Boundedness and convergence of solutions for the string coupled to a nonlinear oscillator

TL;DR

This paper investigates the long-term behavior of a coupled system of an infinite string and a nonlinear oscillator, proving solutions tend to a time-periodic state as time approaches infinity.

Contribution

It establishes the convergence of solutions to a time-periodic solution for the coupled string-oscillator system with periodic initial data.

Findings

Solutions converge to a time-periodic state as t→∞

The system's solutions are bounded and stable over time

The analysis applies to systems with nonlinear oscillators coupled to infinite strings

Abstract

A system of equations consisting of an infinite string coupled to a nonlinear oscillator is considered. The Cauchy problem for the system with the periodic initial data is studied. The main goal is to prove the convergence of the solutions as $t\to\infty$ to a time periodic solution.