On Stabilization of String-Nonlinear Oscillator Interaction

In the present paper, we consider a system of equations that describes the interaction of a nonlinear oscillator with an infinite string. The main result is the stabilization: roughly speaking, each finite energy solution to the system tends to a stationary solution as t → + ∞ (and similarly as t→ − ∞). The proof uses the description of a reversible system by an irreversible. The limit stationary solutions corresponding to t = ± ∞ may be different and arbitrary. The result gives a mathematical model of transitions to stationary states in reversible systems; these transitions are similar to Bohr ones. Such transitions are impossible for finite-dimensional Hamiltonian systems and for linear autonomous Shrödinger equations. The paper contains the complete exposition and an extension of the author′s recent results.