A finite genus solution of the Hirota equation via integrable symplectic maps

Cao, Cungen; Zhang, Guangyao

doi:10.1088/1751-8113/45/9/095203

Cited by 22 publications

(47 citation statements)

References 25 publications

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“…In this paper, we exhibited a new version of the algebro-geometric approach to deal with the partial difference equations of KdV-type, which is different from the existing results in the literatures. 16,41,42 We have presented examples of integrable symplectic maps and finite genus solutions for lattice KdV-type equations. In the lpKdV and lSKdV cases, there are two discrete potentials, and we need to impose constraints between them to construct the algebro-geometric solutions using the technique of nonlinearization.…”

Section: Resultsmentioning

confidence: 99%

“…41, integrable symplectic maps and novel theta function solutions for Equation () were constructed through integrable Hamiltonian systems associated with the continuous sG equation. In contract, in this paper, we start from the Kaup‐Newell spectral problem 45

\partial_{x} χ = V false(λ; v, w false) χ = (\begin{matrix} λ^{2} / 2 & λ v \\ λ w & - λ^{2} / 2 \end{matrix}) χ,

and the associated discrete spectral problem

\tilde{χ} = {(λ^{2} - β^{2})}^{- 1 / 2} D^{(β)} false(λ; a false) χ, D^{(β)} false(λ; a false) = (\begin{matrix} λ a & β \\ β & λ a^{- 1} \end{matrix}) .

In this example, we actually have a different type of situation from the one of the previous examples 16,41,42 since the relation between the discrete potential

a

and the continuous potentials

v, w

is implicit. However, based on the Lax structure of the Kaup‐Newell equation, (), the system can still be nonlinearized, in the sense mentioned above, as an integrable symplectic map.…”

Section: Introductionmentioning

confidence: 99%

“…In contrasts to the usual cases treated in Refs. 16,41,42, where potentials themselves satisfy the corresponding discrete models, the solutions here are expressed in terms of the derivative of a special theta function with respect to the auxiliary Darboux variable.…”

mentioning

confidence: 99%

“…In this example, we actually have a different type of situation from the one of the previous examples 16,41,42 since the relation between the discrete potential and the continuous potentials , is implicit. However, based on the Lax structure of the Kaup-Newell equation, (6), the system can still be nonlinearized, in the sense mentioned above, as an integrable symplectic map.…”

mentioning

confidence: 99%

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Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

Jiang

Nijhoff

2020

Stud Appl Math

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In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg-de Vries-type (KdV-type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.

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Section: Resultsmentioning

confidence: 99%

\partial_{x} χ = V false(λ; v, w false) χ = (\begin{matrix} λ^{2} / 2 & λ v \\ λ w & - λ^{2} / 2 \end{matrix}) χ,

and the associated discrete spectral problem

\tilde{χ} = {(λ^{2} - β^{2})}^{- 1 / 2} D^{(β)} false(λ; a false) χ, D^{(β)} false(λ; a false) = (\begin{matrix} λ a & β \\ β & λ a^{- 1} \end{matrix}) .

In this example, we actually have a different type of situation from the one of the previous examples 16,41,42 since the relation between the discrete potential

a

and the continuous potentials

v, w

is implicit. However, based on the Lax structure of the Kaup‐Newell equation, (), the system can still be nonlinearized, in the sense mentioned above, as an integrable symplectic map.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

Jiang

Nijhoff

2020

Stud Appl Math

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show abstract

“…Adding elliptic information to integrable systems, either to equations or to solutions, is an interesting topic, which brings the study of the integrable systems into a larger area and more insight [26][27][28][29][30][31][32][33]. In this paper, we have shown that the Cauchy matrix approach works for the study of some elliptic integrable systems, i.e.…”

Section: Discussionmentioning

confidence: 99%

The Sylvester equation and the elliptic Korteweg-de Vries system

Sun

Zhang

Nijhoff

2017

Journal of Mathematical Physics

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The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of kM + M k = rc T −gK −1 rc T K −1 where k and K are commutative matrices and obey the matrix relation k 2 = K + 3e 1 I + gK −1 . The obtained elliptic equations, both discrete and continuous, are formulated by the scalar function S (i,j) which is defined using (k, K, M , r, c) and constitute an infinite size symmetric matrix. Lax pairs for both the discrete and continuous system are derived. The explicit solution M of the Sylvester equation and generalized solutions of the obtained elliptic equations are presented according to the canonical forms of matrix k.

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A novel solution to the generalized lattice Liouville equation

Xu,

Yi,

2024

Applied Mathematics Letters

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A finite genus solution of the Hirota equation via integrable symplectic maps

Cited by 22 publications

References 25 publications

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

The Sylvester equation and the elliptic Korteweg-de Vries system

A novel solution to the generalized lattice Liouville equation

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