Solutions of nonlinear nonautonomous Klein–Fock–Gordon equation. The choice of ansatz

Abstract

Methods of construction of exact analytical solutions of the nonlinear nonautonomous Klein–Fock–Gordon (KFG) equation are proposed. The solutions are sought in the form of a composite function U = f(θ). The function f(θ) is found from an ordinary nonlinear differential equation of the second order. The argument (ansatz) θ(x, y, z, t) is a root of an algebraic equation (cubic or square). It is shown that as the algebraic equations it is possible to take equations of families of surfaces, determining various curvilinear coordinates. The found ansatzes are used to construct functions φ(θ) satisfying Laplace’s equation. This result allows us to develop a new method of the solution of the nonlinear nonautonomous KFG equation. The general ways of the solution of the KFG equation are illustrated by consideration of some special cases.