Wilhelm Fushchych

Higher symmetries and exact solutions of linear and nonlinear Schrödinger equation

A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schrodinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painleve, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schrodinger equations.

Time-dependent symmetries of variable-coefficient evolution equations and graded Lie algebras

Polynomial-in-time dependent symmetries are analysed for polynomial-in-time dependent evolution equations. Graded Lie algebras, especially Virasoro algebras, are used to construct nonlinear variable-coefficient evolution equations, both in 1 + 1 dimensions and in 2 + 1 dimensions, which possess higher-degree polynomial-in-time dependent symmetries. The theory also provides a kind of new realization of graded Lie algebras. Some illustrative examples are given.

On the general solution of the d'Alembert equation with a nonlinear eikonal constraint and its applications

We construct the general solutions of the system of nonlinear differential equations ⧠nu=0, uμuμ=0 in the four‐ and five‐dimensional complex pseudo‐Euclidean spaces. The results obtained are used to reduce the multidimensional nonlinear d’Alembert equation ⧠4u=F(u) to ordinary differential equations and to construct its new exact solutions.

On the new approach to variable separation in the time‐dependent Schrödinger equation with two space dimensions

We suggest an effective approach to separation of variables in the Schrodinger equation with two space variables. Using it we classify inequivalent potentials V(x1,x2) such that the corresponding Schrodinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrodinger equation with the anisotropic harmonic oscillator potential V=k1x21+k2x22 and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schrodinger equation (V=0).

Orthogonal and non-orthogonal separation of variables in the wave equation utt-uxx+V(x)u=0utt-uxx+V(x)u=0

We develop a direct approach to the separation of variables in partial differential equations. Within the framework of this approach, the problem of the separation of variables in the wave equation with time-independent potential reduces to solving an over-determined system of nonlinear differential equations. We have succeeded in constructing its general solution and, as a result, all potentials V(x) permitting variable separation have been found. For each of them we have constructed all inequivalent coordinate systems providing separability of the equation under study. It should be noted that the above approach yields both orthogonal and non-orthogonal systems of coordinates.

Symmetry reduction and some exact solutions of nonlinear biwave equations

Symmetry analysis of a class of biwave equations □2u = F(u) and of a system of wave equations which is equivalent to it is performed. Reduction of the nonlinear biwave equations by means of the Ansatze invariant under non-conjugate subalgebras of the extended Poincare algebra AP(1,1) and the conformal algebra AC(1,1) is carried out. Some exact solutions of these equations are obtained.

Conditional symmetry and new classical solutions of the Yang-Mills equations

We suggest an effective method for reducing the Yang-Mills equations to systems of ordinary differential equations. With the use of this method we construct extensive families of new exact solutions of the Yang-Mills equations. Analysis of the solutions thus obtained shows that they correspond to the conditional (non-classical) symmetry of the equations under study.

Symmetry of the Schrödinger Equation with Variable Potential

We study symmetry properties of the Schrodinger equation with the potential as a new dependent variable, i.e., the transformations which do not change the form of the class of equations. We also consider systems of the Schrodinger equations with certain conditions on the potential. In addition we investigate symmetry properties of the equation with convection term. The contact transformations of the Schrodinger equation with potential are obtained.

High-order equations of motion in quantum mechanics and Galilean relativity

Linear partial differential equations of arbitrary order invariant under the Galilei transformations are described. Symmetry classification of potentials for these equations in two- dimensional space is carried out. High-order nonlinear partial differential equations invariant under the Galilei, extended Galilei and full Galilei algebras are studied.

New scale-invariant nonlinear differential equations for a complex scalar field

Abstract We describe all complex wave equations of the form □u = F(u, u ∗ ) invariant under the extended Poincare group. As a result, we have obtained the five new classes of P (1, 3)-invariant nonlinear partial differential equations for the complex scalar field.