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Khazova Y. A., Lukianenko V. A. Application of integral methods for the study of the parabolic problem. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2019, vol. 27, iss. 4, pp. 85-98. DOI: 10.18500/0869-6632-2019-27-4-85-98

# Application of integral methods for the study of the parabolic problem

Topic. The work is devoted to the study of a mathematical model describing an optical system with two-dimensional feedback. An example of such an optical system could be a nonlinear interferometer with a specular reﬂection of the ﬁeld. The mathematical model is a nonlinear functional diﬀerential parabolic equation with the transformation of the spatial variable reﬂection and conditions on the disk. Aim of the work is to study the conditions for the occurrence of spatially inhomogeneous stationary solutions. It is supposed to answer the question of the asymptotic form of the solutions being born and the determination of their stability. Methods. The study is carried out by methods of theoretical analysis, namely, the method of central manifolds, the Fourier method and the method of reducing to an integral equation are used. Using the method of separation of variables, a lemma on eigenvalues and eigenfunctions of the corresponding linearized problem is proved. To determine the asymptotic form of the solution for the linearized and corresponding nonlinear parabolic problems, the method of reduction to an integral equation was used. The necessary calculations on the proof of uniqueness and the continuous dependence of the solution on the initial conditions are given. Results and discussion. Based on the method of central manifolds, a theorem on the existence and stability of a spatially inhomogeneous stationary solution is proved. A representation is obtained for an inhomogeneous structure that is born as a result of a bifurcation of the «fork» type when the bifurcation parameter passes through a critical value. According to the proved theorem, this solution is born asymptotically stable. This theorem is local in nature and works in the vicinity of the bifurcation value of the diﬀusion coeﬃcient. The results presented in this paper are a continuation of research in the ﬁeld of nonlinear optics. The possibility of analyzing the structures being born not only in the vicinity of the bifurcation parameter value, but also throughout the change interval of the selected parameter, remains relevant. The results presented in this paper can be applied both in the theoretical analysis of the problems of nonlinear optics and in the practical processing and interpretation of information obtained in the formulation of computational or physical experiments.

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