The hyperelliptic inverse scattering transform for the periodic, defocusing nonlinear Schroedinger equation

Journal Article · Mon Nov 01 00:00:00 EST 1993 · Journal of Computational Physics; (United States)

DOI:https://doi.org/10.1006/jcph.1993.1202· OSTI ID:7281232

Osborne, A R ^[1]

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1. Istituto di Fisica Generale dell'Universita, Torino (Italy)

The nonlinear Schroedinger (NLS) equation describes the spatio-temporal evolution of the complex envelope function of a narrow-banded, nonlinear wave train. Here I exploit the nonlinear Fourier structure of NLS, known as the inverse scattering transform, to study nonlinear periodic modulations in the spectral domain. Numerical algorithms are presented for both the direct and inverse scattering transforms of the [open quotes]defocusing[close quotes] NLS equation with periodic boundary conditions. A discrete algorithm is given for computing the monodromy matrix of periodic spectral theory and the direct scattering transform is then computed from the elements of this matrix. A fast algorithm for the inverse scattering transform in terms of the hyperelliptic function representation is also given; solutions to the defocusing NLS equation are determined by a linear superposition of these nonlinear oscillation modes. The direct algorithm uses computer time proportional to M[sup 2], where M is the number of points in the discrete envelope function of the wave train. The inverse scattering algorithm is [open quotes]fast[close quotes] in the sense that it is M[sup 3], this contrasts to the periodic theta-function inverse problem for NLS which is M[sup 4]. Several wave train solutions of NLS are considered and their inverse scattering transform spectra and nonlinear Fourier decompositions are discussed. Application of the method to the analysis of computer generated or experimentally measured space or time series is the major motivation for this work. 53 refs., 27 figs.

• OSTI ID: 7281232
• Journal Information: Journal of Computational Physics; (United States), Vol. 109:1; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
SCHROEDINGER EQUATION
FOURIER ANALYSIS
DIFFERENTIAL EQUATIONS
EQUATIONS
PARTIAL DIFFERENTIAL EQUATIONS
WAVE EQUATIONS
661100* - Classical & Quantum Mechanics- (1992-)
990200 - Mathematics & Computers