Time-splitting finite difference method with the wavelet-adaptive grids for semiclassical Gross-Pitaevskii equation in supercritical case

Abstract The Gross–Pitaevskii equation is the model equation of the single-particle wave function in a Bose–Einstein condensation. A computation difficulty of the Gross–Pitaevskii equation comes from the semiclassical problem in supercritical case. In this paper, we apply a diffeomorphism to transform the original one-dimensional Gross–Pitaevskii equation into a modified equation. The adaptive grids are constructed through the interpolating wavelet method. Then, we use the time-splitting finite difference method with the wavelet-adaptive grids to solve the modified Gross–Pitaevskii equation, where the approximation to the second-order derivative is given by the Lagrange interpolation method. At last, the numerical results are given. It is shown that the obtained time-splitting finite difference method with the wavelet-adaptive grids is very efficient for solving the one-dimensional semiclassical Gross–Pitaevskii equation in supercritical case and it is suitable to deal with the local high oscillation of the solution.

[1]  W. Bao,et al.  MATHEMATICAL THEORY AND NUMERICAL METHODS FOR , 2012 .

[2]  Gilles Deslauriers,et al.  Symmetric Iterative Interpolation Processes , 1989 .

[3]  Bengt Fornberg,et al.  Classroom Note: Calculation of Weights in Finite Difference Formulas , 1998, SIAM Rev..

[4]  Jie Shen,et al.  A Fourth-Order Time-Splitting Laguerre-Hermite Pseudospectral Method for Bose-Einstein Condensates , 2005, SIAM J. Sci. Comput..

[5]  Rémi Carles,et al.  Semi-Classical Analysis For Nonlinear Schrodinger Equations , 2008 .

[6]  Songhe Song,et al.  Symplectic wavelet collocation method for Hamiltonian wave equations , 2010, J. Comput. Phys..

[7]  T. Cazenave.  Semilinear Schrodinger Equations , 2003 .

[8]  Leland Jameson,et al.  A Wavelet-Optimized, Very High Order Adaptive Grid and Order Numerical Method , 1998, SIAM J. Sci. Comput..

[9]  Leland Jameson,et al.  ON THE WAVELET OPTIMIZED FINITE DIFFERENCE METHOD , 1994 .

[10]  Shanshan Jiang,et al.  Explicit multi-symplectic methods for Klein-Gordon-Schrödinger equations , 2009, J. Comput. Phys..

[11]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[12]  Christian Lubich,et al.  On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations , 2008, Math. Comput..

[13]  Hanquan Wang,et al.  Numerical studies on the split-step finite difference method for nonlinear Schrödinger equations , 2005, Appl. Math. Comput..

[14]  Erwan Faou,et al.  Geometric Numerical Integration and Schrodinger Equations , 2012 .

[15]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[16]  Qianshun Chang,et al.  Finite difference method for generalized Zakharov equations , 1995 .

[17]  E. Hairer,et al.  Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations , 2004 .

[18]  Christof Sparber,et al.  Mathematical and computational methods for semiclassical Schrödinger equations* , 2011, Acta Numerica.

[19]  Naoki Saito,et al.  Multiresolution representations using the autocorrelation functions of compactly supported wavelets , 1993, IEEE Trans. Signal Process..

[20]  G. Hagedorn.  Semiclassical quantum mechanics , 1980 .

[21]  C. Lubich.  From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , 2008 .

[22]  L. Gauckler,et al.  Convergence of a split-step Hermite method for the Gross–Pitaevskii equation , 2011 .

[23]  W. Bao,et al.  Mathematical Models and Numerical Methods for Bose-Einstein Condensation , 2012, 1212.5341.

[24]  Patrick Fischer,et al.  Numerical Solution of the Schrödinger Equation in a Wavelet Basis for Hydrogen-like Atoms , 1998 .

[25]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[26]  R. Carles,et al.  Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation , 2009, 0912.1939.