Mathematical methods for solution of nonlinear model of deformation of crystal media with complex lattice

Mathematical methods of the solution of the nonlinear equations of deformation of the complex crystal lattice consisting of two sublattices are developed. The nonlinear theory generalizes the classical theory of acoustic and optical deformations to the case of nonlinear interaction of sublattices. The equations describing optical modes represent system of three coupled sine-Gordon (SG) equations with the coefficients preceding the sine - amplitude depending on macrodeformations. They are reduced to SG equation with a constant (homogeneous stresses) or a variable (non-homogeneous stresses) amplitude by taking into account the simplifying assumptions for a one-dimensional case of deformation. The Lamb's modified method is offered for solution of the SG equation with a constant amplitude. Solutions are obtained which are expressed through Jacobi elliptic functions. They are expressed through circular and hyperbolic functions in special cases. The method of finding the solutions of the SG equation with a variable amplitude is offered. Features of the obtained solutions are discussed.