An Explicit Unconditionally Stable Numerical Method for Solving Damped Nonlinear Schrödinger Equations with a Focusing Nonlinearity

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Abstract
This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrödinger equations (NLSs). 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 NLSs in two dimensions with a linear, cubic, or 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 δ is larger than a threshold value δ th. We note that our method can also be applied to solve the three-dimensional Gross-Pitaevskii equation with a quintic damping term to model the dynamics of a collapsing and exploding Bose-Einstein condensate (BEC)