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Analysis of a Crank-Nicolson finite difference scheme for (2+1)D perturbed nonlinear Schrödinger equations with saturable nonlinearity

Anh-Ha Le, Toan T. Huynh, Quan M. Nguyen

TL;DR

The paper analyzes a Crank–Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and cubic loss, establishing boundedness, existence/uniqueness, and a second-order convergence rate in both time and space. The CNFD scheme is formulated on a uniform grid with discrete operators and an energy framework, and a rigorous convergence proof is provided under a mild time-step condition $\tau \lesssim h$. Numerical experiments validate the theoretical results, demonstrate second-order accuracy in $L^2$ and $H^1$ norms, and show good agreement with alternative soliton computation approaches like AITEM and SSFM. The work also discusses the potential to extend the method to 3D solitons (light bullets) and to more complex noisy settings, underscoring practical relevance for simulations in saturable nonlinear media.

Abstract

We analyze a Crank-Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and a perturbation of cubic loss. We show the boundedness, the existence and uniqueness of a numerical solution. We establish the error bound to prove the convergence of the numerical solution. Moreover, we find that the convergence rate is at the second order in both time step and spatial mesh size under a mild assumption. The numerical scheme is validated by the extensive simulations of the (2+1)D saturable nonlinear Schrödinger model with cubic loss. The simulations for travelling solitons are implemented by using an accelerated imaginary-time evolution scheme and the Crank-Nicolson finite difference method.

Analysis of a Crank-Nicolson finite difference scheme for (2+1)D perturbed nonlinear Schrödinger equations with saturable nonlinearity

TL;DR

The paper analyzes a Crank–Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and cubic loss, establishing boundedness, existence/uniqueness, and a second-order convergence rate in both time and space. The CNFD scheme is formulated on a uniform grid with discrete operators and an energy framework, and a rigorous convergence proof is provided under a mild time-step condition . Numerical experiments validate the theoretical results, demonstrate second-order accuracy in and norms, and show good agreement with alternative soliton computation approaches like AITEM and SSFM. The work also discusses the potential to extend the method to 3D solitons (light bullets) and to more complex noisy settings, underscoring practical relevance for simulations in saturable nonlinear media.

Abstract

We analyze a Crank-Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and a perturbation of cubic loss. We show the boundedness, the existence and uniqueness of a numerical solution. We establish the error bound to prove the convergence of the numerical solution. Moreover, we find that the convergence rate is at the second order in both time step and spatial mesh size under a mild assumption. The numerical scheme is validated by the extensive simulations of the (2+1)D saturable nonlinear Schrödinger model with cubic loss. The simulations for travelling solitons are implemented by using an accelerated imaginary-time evolution scheme and the Crank-Nicolson finite difference method.
Paper Structure (11 sections, 10 theorems, 121 equations, 1 figure, 2 tables)

This paper contains 11 sections, 10 theorems, 121 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Finite difference method and main results
  3. Crank–Nicolson finite difference method
  4. Main results
  5. Error Estimates
  6. Boundedness of discrete solutions
  7. Existence and Uniqueness Solution
  8. Convergence
  9. Numerical experiments
  10. Conclusion
  11. Estimate the consistency error

Key Result

Theorem 2.1

For any given $U^n$, there exists a unique solution $U^{n+1}$ of the CNFD discretization in (CNFD).

Figures (1)

  • Figure 1: (Color online) The soliton profile at $t=0$ and its evolution obtained by the simulation of (\ref{['eq:NLS']}) using the CNFD scheme of (\ref{['CNFD']}) with parameters as $h=2^{-4}$ and $\tau=2^{-7}$.

Theorems & Definitions (17)

  • Theorem 2.1: Unique solution
  • Theorem 2.2: Convergence
  • Lemma 3.1: Boundedness of discrete solutions
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: Boundedness of energy
  • proof
  • ...and 7 more