Solitons and Infinite Dimensional Lie Algebras

Jimbo, Michio; Miwa, Tetsuji

doi:10.2977/prims/1195182017

Cited by 1,394 publications

(1,224 citation statements)

References 11 publications

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“…The eigenfunction φ and the adjoint eigenfunction ψ of the KP hierarchy associated with equation (4) are defined by ∂φ ∂t n = B n φ, ∂ψ ∂t n = −B * n ψ,…”

Section: Introductionmentioning

confidence: 99%

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Cheng

Römer

2006

J. High Energy Phys.

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Abstract. We present a systematic way to construct solutions of the (n = 5)-reduction of the BKP and CKP hierarchies from the general τ function τ (n+k) of the KP hierarchy. We obtain the one-soliton, two-soliton, and periodic solution for the bi-directional Sawada-Kotera (bSK), the bi-directional Kaup-Kupershmidt (bKK) and also the bi-directional Satsuma-Hirota (bSH) equation. Different solutions such as left-and right-going solitons are classified according to the symmetries of the 5th roots of e iε . Furthermore, we show that the soliton solutions of the nreduction of the BKP and CKP hierarchies with n = 2j + 1, j = 1, 2, 3, . . ., can propagate along j directions in the 1 + 1 space-time domain. Each such direction corresponds to one symmetric distribution of the nth roots of e iε . Based on this classification, we detail the existence of two-peak solitons of the n-reduction from the Grammian τ function of the sub-hierarchies BKP and CKP. If n is even, we again find two-peak solitons. Last, we obtain the "stationary" soliton for the higher-order KP hierarchy.

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“…The eigenfunction φ and the adjoint eigenfunction ψ of the KP hierarchy associated with equation (4) are defined by ∂φ ∂t n = B n φ, ∂ψ ∂t n = −B * n ψ,…”

Section: Introductionmentioning

confidence: 99%

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Cheng

Römer

2006

J. High Energy Phys.

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“…Subsequently [2,3], it was shown that the KP equation is amenable to the Inverse Scattering Transform (IST) method and hence possesses all the remarkable properties which are commonly associated with soliton equations (see, e.g., [4,5,6]). In particular, there exist several infinite families of soliton equations which are related to the KP equation such as the KP, modified KP (mKP) and Schwarzian KP (SKP) hierarchies.…”

Section: Introductionmentioning

confidence: 99%

Menelaus$apos$ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

Konopelchenko¹,

Schief

2002

J. Phys. A: Math. Gen.

142

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It is shown that the integrable discrete Schwarzian KP (dSKP) equation which constitutes an algebraic superposition formula associated with, for instance, the Schwarzian KP hierarchy, the classical Darboux transformation and quasi-conformal mappings encapsulates nothing but a fundamental theorem of ancient Greek geometry. Thus, it is demonstrated that the connection with Menelaus' theorem and, more generally, Clifford configurations renders the dSKP equation a natural object of inversive geometry on the plane. The geometric and algebraic integrability of dSKP lattices and their reductions to lattices of Menelaus-Darboux, Schwarzian KdV, Schwarzian Boussinesq and Schramm type is discussed. The dSKP and discrete Schwarzian Boussinesq equations are shown to represent discretizations of families of quasi-conformal mappings.

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“…This together with (2.17) makes it possible to retrieve [27,29] the related evolution of elements l n ∈ G n , n ∈ Z N :…”

Section: Integrability Analysis: the Gradient-holonomic Schemementioning

confidence: 99%

Isospectral integrability analysis of dynamical systems on discrete manifolds

Blackmore¹,

Prykarpatsky²,

Prykarpatsky³

2012

OpMath

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Solitons and Infinite Dimensional Lie Algebras

Cited by 1,394 publications

References 11 publications

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Solving bi-directional soliton equations in the KP hierarchy by gauge transformation

Menelaus$apos$ theorem, Clifford configurations and inversive geometry of the Schwarzian KP hierarchy

Isospectral integrability analysis of dynamical systems on discrete manifolds

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