Numerical computation of soliton dynamics for NLS equations in a driving potential

We provide some numerical computations for the soliton dynamics of the nonlinear Schr\"odinger equation with an external potential. After computing the ground state solution $r$ of a related elliptic equation we show that, in the semi-classical regime, the center of mass of the solution with initial datum modelled on $r$ is driven by the solution of a Newtonian type law. Finally, we provide some examples and analyze the numerical errors in the two dimensional case when $V$ is an harmonic potential.

[1]  Terence Tao,et al.  Why are solitons stable , 2008, 0802.2408.

[2]  Eugenio Montefusco,et al.  Soliton dynamics for CNLS systems with potentials , 2010, Asymptot. Anal..

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

[4]  J. Fröhlich,et al.  Colliding Solitons for the Nonlinear Schrödinger Equation , 2008, 0804.1608.

[5]  Mechthild Thalhammer,et al.  High-order time-splitting Hermite and Fourier spectral methods , 2009, J. Comput. Phys..

[6]  A. Soffer,et al.  Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data , 1992 .

[7]  M. Weinstein.  Lyapunov stability of ground states of nonlinear dispersive evolution equations , 1986 .

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

[9]  Weizhu Bao,et al.  Ground-state solution of Bose--Einstein condensate by directly minimizing the energy functional , 2003 .

[10]  Alan Weinstein,et al.  Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential , 1986 .

[11]  H. Yau,et al.  On the Point-Particle (Newtonian) Limit¶of the Non-Linear Hartree Equation , 2002 .

[12]  Vladimir S. Buslaev,et al.  On the stability of solitary waves for nonlinear Schr?odinger equations , 1995 .

[13]  A. Soffer,et al.  Selection of the ground state for nonlinear schrödinger equations , 2004 .

[14]  C. Sulem,et al.  On asymptotic stability of solitary waves for nonlinear Schrödinger equations , 2003 .

[15]  James P. Keener,et al.  Solitons under perturbations , 1977 .

[16]  Mechthild Thalhammer,et al.  A minimisation approach for computing the ground state of Gross-Pitaevskii systems , 2009, J. Comput. Phys..

[17]  Shi Jin,et al.  Computation of the Semiclassical Limit of the Schrödinger Equation with Phase Shift by a Level Set Method , 2008, J. Sci. Comput..

[18]  P. Lions,et al.  Orbital stability of standing waves for some nonlinear Schrödinger equations , 1982 .

[19]  D. J. Kaup,et al.  Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[20]  R'emi Carles,et al.  WKB Analysis for Nonlinear Schrödinger Equations with Potential , 2006, math/0601611.

[21]  Sahbi Keraani,et al.  Semiclassical limit for nonlinear Schrödinger equation with potential. II , 2006, Asymptot. Anal..

[22]  Marlis Hochbruck,et al.  Explicit Exponential Runge-Kutta Methods for Semilinear Parabolic Problems , 2005, SIAM J. Numer. Anal..

[23]  Qiang Du,et al.  Computing the Ground State Solution of Bose-Einstein Condensates by a Normalized Gradient Flow , 2003, SIAM J. Sci. Comput..

[24]  Pierre-Louis Lions,et al.  Nonlinear scalar field equations, I existence of a ground state , 1983 .

[25]  M. Zworski,et al.  Soliton interaction with slowly varying potentials , 2007, 0709.0478.

[26]  Antonio Ambrosetti,et al.  Perturbation Methods and Semilinear Elliptic Problems on R^n , 2005 .

[27]  Tai-Peng Tsai,et al.  Asymptotic dynamics of nonlinear Schrödinger equations: Resonance‐dominated and dispersion‐dominated solutions , 2002 .

[28]  P. Lions.  The concentration-compactness principle in the calculus of variations. The locally compact case, part 1 , 1984 .

[29]  Michael I. Weinstein,et al.  Multichannel nonlinear scattering for nonintegrable equations , 1990 .

[30]  Jared C. Bronski,et al.  Soliton dynamics in a potential , 2000 .

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

[32]  Sahbi Keraani,et al.  SEMICLASSICAL LIMIT OF A CLASS OF SCHRÖDINGER EQUATIONS WITH POTENTIAL , 2002 .

[33]  M. Kwong.  Uniqueness of positive solutions of Δu−u+up=0 in Rn , 1989 .

[34]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[35]  G. Perelman.  Asymptotic stability of solitary waves for nonlinear Schrödinger equations , 2003 .

[36]  M. Squassina.  Soliton dynamics for the nonlinear Schrödinger equation with magnetic field , 2008, 0811.2584.

[37]  S. Gustafson,et al.  Long Time Motion of NLS Solitary Waves in a Confining Potential , 2005, math-ph/0503009.