New approach to solution of sine-Gordon equation with variable amplitude

摘要

Methods of construction of functionally invariant solutions are represented for (3+1) sine-Gordon equation with variant amplitude. The solutions U(x, y, z, t) are expressed through arbitrary function of one ƒ(α) or two ƒ(α, β) ansatzes. Ansatzes (α, β) are defined as roots of the algebraic or mixed (algebraic and the first order differential) equations. The equations, defining ansatzes, also contain arbitrary functions, depending on (α, β). The offered methods allow to find U(x, y, z, t) for private, but wide class of the regular and singular amplitudes. These methods are easily generalized for the cases of spaces with arbitrary dimensions. It is possible to hope, that the found solutions will be useful for the description of the physical processes taking place in the media with real structure.