Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

Ferapontov, E. V.; Novikov, V. S.; Roustemoglou, Ilia

doi:10.1088/1751-8113/46/24/245207

Cited by 9 publications

(23 citation statements)

References 36 publications

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“…Here we skip the long formulas to check a 2 is a symmetry of (33), which can be carried out by organising the terms according to polynomial terms, exponential terms and mixed terms. We can also compare a 2 to the symmetry flows obtained via its Lax representation, which is given in [25,21,14]. We now look at conserved densities for equation (33).…”

Section: The (2+1)-dimensional Volterra Latticementioning

confidence: 99%

“…To verify [a 2 , K] = 0, we can either compute directly or compare it to the symmetry flows obtained via its Lax representation [14].…”

Section: Another (2+1)-dimensional Generalised Volterra Chainmentioning

confidence: 99%

“…In this paper, we give justification of the method explaining where element f is from and give rigorous conditions when [f, u t ] is a master symmetries. We then apply the method to various examples including some new (2 + 1)-dimensional equations presented in [18,14]. Notice that elements e, h and f are all related to linear terms of t in S i , i = 1, 2, 3.…”

Section: Introductionmentioning

confidence: 99%

“…This equation is appeared in [14] when the authors classified a family of equations with the non-locality of intermediate long wave type. In fact, both equations (33) and (35) are listed in the list of integrable equations of this class, and these two equations have coinciding dispersionless limits.…”

mentioning

confidence: 99%

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Representations of sl(2, ℂ) in category O and master symmetries

Wang

2015

Theor Math Phys

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In this paper, we first give a short account on the indecomposable sl(2, C) modules in the Bernstein-Gelfand-Gelfand (BGG) category O. We show these modules naturally arise for homogeneous integrable nonlinear evolutionary systems. We then develop an approach to construct master symmetries for such integrable systems. This method naturally enables us to compute the hierarchy of time-dependent symmetries. We finally illustrate the method using both classical and new examples. We compare our approach to the known existing methods used to construct master symmetries. For the new integrable equations such as a Benjamin-Ono type equation, a new integrable Davey-Stewartson type equation and two different versions of (2+1)-dimensional generalised Volterra Chains, we generate their conserved densities using their master symmetries.

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Section: The (2+1)-dimensional Volterra Latticementioning

confidence: 99%

“…To verify [a 2 , K] = 0, we can either compute directly or compare it to the symmetry flows obtained via its Lax representation [14].…”

Section: Another (2+1)-dimensional Generalised Volterra Chainmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Representations of sl(2, ℂ) in category O and master symmetries

Wang

2015

Theor Math Phys

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confidence: 99%

On the Classification of Discrete Hirota-Type Equations in 3D

Ferapontov

Novikov

Roustemoglou

2014

Int Math Res Notices

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In the series of recent publications [14,15,17,20] we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are 'inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.MSC: 35Q51, 37K10.

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Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

Bury

Михайлов

Wang

2017

Physica D: Nonlinear Phenomena

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In the paper we develop the dressing method for the solution of the two-dimensional periodic Volterra system with a period N . We derive soliton solutions of arbitrary rank k and give a full classification of rank 1 solutions. We have found a new class of exact solutions corresponding to wave fronts which represent smooth interfaces between two nonlinear periodic waves or a periodic wave and a trivial (zero) solution. The wave fronts are non-stationary and they propagate with a constant average velocity. The system also has soliton solutions similar to breathers, which resembles soliton webs in the KP theory. We associate the classification of soliton solutions with the Schubert decomposition of the Grassmanians Gr R (k, N ) and Gr C (k, N ).

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Towards the classification of integrable differential–difference equations in 2 + 1 dimensions

Cited by 9 publications

References 36 publications

Representations of sl(2, ℂ) in category O and master symmetries

Representations of sl(2, ℂ) in category O and master symmetries

On the Classification of Discrete Hirota-Type Equations in 3D

Wave fronts and cascades of soliton interactions in the periodic two dimensional Volterra system

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