Nonlinear model of deformation of crystal media with complex lattice: Mathematical methods of model implementation

New methods for finding solutions to three coupled double sine-Gordon equations are developed. The equations describe the optical mode in the nonlinear theory of a crystalline media with complex lattice with two sublattices developed by the authors. In this model, displacement of the center of inertia of atoms of the elementary cell is described by an acoustic mode while mutual displacement of atoms inside the cell is accounted for by an optical mode. Making use of simplifying assumptions (homogenous deformation, thin-layer approximation) the solution of three coupled equations are reduced to the solution of a single sine-Gordon (double sine-Gordon) equation with constant or variable amplitude. Functionally invariant solutions of the nonlinear Klein–Fock–Gordon equation are constructed, particular cases of which are the sine-Gordon and the double sine-Gordon equations. The approach to determination of functionally invariant solutions of the sine-Gordon equation with variable amplitude is proposed. Equations of our nonlinear theory of crystalline media are derived in the form of two coupled sine-Gordon equations for the case of planar deformation, and functionally invariant solutions for them are obtained. Important features of the solutions obtained are discussed.