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A Dressing Method in Mathematical PhysicsDressing in 2+1 dimensions

A Dressing Method in Mathematical Physics: Dressing in 2+1 dimensions In this chapter we speak again about the origin of the dressing technique, now in multidimensions. The important step was realized in the Moutard papers [340, 341] that the stabilization of the Laplace transformation chain can gen- erate solutions. Notice again (see Chap. 1) that the net of points generated by the transform of the invariants of the gauge transformations has two possible symmetry reductions: the first reduction corresponds to the Moutard case and the second one was discovered by Goursat [192, 193]. The dressing procedure in two spatial dimensions opened a way to apply the Laplace equation in Lax pairs to solve some nonlinear 2+1 equations because their associated spectral problems are expressed in terms of the Laplace equation. The celebrated 2+1 Kadomtsev–Petviashvili (KP) equation for surface water waves (there are lots of other applications [228]; see Chaps. 9, 10) and the corresponding dressing based on the direct extension of the Darboux the- ory (linear Schrod ¨ inger evolution as the first operator in the Lax pair) [313] have been the subject of intense studies [324]. The dressing methods for the Davey–Stewartson (DS) equation were introduced in [277], where, by means of eight Ablowitz–Kaup–Newell–Segur (AKNS) type pairs, ordinary http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Dressing Method in Mathematical PhysicsDressing in 2+1 dimensions

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Publisher
Springer Netherlands
Copyright
© Springer 2007
ISBN
978-1-4020-6138-7
Pages
141 –160
DOI
10.1007/1-4020-6140-4_5
Publisher site
See Chapter on Publisher Site

Abstract

In this chapter we speak again about the origin of the dressing technique, now in multidimensions. The important step was realized in the Moutard papers [340, 341] that the stabilization of the Laplace transformation chain can gen- erate solutions. Notice again (see Chap. 1) that the net of points generated by the transform of the invariants of the gauge transformations has two possible symmetry reductions: the first reduction corresponds to the Moutard case and the second one was discovered by Goursat [192, 193]. The dressing procedure in two spatial dimensions opened a way to apply the Laplace equation in Lax pairs to solve some nonlinear 2+1 equations because their associated spectral problems are expressed in terms of the Laplace equation. The celebrated 2+1 Kadomtsev–Petviashvili (KP) equation for surface water waves (there are lots of other applications [228]; see Chaps. 9, 10) and the corresponding dressing based on the direct extension of the Darboux the- ory (linear Schrod ¨ inger evolution as the first operator in the Lax pair) [313] have been the subject of intense studies [324]. The dressing methods for the Davey–Stewartson (DS) equation were introduced in [277], where, by means of eight Ablowitz–Kaup–Newell–Segur (AKNS) type pairs, ordinary

Published: Jan 1, 2007

Keywords: Laplace Transformation; Liouville Equation; Gordon Equation; MKdV Equation; Reduction Constraint

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