A Dressing Method in Mathematical PhysicsGenerating solutions via ∂ problem

A Dressing Method in Mathematical Physics: Generating solutions via ∂ problem This chapter is devoted to a brief exposition of the ∂ formalism, as applied to nonlinear equations. The ﬁrst three sections deal with the so-called non- linear equations with self-consistent sources (or with nonanalytic dispersion relations ). This class of nonlinear equations is physically interesting because nonanalytic dispersion relations are directly associated with the resonant in- teraction of radiation with matter. In Sects. 10.1 and 10.2 we consider the (1+1)-dimensional nonlinear Schrod ¨ inger (NLS) and modiﬁed NLS equations with self-consistent sources, respectively, along with their gauge equivalents, while Sect. 10.3 is devoted to the Davey–Stewartson I equation with a non- analytic dispersion relation. We analyze these equations by means of the ∂ approach. It should be noted that the Riemann–Hilbert (RH) problem could be applied as well for this aim but, in our opinion, the ∂ approach is frequently the most transparent and leads directly to the ﬁnal results. In the ﬁrst three sections, the ∂ formalism is outlined in a rather unusual setting, but we prove its usefulness for practical calculations. The last two sections comprise examples of nonlinear equations where using the ∂ problem is necessary. Namely, in Sect. 10.4 we consider the Kadomtsev–Petviashvili II (KP http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsGenerating solutions via ∂ problem

Part of the Mathematical Physics Studies Book Series (volume 28)
34 pages

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Publisher
Springer Netherlands
ISBN
978-1-4020-6138-7
Pages
319 –353
DOI
10.1007/1-4020-6140-4_10
Publisher site
See Chapter on Publisher Site

Abstract

This chapter is devoted to a brief exposition of the ∂ formalism, as applied to nonlinear equations. The ﬁrst three sections deal with the so-called non- linear equations with self-consistent sources (or with nonanalytic dispersion relations ). This class of nonlinear equations is physically interesting because nonanalytic dispersion relations are directly associated with the resonant in- teraction of radiation with matter. In Sects. 10.1 and 10.2 we consider the (1+1)-dimensional nonlinear Schrod ¨ inger (NLS) and modiﬁed NLS equations with self-consistent sources, respectively, along with their gauge equivalents, while Sect. 10.3 is devoted to the Davey–Stewartson I equation with a non- analytic dispersion relation. We analyze these equations by means of the ∂ approach. It should be noted that the Riemann–Hilbert (RH) problem could be applied as well for this aim but, in our opinion, the ∂ approach is frequently the most transparent and leads directly to the ﬁnal results. In the ﬁrst three sections, the ∂ formalism is outlined in a rather unusual setting, but we prove its usefulness for practical calculations. The last two sections comprise examples of nonlinear equations where using the ∂ problem is necessary. Namely, in Sect. 10.4 we consider the Kadomtsev–Petviashvili II (KP

Published: Jan 1, 2007

Keywords: Dispersion Relation; Soliton Solution; Spectral Problem; Recursion Operator; Spectral Equation