Ivan Tsyfra

A precise definition of reduction of partial differential equations

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 + 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear wave equations.

On New Reduction of Nonlinear Wave Type Equations via Classical Symmetry Method

The paper is devoted to the construction of an ansatz for the first derivatives of an unknown function which reduces a scalar partial differential equation with three independent variables to a system of equations by using the operators of classical point symmetry. The method is applied to nonlinear wave equation with cubic nonlinearity, Liouville equation and Kadomtsev– Petviashvili equation.

Symmetry Reduction of Evolution and Wave Type Equations

We study the symmetry reduction of nonlinear evolution and wave type differential equations by using operators of non‐point symmetry. In our approach we use both operators of classical and conditional symmetry. It appears that the combination of non‐point and conditional symmetry enables us to construct not only solutions but Backlund transformations too for the equation under study.

Group Transformations with Discrete Parameter and Invariant Schrödinger Equation

We propose a method for the construction of potentials and nonlinearities, for which the Schrodinger equation is invariant with respect to a group transformations with discrete parameters. This is a generalization of the known Lie symmetry of differential equations.

Symmetry of the Maxwell and Minkowski Equations System

Symmetry properties of Maxwell equations in vacuum was studied in detail by Lorentz, Poincare, Bateman, Cuningham [1, 2]. Maximal local Lie group of invariance of linear equations for electromagnetic fields in vacuum is 16 parameters group containing 15 parameter conformal group as a subgroup [3]. It was proved in [4] that the Maxwell equations in the medium, which form a system of first order partial differential equations for vectors ~ D, ~ B, ~ E and ~ H , admit infinite symmetry. Thus, the system of equations