Methods of construction of exact analytical solutions for nonautonomic nonlinear Klein-Fock-Gordon equation

摘要

We develop methods of construction of functionally invariant solutions U(x, y, z, t) for the nonlinear nonautonomic Klein-Fock-Gordon equation. The solutions U(x, y, z, t) are found in the form of an arbitrary function, that depends on one, τ(x, y, z, t), or two, α(x, y, z, t), β(x, y, z, t) specially constructed functions. The functions are called ansatzes. The ansatzes (τ,α,β) are defined as solutions of the special equations (algebraic or mixed type - algebraic and partial differential equations). The equations for defining of the ansatzes contain arbitrary functions, depending on (τ,α, β). The suggested methods allow one to find the solution U(x, y, z, t) for particular, but wide class of the nonautonomic non-linear Klein-Fock-Gordon equations. The methods are illustrated by examples of finding exact analytical solutions of the nonautonomic Liouville equation.