Riemann waves in an elastic medium with small cubic anisotropy

Riemann waves in a weakly non-linear weakly anisotropic elastic material possessing the property of cubic symmetry are considered. The elastic potential is taken in the form of a series expansion in powers of the strain up to the fourth order of smallness. Anisotropy is represented in this expansion by cubic terms with a small coefficient. With that model, a solution is obtained and investigated in the form of quasi-periodic Riemann waves propagating along the principal diagonal of a cube. The characteristic velocities are found, the integral curves on the phase plane are constructed, and the direction in which the parameters vary along the integral curves, resulting in inversion of the solution profile, is indicated.