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Solutions of matrix NLS systems and their discretisations: A unified treatment | | Aristophanes Dimakis
; Folkert Muller-Hoissen
; | | Date: |
31 Dec 2009 | | Abstract: | Using a bidifferential graded algebra approach to integrable partial
differential or difference equations, a unified treatment of continuous,
semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is
presented. These equations originate from a universal equation within this
framework, by specifying a representation of the bidifferential graded algebra
and imposing a reduction. By application of a general result, corresponding
families of exact solutions are obtained that in particular comprise the matrix
soliton solutions in the focusing NLS case. The solutions are parametrised in
terms of constant matrix data subject to a Sylvester equation (which previously
appeared as a rank condition in the integrable systems literature). These data
exhibit a certain redundancy, which we diminish to a large extent.
More precisely, we first consider more general AKNS-type systems from which
two different matrix NLS systems emerge via reductions. In the continuous case,
the familiar Hermitian conjugation reduction leads to a continuous matrix
(including vector) NLS equation, but it is well-known that this does not work
as well in the discrete cases. On the other hand there is a complex conjugation
reduction, which apparently has not been studied previously. It leads to square
matrix NLS systems, but works in all three cases (continuous, semi- and
fully-discrete). A large part of this work is devoted to an exploration of the
corresponding solutions, in particular regularity and asymptotic behaviour of
matrix soliton solutions. | | Source: | arXiv, 1001.0133 | | Services: | Forum | Review | PDF | Favorites | |
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