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Numerical evidence for singularity formation in defocusing fractional NLS in one space dimension

Christian Klein, Christof Sparber

TL;DR

The paper investigates defocusing fractional nonlinear Schrödinger equations in one space dimension with fractional dispersion $0<s\le1$ and power-type nonlinearity. Using a semiclassical scaling with a small parameter $\varepsilon$ and a Fourier spectral numerics coupled to singularity tracing in the complex plane, the authors explore subcritical, critical, and supercritical regimes, focusing on small $s$ where nonlinear effects can induce highly oscillatory singularities. They provide substantial numerical evidence for a novel type of singular behavior in the energy-supercritical regime, characterized by spike-like structures and oscillations with a characteristic decay exponent $\mu\approx -\tfrac{1}{3}$, while mass and energy remain controlled and a turbulent-like cascade to high frequencies is observed in the critical case. The work broadens the understanding of defocusing dispersive dynamics in fractional settings, highlights the occurrence of singular structures in 1D even without focusing nonlinearity, and offers a detailed numerical framework (semiclassical rescaling, Fourier spectral methods, and complex-plane singularity tracing) that can be used to study similar questions in nonlocal dispersive systems.

Abstract

We consider nonlinear dispersive equations of Schrödinger-type involving fractional powers $0<s\le 1$ of the Laplacian and a defocusing power-law nonlinearity. We conduct numerical simulations in the case of small, energy supercritical $s$ and provide evidence for a novel type of highly oscillatory singularity within the solution.

Numerical evidence for singularity formation in defocusing fractional NLS in one space dimension

TL;DR

The paper investigates defocusing fractional nonlinear Schrödinger equations in one space dimension with fractional dispersion and power-type nonlinearity. Using a semiclassical scaling with a small parameter and a Fourier spectral numerics coupled to singularity tracing in the complex plane, the authors explore subcritical, critical, and supercritical regimes, focusing on small where nonlinear effects can induce highly oscillatory singularities. They provide substantial numerical evidence for a novel type of singular behavior in the energy-supercritical regime, characterized by spike-like structures and oscillations with a characteristic decay exponent , while mass and energy remain controlled and a turbulent-like cascade to high frequencies is observed in the critical case. The work broadens the understanding of defocusing dispersive dynamics in fractional settings, highlights the occurrence of singular structures in 1D even without focusing nonlinearity, and offers a detailed numerical framework (semiclassical rescaling, Fourier spectral methods, and complex-plane singularity tracing) that can be used to study similar questions in nonlocal dispersive systems.

Abstract

We consider nonlinear dispersive equations of Schrödinger-type involving fractional powers of the Laplacian and a defocusing power-law nonlinearity. We conduct numerical simulations in the case of small, energy supercritical and provide evidence for a novel type of highly oscillatory singularity within the solution.

Paper Structure

This paper contains 15 sections, 42 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Semiclassical rescaling and numerical methods
  3. Semiclassical rescaling
  4. Numerical method for computing the time-evolution
  5. Singularity tracing in the complex plane
  6. Linear fractional Schrödinger equation
  7. Solution of the fractional NLS: the (sub-)critical regime
  8. The subcritical regime
  9. The energy critical regime: emergence of turbulence
  10. Further examples in the critical regime
  11. Solution of the fractional NLS: the supercritical regime
  12. Cubic nonlinearity
  13. Additional simulations in the supercritical cubic case
  14. Other nonlinearities
  15. Comparison with other numerical methods

Figures (23)

  • Figure 1: Solution to the linear equation \ref{['fschr']} with $\varepsilon=0.1$ and initial data \ref{['inilin']}: on the left for $s=1$, on the right for $s=0.8$.
  • Figure 2: Solution to the linear equation \ref{['fschr']} with $\varepsilon=0.1$ and initial data \ref{['inilin']}: on the left for $s=\frac{1}{2}$, on the right for $s=0.4$.
  • Figure 3: Solution to the linear equation \ref{['fschr']} with $\varepsilon=0.1$ and initial data \ref{['inilin']}: on the left for $s=\frac{1}{4}$, on the right for $s=\frac{1}{5}$.
  • Figure 4: Solution to the cubic fNLS \ref{['critfNLS']} with $\varepsilon=0.1$ and initial data \ref{['ininonlin']}(a): on the left for $s=1$, on the right for $s=0.8$.
  • Figure 5: Solution to the energy-critical cubic fNLS \ref{['critfNLS']} with $s=\frac{1}{4}$, $\varepsilon=0.1$, and initial data \ref{['ininonlin']}(a).
  • ...and 18 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 4.1
  • Remark 4.2
  • Remark 5.1
  • Remark 5.2