A differential quadrature algorithm for nonlinear Schrödinger equation

Numerical solutions of a nonlinear Schrödinger equation is obtained using the differential quadrature method based on polynomials for space discretization and Runge–Kutta of order four for time discretization. Five well-known test problems are studied to test the efficiency of the method. For the first two test problems, namely motion of single soliton and interaction of two solitons, numerical results are compared with earlier works. It is shown that results of other test problems agrees the theoretical results. The lowest two conserved quantities and their relative changes are computed for all test examples. In all cases, the differential quadrature Runge–Kutta combination generates numerical results with high accuracy.

[1]  T. Taha,et al.  Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation , 1984 .

[2]  J. C. Newby,et al.  A finite-difference method for solving the cubic Schro¨dinger equation , 1997 .

[3]  L. Gardner,et al.  B-spline finite element studies of the non-linear Schrödinger equation , 1993 .

[4]  Chang Shu,et al.  Application of GDQ Method for Study of Mixed Convection in Horizontal Eccentric Annuli , 2004 .

[5]  John W. Miles,et al.  An Envelope Soliton Problem , 1981 .

[6]  Richard Bellman,et al.  Differential quadrature and splines , 1975 .

[7]  Michel C. Delfour,et al.  Finite-difference solutions of a non-linear Schrödinger equation , 1981 .

[8]  V. Zakharov,et al.  Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media , 1970 .

[9]  Ö. Civalek.  Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns , 2004 .

[10]  B. Shizgal,et al.  Spectral convergence of the quadrature discretization method in the solution of the Schrodinger and Fokker-Planck equations: comparison with sinc methods. , 2006, The Journal of chemical physics.

[11]  Heli Chen,et al.  The Quadrature Discretization Method (QDM) in comparison with other numerical methods of solution of the Fokker–Planck equation for electron thermalization , 1998 .

[12]  Ghodrat Karami,et al.  Polynomial and harmonic differential quadrature methods for free vibration of variable thickness thick skew plates , 2005 .

[13]  A. Scott,et al.  The soliton: A new concept in applied science , 1973 .

[14]  C. Shu,et al.  Numerical simulation of natural convection between two elliptical cylinders using DQ method , 2004 .

[15]  Chang Shu,et al.  EXPLICIT COMPUTATION OF WEIGHTING COEFFICIENTS IN THE HARMONIC DIFFERENTIAL QUADRATURE , 1997 .

[16]  Ben M. Herbst,et al.  Numerical Experience with the Nonlinear Schrödinger Equation , 1985 .

[17]  Hongzhi Zhong,et al.  Spline-based differential quadrature for fourth order differential equations and its application to Kirchhoff plates , 2004 .

[18]  John Ll Morris,et al.  An investigation into the effect of product approximation in the numerical solution of the cubic nonlinear Schro¨dinger equation , 1988 .

[19]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[20]  Chang Shu,et al.  Integrated radial basis functions‐based differential quadrature method and its performance , 2007 .

[21]  L. Ostrovsky,et al.  PROPAGATION OF MODULATED WAVES IN NONLINEAR DISPERSIVE MEDIA , 1967 .

[22]  Heli Chen,et al.  The Quadrature Discretization Method (QDM) in the solution of the Schrödinger equation , 1998 .

[23]  John Argyris,et al.  An engineer's guide to soliton phenomena: Application of the finite element method , 1987 .

[24]  Ö. Civalek.  Harmonic differential quadrature-finite differences coupled approaches for geometrically nonlinear static and dynamic analysis of rectangular plates on elastic foundation , 2006 .

[25]  Chuei-Tin Chang,et al.  New insights in solving distributed system equations by the quadrature method—I. Analysis , 1989 .

[26]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[27]  C. Shu,et al.  APPLICATION OF GENERALIZED DIFFERENTIAL QUADRATURE TO SOLVE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1992 .

[28]  Xinwei Wang,et al.  Harmonic differential quadrature method and applications to analysis of structural components , 1995 .

[29]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[30]  İdris Dağ,et al.  A quadratic B-spline finite element method for solving nonlinear Schrödinger equation , 1999 .

[31]  Heli Chen,et al.  The quadrature discretization method (QDM) in the solution of the Schrödinger equation with nonclassical basis functions , 1996 .