Functional Differential Equations of Pointwise Type: Bifurcation
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RY DIFFERENTIAL EQUATION
Functional Differential Equations of Pointwise Type: Bifurcation L. A. Beklaryana,* and A. L. Beklaryanb,** a
Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, 117418 Russia b National Research University Higher School of Economics, Moscow, 119049 Russia *e-mail: [email protected] **e-mail: [email protected] Received February 15, 2020; revised February 15, 2020; accepted April 9, 2020
Abstract—The importance of functional differential equations of pointwise type is determined by the fact that their solutions are used to construct traveling-wave solutions for induced infinite-dimensional ordinary differential equations, and vice versa. Solutions of such equations exhibit bifurcation. A theorem on branching bifurcation is obtained for the solution to a linear homogeneous functional differential equation of pointwise type. Keywords: functional differential equation, initial-boundary value problem, bifurcation DOI: 10.1134/S0965542520080047
1. INTRODUCTION Equations of mathematical physics, such as Euler–Lagrange equations of variational problems, have an important class of solutions, namely, soliton solutions [1, 2]. In a number of models, such solutions are well approximated by soliton solutions of finite-difference analogues of the original equations, which describe not a continuous medium, but rather the interaction of medium clumps placed at vertices of a lattice [2, 3]. The resulting systems are infinite-dimensional dynamical ones. The most widely considered class of such problems includes infinite-dimensional systems with Frenkel–Kontorova potentials (periodic and slowly growing) and Fermi–Pasta–Ulam potentials (of exponential growth); a detailed survey of them can be found in [4]. The study of soliton solutions (traveling-wave solutions) is based on the existence of a one-to-one correspondence between soliton solutions of infinite-dimensional dynamical systems and solutions of induced functional differential equations of pointwise type [5–8]. This relationship is a fragment of a more general scheme, which is beyond the scope of this paper [9]. Importantly, the study of soliton solutions to an infinite-dimensional dynamical system is equivalent to the study of solutions to an induced functional differential equation of pointwise type. An existence and uniqueness (existence) theorem for an induced functional differential equation of pointwise type guarantees the existence and uniqueness (existence) of a soliton solution with given initial values. Solutions of functional differential equations of the pointwise type with a quasilinear right-hand side are studied within the formalism based on the group features of such equations and developed in [10, 5, 6, 11]; such solutions can be found in [8]. For linear systems, criteria for the existence of solutions in the form of an analogue of Noether’s theorem and criteria for the pointwise completeness of solutions were obtained in [12]. Easy-to-verify sufficient conditions for the existence of solutions were als
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