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Zbl 1054.35088
Bao, Weizhu; Jaksch, Dieter
An explicit unconditionally stable numerical method for solving damped nonlinear Schrödinger equations with a focusing nonlinearity.
(English)
[J] SIAM J. Numer. Anal. 41, No. 4, 1406-1426 (2003). ISSN 0036-1429; ISSN 1095-7170/e

Summary: This paper introduces an extension of the time-splitting sine-spectral 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 blow-up, while linear damping can arrest blow-up only when the damping parameter ${\delta}$ is larger than a threshold value ${\delta}_{th}$. 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.
MSC 2000:
*35Q55 NLS-like (nonlinear Schroedinger) equations
65T40 Trigonometric approximation and interpolation
65N12 Stability and convergence of numerical methods (BVP of PDE)
65N35 Collocation methods (BVP of PDE)
81-08 Computational methods (quantum theory)

Keywords: damped nonlinear Schrödinger equation; time-splitting sine-spectral method; Gross-Pitaevskii equation; Bose-Einstein condensate; complex Ginzburg-Landau equation

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