Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

Garifullin, Rustem N.; Gubbiotti, Giorgio; Yamilov, R. I.

doi:10.1080/14029251.2019.1613050

Cited by 16 publications

(40 citation statements)

References 40 publications

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“…It is shown in [12] that that equation is Darboux integrable by constructing the first integrals in both directions, and its general solution has been found. The case = 2 is known, see equation (51a) in [2]. The first integrals in both directions have been found there for this equation, see relations (53) in [2].…”

Section: Darboux Integrabilitymentioning

confidence: 90%

“…The case = 2 is known, see equation (51a) in [2]. The first integrals in both directions have been found there for this equation, see relations (53) in [2].…”

Section: Darboux Integrabilitymentioning

confidence: 90%

“…which provides first integrals for all the equations (2). It is easy to see that these integrals (9) and (11) have the orders 1 and 3 , respectively.…”

Section: Darboux Integrabilitymentioning

confidence: 99%

“…For this reason, for any equation of system (16), we can use the first integral 2,1 of equation (2). As a result we get for those equations:…”

Section: Darboux Integrabilitymentioning

confidence: 99%

“…Two integrable equations of this form are known. The case = 1 is presented in [12] and the case = −1 has been found in [2]. Equations in both the cases are Darboux integrable.…”

Section: Introductionmentioning

confidence: 99%

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On series of Darboux integrable discrete equations on square lattice

Garifullin

Yamilov

2019

Ufimsk. Mat. Zh.

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We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers . All the equations have a first integral of the first order in one of directions of the two-dimensional lattice. The minimal order of a first integral in the other direction is equal to 3 for an equation with the number .In the cases = 1, 2, 3 we show that those equations are integrable in quadratures. More precisely, we construct their general solutions in terms of the discrete integrals.We also construct a modified series of Darboux integrable discrete equations which have in different directions the first integrals of the orders 2 and 3 −1, where is the equation number in series. Both first integrals are unobvious in this case.

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Section: Darboux Integrabilitymentioning

confidence: 90%

“…The case = 2 is known, see equation (51a) in [2]. The first integrals in both directions have been found there for this equation, see relations (53) in [2].…”

Section: Darboux Integrabilitymentioning

confidence: 90%

“…which provides first integrals for all the equations (2). It is easy to see that these integrals (9) and (11) have the orders 1 and 3 , respectively.…”

Section: Darboux Integrabilitymentioning

confidence: 99%

“…For this reason, for any equation of system (16), we can use the first integral 2,1 of equation (2). As a result we get for those equations:…”

Section: Darboux Integrabilitymentioning

confidence: 99%

“…Two integrable equations of this form are known. The case = 1 is presented in [12] and the case = −1 has been found in [2]. Equations in both the cases are Darboux integrable.…”

Section: Introductionmentioning

confidence: 99%

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On series of Darboux integrable discrete equations on square lattice

Garifullin

Yamilov

2019

Ufimsk. Mat. Zh.

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Perturbative Symmetry Approach for Differential–Difference Equations

Михайлов

Novikov

Wang

2022

Commun. Math. Phys.

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We propose a new method to tackle the integrability problem for evolutionary differential–difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is integrable or not, and to advance in classification of integrable equations. We define and develop symbolic representation for the difference polynomial ring, difference operators and formal series. In order to formulate necessary integrability conditions, we introduce a novel quasi-local extension of the difference ring. We apply the developed formalism to solve the classification problem of integrable equations for anti-symmetric quasi-linear equations of order $$(-3,3)$$ ( - 3 , 3 ) and produce a list of 17 equations satisfying the necessary integrability conditions. For every equation from the list we present an infinite family of integrable higher order relatives. Some of the equations obtained are new.

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Integrability of Nonabelian Differential–Difference Equations: The Symmetry Approach

Novikov,

Wang

2024

Commun. Math. Phys.

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We extend the approach proposed in Mikhailov et al. (Commun Math Phys 393:1063–1104, 2022) to tackle the integrability problem for evolutionary differential–difference equations (D$$\Delta $$ Δ Es) on free associative algebras, also referred to as nonabelian D$$\Delta $$ Δ Es. This approach enables us to derive necessary integrability conditions, determine the integrability of a given equation, and make progress in the classification of integrable nonabelian D$$\Delta $$ Δ Es. This work involves establishing symbolic representations for the nonabelian difference algebra, difference operators, and formal series, as well as introducing a quasi-local extension for the algebra of formal series within the context of symbolic representations. Applying this formalism, we solve the classification problem of integrable skew-symmetric quasi-linear nonabelian equations of orders $$(-1,1)$$ ( - 1 , 1 ) , $$(-2,2)$$ ( - 2 , 2 ) , and $$(-3,3)$$ ( - 3 , 3 ) , consequently revealing some new equations in the process.

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Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations

Cited by 16 publications

References 40 publications

On series of Darboux integrable discrete equations on square lattice

On series of Darboux integrable discrete equations on square lattice

Perturbative Symmetry Approach for Differential–Difference Equations

Integrability of Nonabelian Differential–Difference Equations: The Symmetry Approach

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