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R. I. Yamilov

Publications and source records attributed to R. I. Yamilov.

16 recordsLinked to original sources
arXiv2020

Modified series of integrable discrete equations on a square lattice with a non-standard symmetry structure

In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by using discrete non-point transformations. We use both non-invertible linearizable transformations and non-point transformations invertible on solutions of the discrete equation. As a result, we get several series of new examples of discrete equations along with their generalized symmetries and master symmetries. The generalized symmetries constructed give new integrable examples of five- and seven-point differential-difference equations together with their master symmetries. In the case of discrete equations, the method of constructing non-invertible linearizable transformations by using conservation laws is considered, apparently, for the first time.

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arXiv2019

On a series of Darboux integrable discrete equations on the square lattice

We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers $M$. 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 $3M$ for an equation with the number $M$. In the cases $M=1,\ 2,\ 3$ 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 $3M-1$, where $M$ is the equation number in series. Both first integrals are unobvious in this case.

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arXiv2019

An unusual series of autonomous discrete integrable equations on the square lattice

We present an infinite series of autonomous discrete equations on the square lattice possessing hierarchies of autonomous generalized symmetries and conservation laws in both directions. Their orders in both directions are equal to $κN$, where $κ$ is an arbitrary natural number and $N$ is equation number in the series. Such a structure of hierarchies is new for discrete equations in the case $N>2$. Symmetries and conservation laws are constructed by means of the master symmetries. Those master symmetries are found in a direct way together with generalized symmetries. Such construction scheme seems to be new in the case of conservation laws. One more new point is that, in one of directions, we introduce the master symmetry time into coefficients of discrete equations. In most interesting case $N=2$ we show that a second order generalized symmetry is closely related to a relativistic Toda type integrable equation. As far as we know, this property is very rare in the case of autonomous discrete equations.

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arXiv2018

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

In this paper we construct the autonomous quad-equations which admit as symmetries the five-point differential-difference equations belonging to known lists found by Garifullin, Yamilov and Levi. The obtained equations are classified up to autonomous point transformations and some simple non-autonomous transformations. We discuss our results in the framework of the known literature. There are among them a few new examples of both sine-Gordon and Liouville type equations.

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arXiv2017

On the integrability of a lattice equation with two continuum limits

We study a new example of lattice equation being one of the key equations of a recent generalized symmetry classification of five-point differential-difference equations. This equation has two different continuum limits which are the well-known fifth order partial-differential equations, namely, the Sawada-Kotera and Kaup-Kupershmidt equations. We justify its integrability by constructing an $L-A$ pair and a hierarchy of conservation laws.

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arXiv2017

Classification of five-point differential-difference equations II

Using the generalized symmetry method we finish a classification, started in the article [R.N. Garifullin, R.I. Yamilov and D. Levi, Classification of five-point differential-difference equations, J. Phys. A: Math. Theor. 50 (2017) 125201 (27pp)], of integrable autonomous five-point differential-difference equations. The resulting list, up to autonomous point transformations, contains 14 equations some of which seem to be new. We have found non-autonomous or non-point transformations relating most of the obtained equations among themselves as well as their generalized symmetries.

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arXiv2017

Darboux integrability of trapezoidal $H^{4}$ and $H^{6}$ families of lattice equations I: First integrals

In this paper we prove that the trapezoidal $H^{4}$ and the $H^{6}$ families of quad-equations are Darboux integrable systems. This result sheds light on the fact that such equations are linearizable as it was proved using the Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization for quad equations consistent on the cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with some suggestions on how first integrals can be used to obtain general solutions.

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arXiv2016

On the integrability of a discrete analogue of the Kaup-Kupershmidt equation

We study a new example of equation obtained as a result of a recent generalized symmetry classification of differential-difference equations defined on five points of one-dimensional lattice. We have established that in the continuous limit this new equation goes into the well-known Kaup-Kupershmidt equation. We have also proved its integrability by constructing an $L-A$ pair and conservation laws. Moreover, we present a possibly new scheme for deriving conservation laws from $L-A$ pairs.

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arXiv2016

Classification of five-point differential-difference equations

Using the generalized symmetry method, we carry out, up to autonomous point transformations, the classification of integrable equations of a subclass of the autonomous five-point differential-difference equations. This subclass includes such well-known examples as the Itoh-Narita-Bogoyavlensky and the discrete Sawada-Kotera equations. The resulting list contains 17 equations some of which seem to be new. We have found non-point transformations relating most of the resulting equations among themselves and their generalized symmetries.

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arXiv2016

Non-invertible transformations for the classification of differential-difference equations

We discuss aspects of the theory of non-invertible transformations which enter in the problem of classification of diffe\-ren\-tial-difference equations and, in particular, the notion of Miura type transformation. We introduce the concept of non--Miura type linearizable transformation and we present techniques which allow one to construct simple linearizable transformations and help us to solve the classification problem. This theory is illustrated by the example of a new integrable differential--difference equation depending on 5 lattice points, interesting from the viewpoint of the non-invertible transformation which relate it to an Itoh--Narita--Bogoyavlensky equation.

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arXiv2014

Integrable Discrete Nonautonomous Quad-equations as Bäcklund Auto-transformations for Known Volterra and Toda Type Semidiscrete Equations

We construct integrable discrete nonautonomous quad-equations as Bäcklund auto-transformations for known Volterra and Toda type semidiscrete equations, some of which are also nonautonomous. Additional examples of this kind are found by using transformations of discrete equations which are invertible on their solutions. In this way we obtain integrable examples of different types: discrete analogs of the sine-Gordon equation, the Liouville equation and the dressing chain of Shabat. For Liouville type equations we construct general solutions, using a specific linearization. For sine-Gordon type equations we find generalized symmetries, conservation laws and $L-A$ pairs.

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arXiv2011

Generalized symmetry integrability test for discrete equations on the square lattice

We present an integrability test for discrete equations on the square lattice, which is based on the existence of a generalized symmetry. We apply this test to a number of equations obtained in different recent papers. As a result we prove the integrability of 7 equations which differ essentially from the $Q_V$ equation introduced by Viallet and thus from the Adler-Bobenko-Suris list of equations therein contained.

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arXiv2010

Integrability Test for Discrete Equations via Generalized Symmetries

In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations obtained by the multiple scale analysis of the general multilinear dispersive difference equation defined on the square.

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arXiv2009

On a nonlinear integrable difference equation on the square 3D-inconsistent

We present a nonlinear partial difference equation defined on a square which is obtained by combining the Miura transformations between the Volterra and the modified Volterra differential-difference equations. This equation is not symmetric with respect to the exchange of the two discrete variables and does not satisfy the 3D-consistency condition necessary to belong to the Adler-Bobenko-Suris classification. Its integrability is proved by constructing its Lax pair.

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arXiv2009

The Generalized Symmetry Method for Discrete Equations

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing and classifying equations of this class. Those conditions are used at the end to test for integrability discretizations of some well-known hyperbolic equations.

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