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Inverse Scattering Transform (inverse + scattering_transform)

Distribution by Scientific Domains
Mathematics and Statistics100%

Selected Abstracts

The Schrödinger equation and a multidimensional inverse scattering transform

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 16-18 2002
Swanhild Bernstein
Abstract The Schrödinger equation is one of the most important equations in mathematics, physics and also engineering. We outline some connections between transformations of non-linear equations, the Schrödinger equation and the inverse scattering transform. After some remarks on generalizations into higher dimensions we present a multidimensional ,¯ method based on Clifford analysis. To explain the method we consider the formal solution of the inverse scattering problem for the n -dimensional time-dependent Schrödinger equations given by A.I. Nachman and M.J. Ablowitz. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Soliton solutions of the Toda hierarchy on quasi-periodic backgrounds revisited

MATHEMATISCHE NACHRICHTEN, Issue 4 2009
Iryna Egorova
Abstract We investigate soliton solutions of the Toda hierarchy on a quasi-periodic finite-gap background by means of the double commutation method and the inverse scattering transform. In particular, we compute the phase shift caused by a soliton on a quasi-periodic finite-gap background. Furthermore, we consider short range perturbations via scattering theory. We give a full description of the effect of the double commutation method on the scattering data and establish the inverse scattering transform in this setting (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Unified approach to KdV modulations

COMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 10 2001
Gennady A. El
We develop a unified approach to integrating the Whitham modulation equations. Our approach is based on the formulation of the initial-value problem for the zero-dispersion KdV as the steepest descent for the scalar Riemann-Hilbert problem [6] and on the method of generating differentials for the KdV-Whitham hierarchy [9]. By assuming the hyperbolicity of the zero-dispersion limit for the KdV with general initial data, we bypass the inverse scattering transform and produce the symmetric system of algebraic equations describing motion of the modulation parameters plus the system of inequalities determining the number the oscillating phases at any fixed point on the (x, t)-plane. The resulting system effectively solves the zero-dispersion KdV with an arbitrary initial datum. © 2001 John Wiley & Sons, Inc. [source]