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

This paper introduces an extension of the time-splitting sine-spectral (TSSP) method for solving damped focusing nonlinear Schrodinger 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 ${\delta}$ is larger than a threshold value ${\delta}_{\rm th}$. 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).

[1]  P. Markowich,et al.  Three-dimensional simulation of jet formation in collapsing condensates , 2003, cond-mat/0307344.

[2]  C. Sulem,et al.  The nonlinear Schrödinger equation : self-focusing and wave collapse , 2004 .

[3]  Fa-yongZhang.  LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED , 2004 .

[4]  P. Markowich,et al.  Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation , 2003, cond-mat/0303239.

[5]  Shi Jin,et al.  Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes , 2003, SIAM J. Sci. Comput..

[6]  Wolfgang Ketterle,et al.  Bose–Einstein condensation of atomic gases , 2002, Nature.

[7]  S. Adhikari.  Mean-field theory for collapsing and exploding Bose-Einstein condensates , 2002 .

[8]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[9]  T. Hänsch,et al.  Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms , 2002, Nature.

[10]  C. Wieman,et al.  Dynamics of collapsing and exploding Bose–Einstein condensates , 2001, Nature.

[11]  R. Duine,et al.  Explosion of a collapsing Bose-Einstein condensate. , 2000, Physical review letters.

[12]  H. Saito,et al.  Intermittent implosion and pattern formation of trapped Bose-Einstein condensates with an attractive interaction. , 2000, Physical review letters.

[13]  Gadi Fibich,et al.  Self-Focusing in the Damped Nonlinear Schrödinger Equation , 2001, SIAM J. Appl. Math..

[14]  Girija Jayaraman,et al.  Variable mesh difference schemes for solving a nonlinear Schrödinger equation with a linear damping term , 2000 .

[15]  R. Temam,et al.  Resolution of a stochastic weakly damped nonlinear Schrödinger equation by a multilevel numerical method. , 2000, Journal of the Optical Society of America. A, Optics, image science, and vision.

[16]  O. Goubet.  Asymptotic Smoothing Effect for a Weakly Damped Nonlinear Schrodinger Equation in T2 , 2000 .

[17]  Cornish,et al.  Magnetic field dependence of ultracold inelastic collisions near a feshbach resonance , 2000, Physical review letters.

[18]  Gadi Fibich,et al.  Self-Focusing in the Perturbed and Unperturbed Nonlinear Schrödinger Equation in Critical Dimension , 1999, SIAM J. Appl. Math..

[19]  N. Akroune.  REGULARITY OF THE ATTRACTOR FOR A WEAKLY DAMPED NONLINEAR SCHRODINGER EQUATION ON R , 1999 .

[20]  F. Dalfovo,et al.  Theory of Bose-Einstein condensation in trapped gases , 1998, cond-mat/9806038.

[21]  Gadi Fibich,et al.  Self-focusing in the complex Ginzburg-Landau limit of the critical nonlinear Schrödinger equation , 1998 .

[22]  C. Gardiner,et al.  Cold Bosonic Atoms in Optical Lattices , 1998, cond-mat/9805329.

[23]  Guy Moebs,et al.  A multilevel method for the resolution of a stochastic weakly damped nonlinear Schro¨dinger equation , 1998 .

[24]  Y. Kagan,et al.  COLLAPSE AND BOSE-EINSTEIN CONDENSATION IN A TRAPPED BOSE GAS WITH NEGATIVE SCATTERING LENGTH , 1998, cond-mat/9801168.

[25]  L. Molinet,et al.  Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbb{R}^2$ , 1996, Advances in Differential Equations.

[26]  O. Goubet.  Approximate inertial manifolds for a weakly damped nonlinear Schrödinger equation , 1997 .

[27]  A. Mielke.  The complex Ginzburg - Landau equation on large and unbounded domains: sharper bounds and attractors , 1997 .

[28]  Eric Cornell,et al.  Very Cold Indeed: The Nanokelvin Physics of Bose-Einstein Condensation , 1996, Journal of research of the National Institute of Standards and Technology.

[29]  C. Wieman,et al.  Observation of Bose-Einstein Condensation in a Dilute Atomic Vapor , 1995, Science.

[30]  Edwards,et al.  Numerical solution of the nonlinear Schrödinger equation for small samples of trapped neutral atoms. , 1995, Physical review. A, Atomic, molecular, and optical physics.

[31]  R. Temam,et al.  An applicatin of approximate inertial manifolds to a weakly damped nonlinear schrödinger equation , 1995 .

[32]  George Papanicolaou,et al.  Singular solutions of the Zakharov equations for Langmuir turbulence , 1991 .

[33]  George Papanicolaou,et al.  Stability of isotropic singularities for the nonlinear schro¨dinger equation , 1991 .

[34]  Masayoshi Tsutsumi,et al.  On global solutions to the initial-boundary value problem for the damped nonlinear Schrödinger equations , 1990 .

[35]  L. S. Peranich,et al.  A finite difference scheme for solving a non-linear schro¨dinger equation with a linear damping term , 1987 .

[36]  Masayoshi Tsutsumi,et al.  Nonexistence of Global Solutions to the Cauchy Problem for the Damped Nonlinear Schrödinger Equations , 1984 .

[37]  E. Gross.  Structure of a quantized vortex in boson systems , 1961 .

[38]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .