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

Filip Ficek

TL;DR

The paper proves the existence of time-periodic solutions to the 1D cubic wave equation with Dirichlet boundaries by an elementary fixed-point argument built on a first-order perturbative expansion. By restricting to frequencies $\Omega=(2k+1)/(2k)$, it constructs an explicit approximate profile $u_k$ and shows that for large $k$ a nearby true solution $u$ exists with $\|u-u_k\|=O(k^{-1/2})$. The method yields explicit information about the frequencies and structure of the solutions and, via a rescaling, also produces a family of focusing solutions with frequencies $\Omega=2k/(2k+1)$. Compared to prior approaches, this contraction-based, constructive framework avoids small-divisor and Nash–Moser techniques while delivering transparent, quantitative results and an explicit description of the solution’s composition.

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.

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

TL;DR

The paper proves the existence of time-periodic solutions to the 1D cubic wave equation with Dirichlet boundaries by an elementary fixed-point argument built on a first-order perturbative expansion. By restricting to frequencies , it constructs an explicit approximate profile and shows that for large a nearby true solution exists with . The method yields explicit information about the frequencies and structure of the solutions and, via a rescaling, also produces a family of focusing solutions with frequencies . Compared to prior approaches, this contraction-based, constructive framework avoids small-divisor and Nash–Moser techniques while delivering transparent, quantitative results and an explicit description of the solution’s composition.

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.
Paper Structure (11 sections, 14 theorems, 103 equations)

This paper contains 11 sections, 14 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Functional setting
  4. Fixed-point formulation
  5. Preliminaries
  6. Space $X$
  7. Operator $L_k^{-1}$
  8. Function $u_k$
  9. Bound on the residue $\mathcal{N}_k(0)$
  10. Operator $A$
  11. Bound on the operator $H_k$

Key Result

Theorem 1

For every $k\geq 79\,675$ there exists a non-trivial solution $u$ to eq:nlw with $\Omega=\frac{2k+1}{2k}$. This solution satisfies $\Vert u-u_k\Vert<\frac{139}{42\,500} k^{-1/2}$, with the norm $\Vert\cdot\Vert$ defined below.

Theorems & Definitions (30)

  • Theorem 1
  • Remark 1
  • Remark 2
  • proof : Proof of Theorem 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 20 more