A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schrödinger Equations with Nonlinear Damping

TL;DR

This paper develops and analyzes a Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. It proves boundedness of the discrete mass and energy, establishes existence and uniqueness of the discrete solution for small time steps, and demonstrates second-order convergence in space and time under the condition $\tau \lesssim h$. Numerical experiments with soliton and Gaussian-beam initial data validate the theoretical results, showing accurate amplitude dynamics and conservation properties when damping is absent. The study provides a rigorous numerical framework for high-dimensional NLS equations with modified nonlinearity and damping, with potential extensions to higher dimensions and alternative time-stepping schemes.

Abstract

Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of $n \geq 2$ suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete $L^2$-norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete $L^2$-norm and $H^1$-norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence.

Figures (4)

• Figure 1: (Color online) The contour plot of the initial soliton profile $|u_0(x,y)|$ of (\ref{['IC']}) and the evolution of its profiles $|u(x,y,t)|$ obtained by simulating (\ref{['eq:NLS']}) with the CNFD scheme (\ref{['CNFD']}) for $h=2^{-4}$ and $\tau=2^{-7}$.
• Figure 2: (Color online) The relative error $E_{2,h}$ in different regions of $h$ and $\tau$ in simulations using the CNFD scheme with the solitonic initial condition of (\ref{['IC']}).
• Figure 3: (Color online) An illustration for the oscillations in the numerical solution using the CNFD scheme. The contour plots represent the initial soliton profile $|u_0(x,y)|$ of (\ref{['IC']}) and the evolution of its profiles $|u(x,y,t)|$ obtained by simulating (\ref{['eq:NLS']}) with $h=2^{-3}$ and $\tau=3h$.
• Figure 4: The dependence of discrete energy (a) and discrete mass (b) with respect on time $t$ with $(\lambda,\nu,\varepsilon)= (10,1,5)$ and $(\lambda,\nu,\varepsilon)= (10,1,0)$

Theorems & Definitions (18)

• Lemma 3.1
• proof
• Lemma 3.2: Boundedness of the discrete solution in the discrete $L^2$-norm
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5: Boundedness of discrete energy
• proof