Two-dimensional stationary structures in a parabolic equation with an inversion transformation of its spatial arguments

Abstract

Properties of stationary structures in a nonlinear optical resonator with a lateral inversions transformer in its feedback are investigated. A mathematical description of optical structures is based on a scalar parabolic equation with an inversion transformation of its spatial arguments and the Neumann condition on a square. The evolution of forms of stationary structures and their stability with decreasing the diffusion coefficient are investigated. It is shown that the number of stable stationary structures increases with decreasing the diffusion coefficient. In this work, the center manifold method and Galerkin method are used. INTRODUCTIONOptical systems with two-dimensional feedback [1, 2] demonstrate ample opportunities for the investigation of processes of birth and evolution of dissipative structures. The feedback makes it possible to exert influence on the dynamics of a system by means of controlled transformation of spatial arguments that is performed by prisms, lenses, dynamic holograms, and other devices. A nonlinear interferometer with lateral inversion of field in its two-dimensional feedback is the simplest optical system that realizes the nonlocal character of interaction of light fields. In this case, a variety of optical structures is experimentally established and the dependence of their number and forms on the diffusion coefficient [3-5] is revealed.Mathematical models of optical systems with two-dimensional feedback are semilinear parabolic equations with transformations of spatial arguments. In this work, the Neumann problem on a square is investigated for a parabolic equation with an inversion transformation of it spatial argument. Questions of existence, stability, and asymptotic form of its spatially inhomogeneous stationary solutions generated from a spatially homogeneous solution are considered. Following [6], a hierarchy of simplified models of the mentioned problem is constructed. This approach allows one to elucidate questions of evolution and interaction of stationary structures in the case of a considerable (rather than a small) deviation of the bifurcational parameter from bifurcational values. The results obtained in this case conform with the results of a numerical investigation of the initial problem [3][4][5]. We note that related problems in the one-dimensional case are considered in [2,7,8].