Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–

Date, Etsurō; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji

doi:10.1143/jpsj.50.3806

Cited by 333 publications

(199 citation statements)

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“…The results obtained with the dressing method guide us to the introduction of the τ -function [22,23] in section 8. We define the τ -functions for the G-CAT models as the orbit of the highest weight state of an integrable representation of the Kac-MoodyĜ obtained under the action of the group element used in the dressing transformation.…”

Section: Introductionmentioning

confidence: 79%

“…Namely, in constructing the general solutions of the model, we have used and, when needed, we have suitably generalized the Leznov-Saveliev method for zero-curvature equations [2,3,15], the dressing-transformation approach [16,17,18,19], and we have also defined the appropriated τ -functions [34,35,24]. The result is a systematic procedure for constructing the one-soliton solutions and an explicit formula for their masses.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

“…The dressing transformations [16,17,18,19] are non-local gauge transformations acting on A µ and leaving its form invariant. Each one of these gauge transformations is performed in two different ways in terms of two different group elements Θ + and Θ − of G (the group whose Lie algebra isĜ)…”

Section: The Dressing Transformationsmentioning

confidence: 99%

“…They are used in the Hirota method to construct exact solutions of soliton equations [34]. In addition, formalisms have been developed to construct soliton equations starting from a given definition of the τ -function [22,23]. The connection between dressing transformations and τ -functions has been explored in the context of the KdV equation [35] and of the generalized integrable hierarchies of the KdV type [24].…”

Section: The τ -Functionsmentioning

confidence: 99%

“…In section 7 we use the dressing transformation method [16,17,18,19] to construct the solutions of the G-CAT model lying in the orbit of the vacuum. The construction is quite simple and leads to a very useful expression for the solutions.…”

Section: Introductionmentioning

confidence: 99%

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Solitons, τ-functions and hamiltonian reduction for non-Abelian conformal affine Toda theories

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We consider the Hamiltonian reduction of the two-loop Wess-Zumino-Novikov-Witten model (WZNW) based on an untwisted affine Kac-Moody algebraĜ. The resulting reduced models, called Generalized Non-Abelian Conformal Affine Toda (G-CAT), are conformally invariant and a wide class of them possesses soliton solutions; these models constitute non-abelian generalizations of the Conformal Affine Toda models. Their general solution is constructed by the Leznov-Saveliev method. Moreover, the dressing transformations leading to the solutions in the orbit of the vacuum are considered in detail, as well as the τ -functions, which are defined for any integrable highest weight representation ofĜ, irrespectively of its particular realization. When the conformal symmetry is spontaneously broken, the G-CAT model becomes a generalized Affine Toda model, whose soliton solutions are constructed. Their masses are obtained exploring the spontaneous breakdown of the conformal symmetry, and their relation to the fundamental particle masses is discussed.

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Section: Introductionmentioning

confidence: 79%

Section: Conclusion and Discussionmentioning

confidence: 99%

Section: The Dressing Transformationsmentioning

confidence: 99%

Section: The τ -Functionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solitons, τ-functions and hamiltonian reduction for non-Abelian conformal affine Toda theories

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Symmetries and Integrability

1987

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Integrable nonlinear evolution equations in one-spatial and one-temporal dimensions possess a remarkably rich algebraic structure: Infinitely many symmetries and conserved quantities, existence of a bi-Hamiltonian formulation etc. A certain operator ~, called recursion operator, plays a central role in investigating the above algebraic properties. Recently the above theory has been extended to equations in two spatial and one temporal dimensions. In particular, the multidimensional analogue of the operator ~ has been found and its bi-Hamiltonian factorization has been established. The above and other aspects of integrable equations, including mastersymmetries, are reviewed in this manuscript. Furthermore, a definition of integrability is proposed based on symmetry considerations.

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Quasifinite Representations of Classical Lie Subalgebras of 1+∞

Kač¹,

Wang²,

Yan³

1998

Advances in Mathematics

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We show that there are precisely two, up to conjugation, anti-involutions _ \ of the algebra of differential operators on the circle preserving the principal gradation. We classify the irreducible quasifinite highest weight representations of the central extension D \ of the Lie subalgebra of this algebra fixed by &_ \ , and find the unitary ones. We realize them in terms of highest weight representations of the central extension of the Lie algebra of infinite matrices with finitely many non-zero diagonals over the algebra C[u]Â(u m+1 ) and its classical Lie subalgebras of B, C and D types. Character formulas for positive primitive representations of D \ (including all the unitary ones) are obtained. We also realize a class of primitive representations of D \ in terms of free fields and establish a number of duality results between these primitive representations and finite-dimensional irreducible representations of finitedimensional Lie groups and supergroups. We show that the vacuum module V c of D + carries a vertex algebra structure and establish a relationship between V c for c # 1 2 Z and W-algebras. Academic Press

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Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–

Cited by 333 publications

References 3 publications

Solitons, τ-functions and hamiltonian reduction for non-Abelian conformal affine Toda theories

Solitons, τ-functions and hamiltonian reduction for non-Abelian conformal affine Toda theories

Symmetries and Integrability

Quasifinite Representations of Classical Lie Subalgebras of 1+∞

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