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Cnoidal Waves for the cubic nonlinear Klein-Gordon and Schrödinger Equations

Guilherme de Loreno, Gabriel E. B. Moraes, Fábio Natali, Ademir Pastor

TL;DR

The paper analyzes periodic cnoidal waves for the cubic Klein–Gordon and cubic NLS equations in the zero-mean periodic energy space. By combining Floquet spectral theory, Morse index arguments, and Grillakis–Shatah–Strauss theory, it proves that KG cnoidal waves are orbitally unstable while NLS cnoidal waves are orbitally stable, with stability established via a Lyapunov functional on the constrained zero-mean space. Existence of explicit cnoidal wave profiles is leveraged to compute spectral data and convexity, enabling precise instability/stability results in regimes determined by the elliptic modulus. The results illuminate a dichotomy between KG and NLS in periodic settings and extend stability theory for zero-mean periodic waves using constrained operators and energy–mass functionals.

Abstract

In this paper, we establish orbital stability results for \textit{cnoidal} periodic waves of the cubic nonlinear Klein-Gordon and Schrödinger equations in the energy space restricted to zero mean periodic functions. More precisely, for one hand, we prove that the cnoidal waves of the cubic Klein-Gordon equation are orbitally unstable as a direct application of the theory developed by Grillakis, Shatah, and Strauss. On the other hand, we show that the cnoidal waves for the Schrödinger equation are orbitally stable by constructing a suitable Lyapunov functional restricted to the associated zero mean energy space. The spectral analysis of the corresponding linearized operators, restricted to the periodic Sobolev space consisting of zero mean periodic functions, is performed using the Floquet theory and a Morse Index Theorem.

Cnoidal Waves for the cubic nonlinear Klein-Gordon and Schrödinger Equations

TL;DR

The paper analyzes periodic cnoidal waves for the cubic Klein–Gordon and cubic NLS equations in the zero-mean periodic energy space. By combining Floquet spectral theory, Morse index arguments, and Grillakis–Shatah–Strauss theory, it proves that KG cnoidal waves are orbitally unstable while NLS cnoidal waves are orbitally stable, with stability established via a Lyapunov functional on the constrained zero-mean space. Existence of explicit cnoidal wave profiles is leveraged to compute spectral data and convexity, enabling precise instability/stability results in regimes determined by the elliptic modulus. The results illuminate a dichotomy between KG and NLS in periodic settings and extend stability theory for zero-mean periodic waves using constrained operators and energy–mass functionals.

Abstract

In this paper, we establish orbital stability results for \textit{cnoidal} periodic waves of the cubic nonlinear Klein-Gordon and Schrödinger equations in the energy space restricted to zero mean periodic functions. More precisely, for one hand, we prove that the cnoidal waves of the cubic Klein-Gordon equation are orbitally unstable as a direct application of the theory developed by Grillakis, Shatah, and Strauss. On the other hand, we show that the cnoidal waves for the Schrödinger equation are orbitally stable by constructing a suitable Lyapunov functional restricted to the associated zero mean energy space. The spectral analysis of the corresponding linearized operators, restricted to the periodic Sobolev space consisting of zero mean periodic functions, is performed using the Floquet theory and a Morse Index Theorem.

Paper Structure

This paper contains 14 sections, 20 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Existence of Periodic Waves of Cnoidal Type
  4. Klein-Gordon equation
  5. Schrödinger equation
  6. Spectral Analysis.
  7. Spectral analysis for the Klein-Gordon equation
  8. Spectral analysis for the Schrödinger equation
  9. Orbital instability of the cnoidal waves for the KG equation
  10. Instability for the cnoidal wave.
  11. Orbital stability of the cnoidal waves for the NLS equation
  12. Global well-posedness
  13. Positivity of the operator $\boldsymbol{\mathcal{G}"(\Phi)}$
  14. Stability of the cnoidal waves

Key Result

Proposition 3.1

Let $L>0$ be fixed. For each $c \in (-1,1)$ there exists a unique $\beta_2 \in (\sqrt{2}, \infty)$ such that the cnoidal wave with is an $L$-periodic solution of ode1. In addition, the curve $c\in(-1,1)\longmapsto\varphi_c\in H^2_{per}$ is smooth.

Figures (2)

  • Figure 4.1: Graphic of $D_3$ for $L=2\pi$.
  • Figure 5.1: Graphic of $\mathsf{d}"$ as a function of $k\in\left(\frac{1}{\sqrt2},1\right)$.

Theorems & Definitions (47)

  • Proposition 3.1: Smooth Curve of Cnoidal Waves
  • proof
  • Remark 3.2
  • Proposition 3.3: Smooth Curve of Cnoidal Waves
  • Remark 3.4
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • ...and 37 more