Transformation Groups for Soliton Equations – Euclidean Lie Algebras and Reduction of the KP Hierarchy –

“…Finally we remark that the G R[w] -action discussed in sections 6 and 10 is very reminiscient of the generation of solutions to the KdV hierarchies through dressing transformations, [6,12,19]. The construction involves a vertex operator representation of a Kac Moody algebra, and generates tau-functions from the vacuum.…”

Section: Discussionmentioning

confidence: 99%

Co-adjoint orbits of the generalised S1(2), S1(3) KdV hierarchies

Burroughs¹

1992

Nuclear Physics B

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In this paper we develop two coadjoint orbit constructions for the phase spaces of the generalised Sl(2) and Sl(3) KdV hierachies. This involves the construction of two group actions in terms of Yang Baxter operators, and an Hamiltonian reduction of the coadjoint orbits. The Poisson brackets are reproduced by the Kirillov construction. From this construction we obtain a 'natural' gauge fixing proceedure for the generalised hierarchies.

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“…In particular, for n = 1 this is the principal realization [KKLW], and for m = 1 this is the homogeneous realization [FK]. The m-th reduced 1-component KP was studied in a great detail in [DJKM2] (see also [KP2]). …”

Section: 1mentioning

confidence: 99%

“…As shown in [S], [DJKM2], the m-th reduction of the KP hiearchy, i.e. the requirement that L m is a differential operator, leads to the classical formulation of the celebrated KdV hierarchy for m = 2, Boussinesq for m = 3 and all the Gelfand-Dickey hierarchies for m > 3.…”

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confidence: 99%

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The n-component KP hierarchy and representation theory

Kač

Leur

2003

Journal of Mathematical Physics

132

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It is the aim of the present article to give all formulations of the n-component KP hierarchy and clarify connections between them. The generalization to the n-component KP hierarchy is important because it contains many of the most popular systems of soliton equations, like the Davey–Stewartson system (for n=2), the two-dimensional Toda lattice (for n=2), the n-wave system (for n⩾3) and the Darboux–Egoroff system. It also allows us to construct natural generalizations to the Davey–Stewartson and Toda lattice systems. Of course, the inclusion of all these systems in the n-component KP hierarchy allows us to construct their solutions by making use of vertex operators.

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Transformation Groups for Soliton Equations – Euclidean Lie Algebras and Reduction of the KP Hierarchy –

Cited by 244 publications

References 15 publications

Note on the Two-Component Analogue of Two-Dimensional Long Wave – Short Wave Resonance Interaction System

Note on the Two-Component Analogue of Two-Dimensional Long Wave – Short Wave Resonance Interaction System

Co-adjoint orbits of the generalised S1(2), S1(3) KdV hierarchies

The n-component KP hierarchy and representation theory

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