The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra

Mas, Javier; Seco, M.

doi:10.1063/1.531745

Cited by 32 publications

(25 citation statements)

References 45 publications

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“…The author (Jingsong He) would like to thank the Centre for Scientific Computing and University of Warwick for supporting him to visit there. Special thanks go to Dr. Rudolf A. Römer at Warwick for the numerous helpful discussions, Dr. P. Iliev for explaining few parts of his work [20] and Professor J. Mas for answering questions on his paper [18]. Jingsong He is also grateful to Professors F. Calogero, A. Degasperis, D. Levi and P.M. Santini of University of Rome "La Sapienza" for their hospitality during his visit to Rome.…”

Section: Acknowledgementsmentioning

confidence: 99%

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q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

He¹

2006

SIGMA

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Abstract. Using the determinant representation of gauge transformation operator, we have shown that the general form of τ function of the q-KP hierarchy is a q-deformed generalized Wronskian, which includes the q-deformed Wronskian as a special case. On the basis of these, we study the q-deformed constrained KP (q-cKP) hierarchy, i.e. l-constraints of q-KP hierarchy. Similar to the ordinary constrained KP (cKP) hierarchy, a large class of solutions of q-cKP hierarchy can be represented by q-deformed Wronskian determinant of functions satisfying a set of linear q-partial differential equations with constant coefficients. We obtained additional conditions for these functions imposed by the constraints. In particular, the effects of q-deformation (q-effects) in single q-soliton from the simplest τ function of the q-KP hierarchy and in multi-q-soliton from one-component q-cKP hierarchy, and their dependence of x and q, were also presented. Finally, we observe that q-soliton tends to the usual soliton of the KP equation when x → 0 and q → 1, simultaneously.

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Section: Acknowledgementsmentioning

confidence: 99%

“…The second type is the q-KP hierarchy [17,22]. Its τ function, bi-Hamiltonian structure and additional symmetries have already been reported in [20,21,18,22]. The third type is the q-AKNS-D hierarchy, and its bilinear identity and τ function were obtained in [23].…”

Section: Introductionmentioning

confidence: 99%

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

He¹

2006

SIGMA

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“…The q-deformation of classical nonlinear integrable system started in 1990's by means of q-derivative ∂ q instead of usual derivative with respect spatial variable in the classical system. Several q-deformed integrable systems have been presented, for example the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy is a subject of intensive study in the literatures [16]- [14]. The q-Toda equation was also studied in [17,21] but not for a whole hierarchy.…”

Section: Introductionmentioning

confidence: 99%

Integrability on generalized q-Toda equation and hierarchy

Meng

Huang

2021

JNMP

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In this paper, we construct a new integrable equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized q-Toda equation and a whole integrable generalized q-Toda hierarchy are also constructed. To show the integrability, the Bi-Hamiltonian structure and tau symmetry of the generalized q-Toda hierarchy are given and this leads to the tau function.

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“…Several q-deformed integrable systems have been presented, for example the q-deformed Kadomtsev-Petviashvili (q-KP) hierarchy is a subject of intensive study in the literature [17]- [24]. Basing on a similar q-operator as q-KP hierarchy in [20,21], the q-Toda equation was studied in [25,26] but not for a whole hierarchy.…”

Section: Introductionmentioning

confidence: 99%

Sato theory on the q-Toda hierarchy and its extension

2015

Chaos, Solitons & Fractals

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In this paper, we construct the Sato theory including the Hirota bilinear equations and tau function of a new q-deformed Toda hierarchy(QTH). Meanwhile the Block type additional symmetry and bi-Hamiltonian structure of this hierarchy are given. From Hamiltonian tau symmetry, we give another definition of tau function of this hierarchy. Afterwards, we extend the q-Toda hierarchy to an extended q-Toda hierarchy(EQTH) which satisfy a generalized Hirota quadratic equation in terms of generalized vertex operators. The Hirota quadratic equation might have further application in Gromov-Witten theory. The corresponding Sato theory including multi-fold Darboux transformations of this extended hierarchy is also constructed. At last, we construct the multicomponent extension of the q-Toda hierarchy and show the integrability including its bi-Hamiltonian structure, tau symmetry and conserved densities.

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The algebra of q-pseudodifferential symbols and the q-WKP(n) algebra

Cited by 32 publications

References 45 publications

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

q-Deformed KP Hierarchy and q-Deformed Constrained KP Hierarchy

Integrability on generalized q-Toda equation and hierarchy

Sato theory on the q-Toda hierarchy and its extension

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