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Continuum limit of 3D fractional nonlinear Schrödinger equation

Jiajun Wang

TL;DR

The paper establishes a strong $L^2$ continuum limit for the 3D fractional nonlinear Schrödinger equation by showing that solutions of the discrete FNLS on $h\mathbb{Z}^3$ converge to the continuous FNLS on $\mathbb{R}^3$ as $h\to0$, in the energy-subcritical regime with $3\le p<5$ and $\frac{3(p-1)}{p+1}<\alpha<2$. The approach hinges on proving a uniform-in-$h$ Strichartz estimate for the lattice propagator $U_h(t)$, achieved through detailed oscillatory-integral analysis and Newton polyhedron techniques, and a careful comparison framework via discretization and interpolation operators $d_h$ and $p_h$. The main result provides an explicit convergence rate $h^{\alpha/(2+\alpha)}$ on a time interval $[0,T]$ depending on the initial data, and clarifies the dimensional barriers by highlighting why the method currently applies to $d=3$ in the energy-subcritical range. The work extends prior 1D/2D continuum-limit results to 3D, offering a rigorous justification for discretizations of long-range FNLS in higher dimensions and informing numerical simulations.

Abstract

In this paper, we investigate the continuum limit theory of the fractional nonlinear Schrödinger equation in dimension 3. We show that the solution of discrete fractional nonlinear Schrödinger equation on hZ^3 will converge strongly in L^2 to the solution of fractional nonlinear Schrödinger equation on R^3, when h->0. The key is proving the uniform-in-h Strichartz estimate for discrete fractional nonlinear Schrödinger equation, by using the uniform estimate of oscillatory integral and Newton polyhedron techniques.

Continuum limit of 3D fractional nonlinear Schrödinger equation

TL;DR

The paper establishes a strong continuum limit for the 3D fractional nonlinear Schrödinger equation by showing that solutions of the discrete FNLS on converge to the continuous FNLS on as , in the energy-subcritical regime with and . The approach hinges on proving a uniform-in- Strichartz estimate for the lattice propagator , achieved through detailed oscillatory-integral analysis and Newton polyhedron techniques, and a careful comparison framework via discretization and interpolation operators and . The main result provides an explicit convergence rate on a time interval depending on the initial data, and clarifies the dimensional barriers by highlighting why the method currently applies to in the energy-subcritical range. The work extends prior 1D/2D continuum-limit results to 3D, offering a rigorous justification for discretizations of long-range FNLS in higher dimensions and informing numerical simulations.

Abstract

In this paper, we investigate the continuum limit theory of the fractional nonlinear Schrödinger equation in dimension 3. We show that the solution of discrete fractional nonlinear Schrödinger equation on hZ^3 will converge strongly in L^2 to the solution of fractional nonlinear Schrödinger equation on R^3, when h->0. The key is proving the uniform-in-h Strichartz estimate for discrete fractional nonlinear Schrödinger equation, by using the uniform estimate of oscillatory integral and Newton polyhedron techniques.
Paper Structure (4 sections, 16 theorems, 151 equations)

This paper contains 4 sections, 16 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}
  3. Proof of Lemma \ref{['3.4']}
  4. Appendix

Key Result

Theorem 1.1

For $3\le p<5$, $\frac{3(p-1)}{p+1}<\alpha<2$, $u_{0}\in H^{\frac{\alpha}{2}}(\mathbb{R}^{3})$ and $u_{0,h}=d_{h}u_{0}$, we have $u\in C([0,T];H^{\frac{\alpha}{2}}(\mathbb{R}^{3}))$ and $u_{h}\in C([0,T];L_{h}^{2})$ are the solutions of equations (CFNLS) and (FNLS), for some $T=T(\|u_{0}\|_{H^{\frac where $C_{1}, C_{2}$ is positive constants that only depend on $\|u_{0}\|_{H^{\frac{\alpha}{2}}(\ma

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • ...and 32 more