Traveling waves solution in parabolic problem with a rotation

Optical systems with two-dimensional feedback demonstrate wide possibilities for emergence of dissipative structures. Feedback allows to influence on dynamics of the optical system by controlling the transformation of spatial variables performed by prisms, lenses, dynamic holograms and other devices. Nonlinear interferometer with mirror reflection of a field in two-dimensional feedback is one of the simplest optical systems in which the nonlocal interaction of light fields is realized. A mathematical model of optical systems with two-dimensional feedback is the nonlinear parabolic equation with rotation transformation of a spatial variable and periodicity conditions on a circle. Such problems are investigated: conditions of occurrence the traveling wave solution, how the form of the solution changes as the diffusion coefficient decreases, dynamics of the solution’s stability when the value of bifurcation parameter is decrease. As a bifurcation parameter was taken diffusion coefficient. The method of central manifolds and the Galerkin’s method are used in this paper. The method of central manifolds allows to prove the theorem on the existence and form of the traveling wave solution in the neighborhood of the critical bifurcation value. The first traveling wave born as a result of the Andronov-Hopf bifurcation. According to the central manifold theorem, the first traveling wave is born orbitally stable. The theorem gives the opportunity to explore solutions near the critical values of the bifurcation parameter. The Galerkin’s method was used by further research of traveling waves when bifurcation parameter was decrease. If the bifurcation parameter decreases and transition through the critical value, the zero solution of the problem loses stability. As a result, a periodic solution of the traveling wave type branches off from the zero solution. This wave is born orbitally stable. Further, the bifurcation parameter and its passage through the next critical value from the zero solution, the second solution of the traveling wave type is arise as a result of the Andronov-Hopf bifurcation. This wave is born unstable. Numerical calculations have shown that the application of the Galerkin’s method leads to correct results. The results can be used by experiments on the study of phenomena in optical systems with feedback.