The buffer property in resonance systems of non-linear hyperbolic equations

A. Yu. Kolesov ; E. F. Mishchenko ; N. Kh. Rozov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 55 (2000) no. 2, pp. 297-321

Résumé

We study hyperbolic boundary-value problems for systems of telegraph equations with non-linear boundary conditions at the endpoints of a finite interval. The buffer property is established, that is, the existence of an arbitrary given finite number of stable time-periodic solutions for appropriately chosen parameter values, for this class of systems. For the case of a resonance spectrum of eigenfrequencies, the study of self-induced oscillations in various systems is shown to lead to one of the following two model problems, which are a kind of invariant: \begin{gather*} \frac{\partial^2w}{\partial t\partial x}=w+\lambda(1-w^2)\frac{\partial w}{\partial x}\,, \qquad w(t,x+1)\equiv-w(t,x), \qquad \lambda>0; \\ \frac{\partial w}{\partial t}+a^2\frac{\partial^3w}{\partial x^3}=w-w^3, \qquad w(t,x+1)\equiv-w(t,x), \qquad a\ne 0. \end{gather*} Informative examples from radiophysics are considered.