Construction of q-discrete two-dimensional Toda lattice equation with self-consistent sources

Wang, Hong Yan; Hu, Xing; Tam, Hon Wah

doi:10.2991/jnmp.2007.14.2.8

Cited by 5 publications

(2 citation statements)

References 11 publications

(14 reference statements)

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“…According to the definition of quantum integrals, the Grammian solution (4.26) can be expressed in terms of formal series, too. A similar Gramm-type determinant solution for the bilinear q-2DTL equation (2.4) was reported in [23]. We would like to point out that these two Grammian solutions are equivalent under certain transformations.…”

Section: Quantum Integral Representationsupporting

confidence: 72%

An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

Li¹,

Wang²,

Yao³

et al. 2022

Preprint

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In our previous work [1], we constructed quasi-Casoratian solutions to the noncommutative q-difference two-dimensional Toda lattice (q-2DTL) equation by Darboux transformation, which we can prove produces the existing Casoratian solutions to the bilinear q-2DTL equation obtained by Hirota's bilinear method in commutative setting. It is actually true that one can not only construct solutions to soliton equations but also solutions to their corresponding Bäcklund transformations by their Darboux transformations and binary Darboux transformations. To be more specific, eigenfunctions produced by iterating Darboux transformations and binary Darboux transformations for soliton equations give nothing but determinant solutions to their Bäcklund transformations, individually. This reveals the profound connections between Darboux transformations and Hirota's bilinear method. In this paper, we shall expound this viewpoint in the case of the q-2DTL equation. First, we derive a generalized bilinear Bäcklund transformation and thus a generalized Lax pair for the bilinear q-2DTL equation. And then we successfully construct the binary Darboux transformation for the q-2DTL equation, based on which, Grammian solutions expressed in terms of quantum integrals are established for both the bilinear q-2DTL equation and its bilinear Bäcklund transformation. In the end, by imposing the 2-periodic reductions on the corresponding results of the q-2DTL equation, we derive a qdifference sine-Gordon equation, a modified q-difference sine-Gordon equation and obtain their corresponding solutions.

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Section: Quantum Integral Representationsupporting

confidence: 72%

An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

Li¹,

Wang²,

Yao³

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Very recently, we have proposed a new method called 'source generation procedure' to systematically construct and solve SESCSs [10][11][12]. One of the advantages of this new approach is that SESCSs and their soliton solutions can be generated simultaneously from the procedure.…”

Section: Introductionmentioning

confidence: 99%

New type of Kadomtsev–Petviashvili equation with self-consistent sources and its bilinear Bäcklund transformation

Hu¹,

Wang²

2007

Inverse Problems

Self Cite

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A new type of the KP equation with self-consistent sources (KPESCS) first found by Mel'nikov (1983 Lett. Math. Phys. 7 129–36) is re-constructed via source generation procedure. A new feature of the obtained KPESCS is that we allow y-dependence of the arbitrary constants in the determinantal solution for the KP equation while applying the source generation procedure. We also propose a new idea of commutativity of source generation procedure and Bäcklund transformations to generate a BT for the new KPESCS which indicates the integrability of the KPESCS.

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Interpolation function of the (h,q)-extension of twisted Euler numbers

Özden

Şimşek

2008

Computers & Mathematics with Applications

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Construction of q-discrete two-dimensional Toda lattice equation with self-consistent sources

Cited by 5 publications

References 11 publications

An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

An insight into the $q$-difference two-dimensional Toda lattice equation, $q$-difference sine-Gordon equation and their integrability

New type of Kadomtsev–Petviashvili equation with self-consistent sources and its bilinear Bäcklund transformation

Interpolation function of the (h,q)-extension of twisted Euler numbers

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