# Periodic Solutions to Nonlinear Wave equation with $X$-dependent Coefficients under the General Boundary Conditions

## TL;DR

This paper proves the existence of time-periodic solutions for a nonlinear wave equation with spatially varying coefficients, relevant to nonhomogeneous strings and seismic wave propagation, using advanced mathematical techniques.

## Contribution

It introduces a novel approach combining Lyapunov-Schmidt reduction and Nash-Moser iteration to establish solutions under general boundary conditions.

## Key findings

- Existence of families of time-periodic solutions proven.
- Applicable to models of nonhomogeneous string vibrations.
- Relevant to seismic wave propagation in nonisotropic media.

## Abstract

In this paper we consider a class of nonlinear wave equation with $x$-dependent coefficients and prove existence of families of time-periodic solutions under the general boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The proofs are based on a Lyapunov-Schmidt reduction together with a differentiable Nash-Moser iteration scheme.