Numerical solution of the nonlinear Schrödinger equation, starting from the scattering data

Starting from the scattering data, the initial-value problem for the focusing nonlinear Schrödinger equation is solved numerically by following the path of the inverse scattering transform. The numerical results of an extensive experimentation suggest that: (a) our method is very effective, whenever the scattering data are analytically known; (b) the split-step Fourier method is not really effective if the exact solution is not smooth enough.

[1]  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.

[2]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

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

[4]  H. Yuen,et al.  Nonlinear deep water waves: Theory and experiment , 1975 .

[5]  Mark P. Robinson,et al.  The solution of nonlinear Schrödinger equations using orthogonal spline collocation , 1997 .

[6]  Thiab R. Taha,et al.  Parallel Split-Step Fourier Methods for Nonlinear Schrödinger-Type Equations , 2003, J. Math. Model. Algorithms.

[7]  W. Ferguson,et al.  Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train , 1977, Journal of Fluid Mechanics.

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

[9]  C. Mee.  Direct and Inverse Scattering for Skewselfadjoint Hamiltonian Systems , 2004 .

[10]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[11]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .

[12]  Akira Hasegawa,et al.  Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion , 1973 .

[13]  Akira Hasegawa,et al.  Optical solitons in fibers , 1993, International Commission for Optics.

[14]  Tuncay Aktosun,et al.  Exact solutions to the focusing nonlinear Schrödinger equation , 2007 .

[15]  B. Herbst,et al.  Split-step methods for the solution of the nonlinear Schro¨dinger equation , 1986 .

[16]  William L. Briggs,et al.  The DFT : An Owner's Manual for the Discrete Fourier Transform , 1987 .

[17]  G. Fairweather,et al.  On the numerical solution of the cubic Schro¨dinger equation in one space variable , 1993 .

[18]  J. Helton,et al.  Current trends in operator theory and its applications , 2004 .

[19]  C. Mee,et al.  Structured matrix numerical solution of the nonlinear Schrodinger equation by the inverse scattering transform , 2009 .

[20]  Tuncay Aktosun,et al.  Direct and inverse scattering for selfadjoint Hamiltonian systems on the line , 2000 .

[21]  R. H. Hardin.  Application of the split-step Fourier method to the numerical solution of nonlinear and variable coefficient wave equations , 1973 .

[22]  J. K. Shaw,et al.  On the Eigenvalues of Zakharov-Shabat Systems , 2003, SIAM J. Math. Anal..

[23]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[24]  M. Ablowitz,et al.  On the solution of a class of nonlinear partial di erence equations , 1977 .

[25]  Graeme Fairweather,et al.  Orthogonal spline collocation methods for Schr\"{o}dinger-type equations in one space variable , 1994 .

[26]  Mark J. Ablowitz,et al.  A Nonlinear Difference Scheme and Inverse Scattering , 1976 .

[27]  Thiab R. Taha,et al.  Analytical and numerical aspects of certain nonlinear evolution equations. I. Analytical , 1984 .