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Nonlinear dynamics of periodic Lugiato-Lefever waves against sums of co-periodic and localized perturbations

Joannis Alexopoulos

TL;DR

The paper addresses nonlinear stability of $T$-periodic standing waves for the Lugiato-Lefever equation against mixed co-periodic and localized perturbations in the tooth-space $L^2_{ ext{per}}(0,T)\oplus L^2(\mathbb{R})$ under diffusive spectral stability. It develops a modulation-based nonlinear iteration that employs both temporal and localized spatio-temporal phase modulations, combined with inverse- and forward-modulated perturbations and $L^2$–$L^\infty$ damping estimates. Under the stated spectral assumptions, it proves global existence and decay to a modulated standing profile, with explicit rates $||u(t)-\phi||_{L^{\infty}}\lesssim E_0$ and $||u(\cdot,t)-\phi_0(\cdot+\sigma_*+\gamma(\cdot,t))||_{L^{\infty}}\lesssim (1+t)^{-3/4}E_0$, along with precise control of the modulational parameters $\sigma$ and $\gamma$. The work discusses extensions to viscous conservation laws, uniformly subharmonic-plus-localized perturbations, nonlocalized phase modulations, and fully nonlocalized data via modulation spaces, highlighting both robustness and current limitations of the approach. Overall, it unifies prior stability theories by accommodating sums of co-periodic and localized perturbations within a single nonlinear framework.

Abstract

In recent years, essential progress has been made in the nonlinear stability analysis of periodic Lugiato-Lefever waves against co-periodic and localized perturbations. Inspired by considerations from fiber optics, we introduce a novel iteration scheme which allows to perturb against sums of co-periodic and localized functions. This unifies previous stability theories in a natural manner.

Nonlinear dynamics of periodic Lugiato-Lefever waves against sums of co-periodic and localized perturbations

TL;DR

The paper addresses nonlinear stability of -periodic standing waves for the Lugiato-Lefever equation against mixed co-periodic and localized perturbations in the tooth-space under diffusive spectral stability. It develops a modulation-based nonlinear iteration that employs both temporal and localized spatio-temporal phase modulations, combined with inverse- and forward-modulated perturbations and – damping estimates. Under the stated spectral assumptions, it proves global existence and decay to a modulated standing profile, with explicit rates and , along with precise control of the modulational parameters and . The work discusses extensions to viscous conservation laws, uniformly subharmonic-plus-localized perturbations, nonlocalized phase modulations, and fully nonlocalized data via modulation spaces, highlighting both robustness and current limitations of the approach. Overall, it unifies prior stability theories by accommodating sums of co-periodic and localized perturbations within a single nonlinear framework.

Abstract

In recent years, essential progress has been made in the nonlinear stability analysis of periodic Lugiato-Lefever waves against co-periodic and localized perturbations. Inspired by considerations from fiber optics, we introduce a novel iteration scheme which allows to perturb against sums of co-periodic and localized functions. This unifies previous stability theories in a natural manner.
Paper Structure (28 sections, 17 theorems, 149 equations, 1 figure)

This paper contains 28 sections, 17 theorems, 149 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preparation and main result
  3. Reformulation as real system
  4. The perturbed solution in Lg
  5. Unmodulated perturbation equations
  6. Spectral assumptions on Lg
  7. Formulation of main result
  8. Strategy of proof
  9. Linear estimates
  10. Semigroup decomposition and estimates on Lg
  11. Semigroup decomposition and estimates on Lg
  12. Nonlinear iteration scheme
  13. Local existence of the solutions
  14. Inverse-modulated perturbations
  15. Modulation in the purely co-periodic setting
  16. ...and 13 more sections

Key Result

Theorem 2.3

Assume assH1 and assD1-assD3. There exist constants $C,\varepsilon>0$ such that for initial data $\mathbf{w}_0 \in H_{{\mathrm{per}}}^6(0,T)$ and $\mathbf{v}_0 \in H^3(\mathbb R)$ with there exist a unique solution of (LLE_real) with initial condition $\mathbf{u}(0) = \phi + \mathbf{w}_0 + \mathbf{v}_0$, some smooth function $\gamma \in C([0,\infty), H^5(\mathbb R))$ and a constant $\sigma_* \in

Figures (1)

  • Figure 1: For the sake of illustration, we reduce to the real part of an initial perturbation $\mathbf{w}_0 + \mathbf{v}_0$ with $\mathbf{w}_0 \in L_{\textrm{per}}^2(0,T)$ and $\mathbf{v}_0 \in L^2(\mathbb R)$. This figure demonstrates that $\mathbf{v}_0$ can in particular be chosen such that $\mathbf{v}_0 + \mathbf{w}_0$ coincides with $\mathbf{w}_0$ except for finitely many periods for which the signal vanishes, which explains the name tooth space for $L^2_{\textrm{per}}(0,T) \oplus L^2(\mathbb R)$ ("knocked out teeth").

Theorems & Definitions (33)

  • Remark 2.1: Interpretation from fiber optics
  • Remark 2.2: Spectrum on modulation space
  • Theorem 2.3
  • Remark 2.4: Uniqueness of solutions
  • Proposition 3.1: LLE_periodic, perkins21
  • Proposition 3.2
  • proof
  • Proposition 3.3: $L^\infty$-estimates
  • proof
  • Proposition 4.1
  • ...and 23 more