The solutions of equations for nonlinear model of deformation of the crystal media allowing martensitic transformations

Mathematical methods are developed for the solution of equations of statics for nonlinear model of deformation of the crystal media with the complex lattice allowing martensitic transformations. Vector of acoustic mode U(t, x, y, z) and vector of optical mode u(t, x, y, z) describe deformation in the nonlinear theory. They can be found from system of six connected nonlinear equations. It is transformed to a system of separate equations, namely, to one sine-Gordon equation with variable factor (amplitude) before sine and to two Poisson equations. In the case of flat deformation, the class of doubly periodic solutions which are expressed through Jacobi elliptic functions is constructed. Analysis of the obtained solutions is carried out.