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Dynamics of periodic fractional discrete NLS in the continuum limit

Brian Choi

TL;DR

This work establishes a rigorous link between the fractional discrete NLS on a periodic lattice and its continuum counterpart by proving convergence to the fractional NLS as the mesh size vanishes, using discrete Strichartz estimates to control the error in $L^2(\mathbb{T})$ for low-regularity data. It provides a sharp convergence rate $O\big(h^{\frac{\alpha}{2+\alpha}}\big)$ in the energy space and shows that this rate is optimal, revealing the subtle dispersive effects introduced by the lattice and the nonlocal operator. In parallel, the paper analyzes modulational instability of CW solutions, deriving explicit gain spectra that depend on the Lévy index $\alpha$, amplitude $A$, and lattice spacing $h$, and demonstrates transitions toward more complex dynamics as $\alpha$ decreases. Overall, the results illuminate how nonlocal long-range coupling and discreteness interact to shape continuum limits and nonlinear wave stability in fractional lattice systems, with implications for numerical discretizations and dispersive dynamics on periodic domains.

Abstract

The fractional discrete nonlinear Schrödinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal Lévy index $α\in (0,2]$. We show that the discrete system converges to the fractional NLS as $h \rightarrow 0$ below the energy space by directly estimating the difference between the discrete and continuum solutions in $L^2(\mathbb{T})$ using the discrete periodic Strichartz estimates. The sharp convergence rate via the finite difference method (FDM) is shown to be $O(h^{\fracα{2+α}})$ in the energy space. To further illustrate the convergent behavior of fDNLS, we survey various dynamical behaviors of the continuous wave (CW) solutions in the context of modulational instability, emphasizing the interplay between linear dispersion (or lattice diffraction), characterized by the nonlocal lattice coupling, and nonlinearity. In particular, the transition as $h \rightarrow 0$ from the linear dependence of maximum gain $Ω_m$ on the amplitude $A$ of CW solutions to the quadratic dependence is shown analytically and numerically.

Dynamics of periodic fractional discrete NLS in the continuum limit

TL;DR

This work establishes a rigorous link between the fractional discrete NLS on a periodic lattice and its continuum counterpart by proving convergence to the fractional NLS as the mesh size vanishes, using discrete Strichartz estimates to control the error in for low-regularity data. It provides a sharp convergence rate in the energy space and shows that this rate is optimal, revealing the subtle dispersive effects introduced by the lattice and the nonlocal operator. In parallel, the paper analyzes modulational instability of CW solutions, deriving explicit gain spectra that depend on the Lévy index , amplitude , and lattice spacing , and demonstrates transitions toward more complex dynamics as decreases. Overall, the results illuminate how nonlocal long-range coupling and discreteness interact to shape continuum limits and nonlinear wave stability in fractional lattice systems, with implications for numerical discretizations and dispersive dynamics on periodic domains.

Abstract

The fractional discrete nonlinear Schrödinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size and the nonlocal Lévy index . We show that the discrete system converges to the fractional NLS as below the energy space by directly estimating the difference between the discrete and continuum solutions in using the discrete periodic Strichartz estimates. The sharp convergence rate via the finite difference method (FDM) is shown to be in the energy space. To further illustrate the convergent behavior of fDNLS, we survey various dynamical behaviors of the continuous wave (CW) solutions in the context of modulational instability, emphasizing the interplay between linear dispersion (or lattice diffraction), characterized by the nonlocal lattice coupling, and nonlinearity. In particular, the transition as from the linear dependence of maximum gain on the amplitude of CW solutions to the quadratic dependence is shown analytically and numerically.
Paper Structure (9 sections, 13 theorems, 89 equations, 4 figures)

This paper contains 9 sections, 13 theorems, 89 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Mathematical background.
  3. Strichartz estimates.
  4. Convergence as $h \rightarrow 0$.
  5. Modulational instability of CW solutions.
  6. Conclusion.
  7. Appendix: well-posedness and uniform estimates.
  8. Appendix: exact solutions.
  9. Introduction

Key Result

Theorem 1.1

Let $\alpha \in (1,2]$. For any $s > s_0 := \max(\frac{3-\alpha}{4},\frac{1}{3})$ and $u_0 \in H^s(\mathbb{T})$, let $S(t)u_0$ and $S_h(t) d_h u_0$ denote the well-posed solutions constructed in lwp_fnls and lwp_fdnls, respectively. Then there exists $C(\| u_0 \|_{H^s},\alpha) > 0$ such that the err holds for all $t \in [0,T]$ where $T = T(\| u_0 \|_{H^s},\alpha)>0$. If $s = \frac{\alpha}{2}$ (ene

Figures (4)

  • Figure 1: The region of linear instability given by \ref{['region_instability']} (in blue) is plotted in $(\xi,A)$ and $(\xi,\alpha)$ for $h = \frac{\pi}{5}$.
  • Figure 2: The log plots used $(\alpha,h,k) = (0.25,\frac{\pi}{50},3)$ and $u_0(x) = A + 10^{-3} e^{ikx}$ where the right plot is the contour of the left plot.
  • Figure 3: Plots of $|u_h(x,t)|^2$ with $h = \frac{\pi}{50}$ and $u_0(x) = 1 + 10^{-6}(e^{ix}+e^{-ix})$.
  • Figure 4: Parameters used for the plot: $h = \frac{\pi}{50},\ u_0(x) = A + 10^{-5}e^{50 i x}$. The quadratic growth of the maximum gain for $|A| \ll 1$ (continuum regime) deviates as the amplitude increases.

Theorems & Definitions (26)

  • Theorem 1.1
  • Proposition 3.1
  • Lemma 3.1: zygmund2002trigonometric
  • proof : Proof of \ref{['prop:main']}
  • Corollary 3.1
  • proof
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Lemma 4.1
  • ...and 16 more