Solutions of the sine-Gordon equation with a variable amplitude

We propose methods for constructing functionally invariant solutions u(x, y, z, t) of the sine-Gordon equation with a variable amplitude in 3+1 dimensions. We find solutions u(x, y, z, t) in the form of arbitrary functions depending on either one (α(x, y, z, t)) or two (α(x, y, z, t), β(x, y, z, t)) specially constructed functions. Solutions f(α) and f(α, β) relate to the class of functionally invariant solutions, and the functions α(x, y, z, t) and β(x, y, z, t) are called the ansatzes. The ansatzes (α, β) are defined as the roots of either algebraic or mixed (algebraic and first-order partial differential) equations. The equations defining the ansatzes also contain arbitrary functions depending on (α, β). The proposed methods allow finding u(x, y, z, t) for a particular, but wide, class of both regular and singular amplitudes and can be easily generalized to the case of a space with any number of dimensions.