Time-periodic solutions to the cubic wave equation: an elementary constructive approach

2025/10/22 by Filip Ficek, Ficek, Filip
Mathematics · Physics and Astronomy · #35B10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2510.20054

openalex publication_date 2025/10/22 · openalex created_date 2025/10/25 · openalex updated_date 2026/07/28

Abstract

We present an elementary proof of existence of infinite family of time-periodic solutions to the one-dimensional nonlinear cubic wave equation with Dirichlet boundary conditions. It relies on the first order perturbative expansion and uses the Banach contraction principle to show existence of nearby solutions. In contrast to the previous results, this approach provides us explicit information about the frequencies and structures of the obtained solutions.

Citations

• Nonlinear oscillations of strings and beams
• New class of time-periodic solutions to the 1D cubic wave equation
• Complex structure of time-periodic solutions decoded in Poincaré-Lindstedt series: the cubic conformal wave equation on \mathbbS³
• Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to u_tt-u_xx± u³=0
• Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions
• Cantor families of periodic solutions for completely resonant nonlinear wave equations

Related

• Periodic Solutions for the 1d Cubic Wave Equation with Dirichlet Boundary Conditions
• New class of time-periodic solutions to the 1D cubic wave equation
• Trees, trunks, and branches -- bifurcation structure of time-periodic solutions to u_tt-u_xx± u³=0
• Newton's method and periodic solutions of nonlinear wave equations
• Construction of periodic solutions of multi-dimensional nonlinear wave equations with unbounded perturbation