Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Abstract

Properties of stationary structures in a nonlinear optical resonator with an inversion transformer in its two-dimensional feedback are investigated. This system is mathematically described by a scalar parabolic equation with an inversion transformation of its spatial argument and with Neumann conditions on a segment. The evolution of forms of stationary structures and their stability are investigated. It is proved that the number of stable stationary structures increases with lengthening the segment. The central manifold method and Galerkin method are used. INTRODUCTIONAt the present time, an extension of investigations in the field of nonlinear optics is caused by the intensive use of optical systems in information technologies (see [1][2][3][4] and bibliographies of these works).A system consisting of a thin layer of a Kerr-type nonlinear medium and a variously organized two-dimensional feedback loop is acknowledged to be among the most popular nonlinear optical systems. A basic distinctive feature of such systems is that an external feedback loop can be used for the direct influence on the nonlinear dynamics of a system by means of controlled transformation of spatial variables that is performed by prisms, lenses, dynamic holograms, and other devices.Parabolic functional-differential equations with transforming the arguments of the sought-for function that are used for modelling optical systems with a two-dimensional feedback form a new class of equations for investigating the structure formation phenomenon. The observable autowave phenomena for the transformation of rotation by a fixed angle in a circle or a ring is mathematically substantiated in [5-7] on the basis of the Andronov-Hopf bifurcation theory. Methods of construction of periodic solutions for an arbitrary domain and a nondegenerate smooth transformation are developed in [8,9]. The method of quasinormal forms for a parabolic functional-differential equation with rotation for the case of small diffusion is used in [10,11] for the description of the dynamics of running waves and slowly varying structures. The optical bufferness of running waves is established in [12][13][14][15]. Following [12,13], recall that, in the phase space of some infinitely dimensional dynamic system, the bufferness phenomenon is realized if, by a suitable choice of parameters, it is possible to guarantee the existence of any fixed number of attractors of the same type in it. The method of central manifolds is used in [16,17] for the investigation of bifurcations of rotating structures in a ring and a circle for the case of rotation and also in a circle for rotation transformation together with radial compression. Based on [18], rotating structures are analyzed in [19,20] by the method of construction of approximate periodic solutions. A local bifurcation theory was used in [21,3] to construct stationary structures and analyze their stability for a parabolic equation on a segment with reflection transformation.A nonlinear interferometer with the mirror reflection of the fiel...

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

“…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].…”

Traveling Waves Solution in Parabolic Problem With a Rotation

“…This wave is born unstable. Введение В оптических системах с обратной связью реализуются два типа взаимодействия поля: локальное, связанное с диффузией частиц нелинейной среды и с самовоздействием поля, и крупномасштабное, вызванное преобразованием поперечных пространственных аргументов поля внутри резонатора, например, поворотом на угол h, сжатием или отражением пространственной переменной в контуре обратной связи [1][2][3][4]. Как следует из эксперимента, описанного в работах [5,6], наличие крупномасштабного взаимодействия существенно обогащает пространственно-временную динамику фазовой модуляции.…”

“…и методе Галеркина[1][2][3][4]. Для описания различных автоволновых процессов, протекающих в нелинейном интерферометре (то есть в нелинейной оптической системе, состоящей из тонкого слоя…”