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Stability of periodic waves in the model with intensity--dependent dispersion

Fábio Natali, Dmitry E. Pelinovsky, Shuoyang Wang

Abstract

We study standing periodic waves modeled by the nonlinear Schrodinger equation with the intensity-dependent dispersion coefficient. Spatial periodic profiles are smooth if the frequency of the standing waves is below the limiting frequency, for which the profiles become peaked (piecewise continuously differentiable with a finite jump of the first derivative). We prove that there exist two families of the periodic waves with smooth profiles separated by a homoclinic orbit and the period function (the energy-to-period mapping) is monotonically increasing for the family inside the homoclinic orbit and decreasing for the family outside the homoclinic orbit. This property allows us to derive a sharp criterion for the energetic stability of such standing periodic waves under time evolution if the perturbations are periodic with the same period for both families and, additionally, for the family outside the homoclinic orbit, spatially odd with respect to the half-period. By numerically approximating the sharp stability criterion, we show that both families are energetically stable for small frequencies but become unstable when the frequency approaches the limiting frequency of the peaked waves.

Stability of periodic waves in the model with intensity--dependent dispersion

Abstract

We study standing periodic waves modeled by the nonlinear Schrodinger equation with the intensity-dependent dispersion coefficient. Spatial periodic profiles are smooth if the frequency of the standing waves is below the limiting frequency, for which the profiles become peaked (piecewise continuously differentiable with a finite jump of the first derivative). We prove that there exist two families of the periodic waves with smooth profiles separated by a homoclinic orbit and the period function (the energy-to-period mapping) is monotonically increasing for the family inside the homoclinic orbit and decreasing for the family outside the homoclinic orbit. This property allows us to derive a sharp criterion for the energetic stability of such standing periodic waves under time evolution if the perturbations are periodic with the same period for both families and, additionally, for the family outside the homoclinic orbit, spatially odd with respect to the half-period. By numerically approximating the sharp stability criterion, we show that both families are energetically stable for small frequencies but become unstable when the frequency approaches the limiting frequency of the peaked waves.

Paper Structure

This paper contains 18 sections, 24 theorems, 165 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results
  4. Methodology and organization of the paper
  5. Existence of periodic waves
  6. Preliminary facts
  7. Existence of (positive) even periodic waves
  8. Existence of odd periodic waves
  9. Monotonicity of the period function
  10. Monotonicity for even periodic waves
  11. Monotonicity for odd periodic waves
  12. Spectral analysis near the periodic waves
  13. Spectral analysis of even periodic waves
  14. Spectral analysis of odd periodic waves
  15. Constrained energy minimization of periodic waves
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix the spatial period $L > 0$ for the periodic domain $\mathbb{T}_L$ and define For any $\omega \in (\omega_L,1)$, there exists a periodic orbit of system (odephi) with the smooth profile $\phi$ satisfying For any $\omega \in (\Omega_L,1)$, there exists a periodic orbit of system (odephi) with the smooth profile $\phi$ satisfying For both families, $x_0$ is an arbitrary translational parameter

Figures (7)

  • Figure 1.1: The phase portrait of system (\ref{['odephi']}) for $\omega = 0.5$.
  • Figure 1.2: The period function $T(\mathcal{E},\omega )$ versus $\mathcal{E}$ for fixed values of $\omega$. The dots denote the cutoff value of $\mathcal{E}$ satisfying $T(\mathcal{E},\omega) = \pi \sqrt{2(1-\omega)/\omega}$ for $\omega = \omega_L$. The vertical lines show divergence of $T(\mathcal{E},\omega)$ at $\mathcal{E} = \mathcal{E}_{\omega}$.
  • Figure 1.3: Numerical approximations for the even waves satisfying (\ref{['even-wave']}) with $x_0 = 0$. Left: the dependence of $\tilde{\mathcal{E}}_L$ versus $\omega$ for $L = 2\pi,3\pi,4\pi$. Right: the spatial profile $\phi$ versus $x$ for $\omega = 0.3,0.6,0.9$ and $L = 4\pi$.
  • Figure 1.4: Numerical approximations for the odd waves satisfying (\ref{['odd-wave']}) with $x_0 = 0$. Left: the dependence of $\tilde{\mathcal{E}}_L$ versus $\omega$ for $L = 2\pi,3\pi,4\pi$. Right: the spatial profile $\phi$ versus $x$ for $\omega = 0.3,0.6,0.9$ and $L = 4\pi$.
  • Figure 1.5: Dependence of $Q(\phi)$ versus $\omega$ for $L = 2\pi,3\pi,4\pi$ and in the limit $L \to \infty$ (dashed line). Left panel: the even wave satisfying (\ref{['even-wave']}). Right panel: the odd wave satisfying (\ref{['odd-wave']}).
  • ...and 2 more figures

Theorems & Definitions (50)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Theorem 2.3
  • ...and 40 more