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 $(\alpha(x,y,z,t))$ or two $(\alpha(x,y,z,t),\beta(x,y,z,t))$ specially constructed functions. Solutions $f(\alpha)$ and $f(\alpha,\beta)$ relate to the class of functionally invariant solutions, and the functions $\alpha(x,y,z,t)$ and $\beta(x,y,z,t)$ are called the ansatzes. The ansatzes $(\alpha,\beta)$ 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 $(\alpha,\beta)$. 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.