Transformation groups for soliton equations

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

doi:10.1016/0167-2789(82)90041-0

Cited by 595 publications

(921 citation statements)

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“…In the case that DP S is the set of all strict partitions, namely, DP, the series (85) and (86) may be considered as multi-variable generalization of hypergeometric function (respectively, basic hypergeometric function), which we obtain when (λ) = 1 and t o is of form (2) where…”

Section: Useful Lemma Let Us Introduce the Following Notationmentioning

confidence: 99%

“…Introduction. The BKP hierarchy was introduced in [1,2] as a particular reduction of the KP hierarchy of integrable equations [1,7]. Like the well-known KP hierarchy, the BKP hierarchy possesses multi-soliton and rational solutions.…”

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confidence: 99%

“…We note that a special case of this series (1), where times were chosen as in (2), and e ξ m was chosen as a step function, was considered in [16] in a combinatorial context, not related to integrable systems.…”

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confidence: 99%

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A Relationship Between Rational and Multi-Soliton Solutions of the BKP Hierarchy

Nimmo

Orlov

2005

Glasgow Math. J.

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Abstract. We consider a special class of solutions of the BKP hierarchy which we call τ -functions of hypergeometric type. These are series in Schur Q-functions over partitions, with coefficients parameterised by a function of one variable ξ , where the quantities ξ (k), k ∈ ‫ޚ‬ + , are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables ξ (k) are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur Q-function expansion of the τ -function of hypergeometric type. We also show that the KdV and the NLS soliton τ -functions coinside the BKP τ -functions of hypergeometric type, evaluated at special point of BKP higher time; the variables ξ (which are BKP integrals of motions) being related to KdV and NLS higher times.2000 Mathematics Subject Classification. 35Q51, 35Q58, 05E05.1. Introduction. The BKP hierarchy was introduced in [1, 2] as a particular reduction of the KP hierarchy of integrable equations [1,7]. Like the well-known KP hierarchy, the BKP hierarchy possesses multi-soliton and rational solutions. In [3,4], the role of projective Schur functions (Q-functions) [6] in obtaining rational solutions of the BKP hierarchy was explained. In fact, the Q-functions are polynomial τ -function solutions of the BKP hierarchy Hirota equations and these are connected to the rational solutions through a change of dependent variables.In [9], certain hypergeometric series in Q-functions (see (85) below) were shown to be τ -functions of the BKP hierarchy. These τ -functions are series of the form

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Section: Useful Lemma Let Us Introduce the Following Notationmentioning

confidence: 99%

mentioning

confidence: 99%

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A Relationship Between Rational and Multi-Soliton Solutions of the BKP Hierarchy

Nimmo

Orlov

2005

Glasgow Math. J.

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“…The basic purpose of them is to construct new solitary wave solutions and periodic solutions. (2 + 1) dimensional Boussinesq and KdomtsevPetviashvili (BKP) equation is an important nonlinear partial differential equation in mathematical physics, which had been mentioned in literatures [14][15][16]. The equation belongs to a symmetry and integrable system.…”

Section: Introductionmentioning

confidence: 99%

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

Xiang¹

2011

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Abstract(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.

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“…Schur polynomials appear not only in representation theory but also in many contexts in mathematical physics, especially in integrable systems. For example, the tau functions of the KP hierarchy have Schur polynomial expansions [2]. Schur polynomials also appear as the singular vectors in conformal field theory [3,4], the Green function of the vicious walkers [5,6], the domain wall boundary partition function of the six vertex model [7,8], the Schur processes as one of the most fundamental examples of determinantal processes [9], to list a few.…”

Section: Introductionmentioning

confidence: 99%

Vertex models, TASEP and Grothendieck polynomials

Motegi

Sakai

2013

J. Phys. A: Math. Theor.

106

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We examine the wavefunctions and their scalar products of a one-parameter family of integrable five vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on/off-shell wavefunctions of the five vertex models are represented as a certain determinant form. Up to some normalization factors, we find the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of Grothendieck polynomials.

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Transformation groups for soliton equations

Cited by 595 publications

References 4 publications

A Relationship Between Rational and Multi-Soliton Solutions of the BKP Hierarchy

A Relationship Between Rational and Multi-Soliton Solutions of the BKP Hierarchy

Jacobi Elliptic Function Solutions for (2 + 1) Dimensional Boussinesq and Kadomtsev-Petviashvili Equation

Vertex models, TASEP and Grothendieck polynomials

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