An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity
Weizhu Bao, Dieter Jaksch
TL;DR
The paper addresses solving damped focusing nonlinear Schrödinger equations with damping terms modeling inelastic collisions in Bose-Einstein condensates. It extends the time-splitting sine-spectral (TSSP) method to this setting, delivering an explicit, unconditionally stable, time-transversal invariant scheme that preserves the exact decay rate for linear damping and provides spectral accuracy in space. Through 2D numerical experiments with linear, cubic, and quintic damping, it shows quintic damping always arrests blow-up, while linear damping arrests blow-up only when the damping exceeds a threshold $\delta_{\rm th}$ that scales with the initial energy $E(0)$ or the nonlinearity parameter $\beta$. The framework also applies to the 3D Gross-Pitaevskii equation with quintic damping, enabling simulations of collapsing and exploding BEC dynamics and suggesting broader utility in nonlinear wave dynamics.
Abstract
This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLS). The method is explicit, unconditionally stable and time transversal invariant. Moreover, it preserves the exact decay rate for the normalization of the wave function if linear damping terms are added to the NLS. Extensive numerical tests are presented for cubic focusing nonlinear Schrödinger equations in 2d with a linear, cubic or a quintic damping term. Our numerical results show that quintic or cubic damping always arrests blowup, while linear damping can arrest blowup only when the damping parameter $\dt$ is larger than a threshold value $\dt_{\rm th}$. We note that our method can also be applied to solve the 3d Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC).