Approximate solutions of the equations which are simulating nonlinear processes

V. A. Lukyanenko

Taurida Journal of Computer Science Theory and Mathematics, no. 4 (2023), pp. 21-55

Résumé

Nonlinear ordinary differential equations and partial differential equations have found application in many sections of physics: photonics and plasmonics, nonlinear optics, gasfree combustion theory, hydrodynamics and electrodynamics; biophysics; nonlinear population dynamics; nonlinear wave theory, etc. In terms of system analysis, the considered models contain evolutionary blocks, diffusion, diffraction, blocks of interaction, nonlinear blocks and others. Within the framework of mathematics with broad interdisciplinary topics in the problems of applied nonlinear dynamics (AND) the existence and behavior of solutions (particularly, periodic and quasi-periodic solutions) of nonlinear equations with parameters, their stability are investigated, as well as spatially heterogeneous structures which are born by bifurcation. Some historical information related to the research of the AND in Crimea is given, for example here was held the famous international Lyapunov Conference about stability problem, which was chaired by G. A. Leonov in 2018. Research in the field of linear and nonlinear equations such as equations of convolution type, singular integral equations with argument shifts, nonlinear Urysohn type equations have their origin from F. D. Gakhov students, namely G. S. Litvinchuk and Yu. I. Chersky. Here, in the Crimea, a monograph "Equations of convolution type" was written by F. D. Gakhov and Yu. I. Chersky. Research in the field of applied non-linear dynamics in V. I. Vernadsky Crimean Federal university which were started by E. P. Belan, now develop in the following directions: quasi-linear parabolic equations of nonlinear optics with transformations of spatial variables (V. A. Lukianenko, Yu. A. Khazova and A. A. Kornuta); nonlinear models of propagation of surface plasmon-polaritons (V. A. Lukinaneko, M. S. Germanchuk and S. P. Plyshevskaya); mathematical model of the phenomenological equation of gas-free spin combustion as a singularly perturbed nonlinear parabolic Van der Pole equation (O. V. Shyian, V. A. Lukianenko and A. A. Grebeneva); nonlinear integral equation of the first Urysohn type and their approximate solutions (V. A. Lukianenko, M. G. Kozlova, V. A. Belozub and Yu. A. Khazova). For the problem of propagation of phase wave of light modulation with transformation of reflection of spatial variable using the method of integral (central) manifolds the theorem about existence of spatially heterogeneous stationary solutions has been proved; applying the Galerkin method, the form and stability of rotating wave solutions, which are born as a result of AdronovHopf bifurcation, metastable structures have been investigated; integral representation of the problem with transformation of involution type on an infinite strip with boundary conditions with oblique derivative has been obtained. Surface plasmon-polariton wave propagation is considered on the example of a system of two related nonlinear Schrodinger equations with cubic nonlinearity Kerr’s type, as well as generalizing model of spin combustion in annular regions (circle, ring, thin ring and circumference) and the quasi-normal form of the problem is constructed and investigated. The dependence of the first spin waves which are born after bifurcation from the time is visualized. The algorithm of solving of nonlinear integral equations of 1st kind of convolution type on example of Urysohn type equation arising in problems of applied nonlinear dynamics is introduced. Constructed structures are consistent with light structures obtained during physical experiments. This section of mathematics is also reflected in the works oby Moscow, Yaroslavl, Rostov, Nizhny Novgorod and other schools of non-linear dynamics. As the main methods of solving the studied problems of applied nonlinear dynamics the method of central manifold, bifurcation analysis, method of Krylov-Bogolyubov-MitropolskySamoilenko (KBMS), asymptotic decomposition of the solution according to eigen functions, averaging method, Jacobi elliptic function method, Galerkin method are offered. The solvability of the problems is based on the operator’s approach of studying equations in Banach spaces. The considered mathematical models are the basis of fundamental research on the development of new technologies of elemental electronics base, aimed at storage, transformation of information and creation of computational systems of intellectual processing of Big Data. The results presented in the paper reflect the research about AND and integral equations within the development program “Crimean mathematical center”.

Détail

Keywords: nonlinear equation, nonstationary effects, rotating waves, bifurcation analysis, integral representation.