includes an interwining operator for the auxiliary linear equations and additional nonlinear algebraic conditions. Symmetry operators for the one-dimensional equation with a constant influence function have been constructed in explicit form and used to obtain a countable set of exact solutions. The symmetry properties of a differential equation (DE) are associated with transforms leaving invariant the set of solutions of the equation. We shall refer to these transforms as symmetry operators (SOs). Symmetry operators make it possible to generate new solutions of an equation by using them sequentially to act on a known solution. The solution generation procedure only illustrates the potentialities of symmetry operators, but does not exhaust them. The problem is how to find symmetry operators or other symmetry transforms in explicit form. The basic mathematical apparatus used in a comprehensive study of symmetries of differential equations is the Lie group theory of continuous transformations leaving an (ordinary or partial) differential equation invariant. The invariance group of an equation is also termed a symmetry group, or a group admitted by the equation. Seeking a symmetry group is reduced to solving a system of linear constitutive equations for an infinitesimal Lie group operator (generator). The Lie group of finite transformations constructed with a generator can be applied to any solution of the equation admitting the group, and thus generate parametric sets of new solutions to the equation from a known one. A Lie group of pointwise invariance transformations of a differential equation can be considered as a set of symmetry operators of the equation. A detailed description of the application of Lie groups to ordinary DEs can be found, for instance, in Refs. 1-3. For partial differential equations (PDEs), the application of Lie group methods leans upon a procedure of prolongation of the action of a Lie group on higher partial derivatives (1-3). The generator of a prolonged Lie group has special structure and is a base object of examination in a study of the PDE symmetry properties. For a Lie group admitted by a PDE, the generator is determined by a linear partial differential equation in the generator coefficients that specify the so-called symmetries of the equation. A set of symmetries possesses algebraic properties, which are used to examine the properties of equations and to find sets of their solutions (1-3). However, the problem of finding SOs of nonlinear equations in explicit form by direct methods not related to Lie groups is poorly studied. In general statement, this problem has no constructive solution due to significant