Numerical Integrators for Continuous Disordered Nonlinear Schrödinger Equation

Abstract

In this paper, we consider the numerical solution of the continuous disordered nonlinear Schrödinger equation, which contains a spatial random potential. We address the finite time accuracy order reduction issue of the usual numerical integrators on this problem, which is due to the presence of the random/rough potential. By using the recently proposed low-regularity integrator (LRI) from (33, SIAM J Numer Anal, 2019), we show how to integrate the potential term by losing two spatial derivatives. Convergence analysis is done to show that LRI has the second order accuracy in $L^2$-norm for potentials in $H^2$. Numerical experiments are done to verify this theoretical result. More numerical results are presented to investigate the accuracy of LRI compared with classical methods under rougher random potentials from applications.

Appendix A. Schemes for the Discrete Model

The splitting for (4.1) reads

$$\begin{aligned} \Phi _T^t:\quad \left\{ \begin{aligned}&i{\dot{v}}_l(t)=-J[ v_{l+1}(t)+v_{l-1}(t)],\quad t>0,\ -N\le l\le N,\\&v_l(0)=v_l^0, \quad -N\le l\le N, \end{aligned}\right. \end{aligned}$$

(A.2)

and

$$\begin{aligned} \Phi _V^t:\quad \left\{ \begin{aligned}&i{\dot{w}}_l(t)= \left( \xi _l+\lambda |w_l(t)|^2\right) w_l(t), \quad t>0,\ -N\le l\le N,\\&w_l(0)=w_l^0,\quad -N\le l\le N. \end{aligned}\right. \end{aligned}$$

(A.3)

For $\Phi _T^t$, by taking Fourier transform

$$\begin{aligned} {\widehat{v}}_j(t)=\frac{1}{2N}\sum _{l=-N}^{N-1}\mathrm {e}^{-ijl\pi /N}v_l(t),\ -N\le j<N,\quad v_l(t)=\sum _{j=-N}^{N-1}\mathrm {e}^{ijl\pi /N}{\widehat{v}}_j(t),\ -N\le l<N, \end{aligned}$$

the equation in (A.2) is diagonalized and the exact solution is

$$\begin{aligned} v_l(t)=\sum _{j=-N}^{N-1}\mathrm {e}^{2itJ\cos (j\pi /N)} \mathrm {e}^{ijl\pi /N}\widehat{(v^0)}_j, \quad t\ge 0,\ -N\le l\le N, \end{aligned}$$

with $v^0=(v^0_{-N},\ldots ,v^0_{N})$. For $\Phi _T^t$, noting that $|w_l(t)|$ is constant in t for all l in (A.3), so the exact solution is

$$\begin{aligned} w_l(t)=\mathrm {e}^{-it(\xi _l+\lambda |w_l^0|^2)}w_l^0,\quad t\ge 0,\ -N\le l\le N. \end{aligned}$$

Then the Strang splitting scheme can be written down same as the composition in (2.1).

The corresponding semi-implicit finite difference method is simply

$$\begin{aligned}&i\frac{u_l^{n+1}-u_l^{n-1}}{2\tau }=-\frac{J}{2} \left( u_{l+1}^{n+1}+u_{l-1}^{n+1}+u_{l+1}^{n-1}+u_{l-1}^{n-1}\right) +\xi _lu_l^{n}+\lambda |u_l^n|^2u_l^n,\quad n\ge 1, \end{aligned}$$

for $-N\le l<N$ with

$$\begin{aligned}&u_l^1=u^0_l+iJ\tau \left( u^0_{l+1}+u^0_{l-1}\right) -i\tau \left( \xi _l+\lambda |u^0_l|^2\right) u^0_l,\quad -N\le l\le N,\\&u^n_{-N}=u^n_{N},\quad u^n_{-N-1}=u^n_{N-1},\quad n\ge 0. \end{aligned}$$