arXiv subjects

Ismagil T. Habibullin

Publications and source records attributed to Ismagil T. Habibullin.

8 recordsLinked to original sources

arXiv2026

Discrete integrable equations with three independent variables

In this paper, we study nonlinear integrable equations with three independent variables of the following types: Toda-type lattices, semi-discrete lattices, and fully discrete Hirota-Miwa type models. It is shown that integrable equations of all three types admit reductions in the form of Darboux-integrable hyperbolic systems. It is important that the transition from one class to another is carried out by means of discretization (continualization) of the above-mentioned reductions with preservation of characteristic integrals. In other words, at the level of reductions, one can establish some correspondence between the classes of 3D models under consideration. In the context of this correspondence, the authors managed to conduct a comparative analysis of the well-known list of integrable Hirota-Miwa type equations, containing 13 equations. It was established that some equations from this list are related by point changes of variables. As a result, the final list of known integrable Hirota-Miwa type equations was reduced to seven. One equation was obtained by discretizing the list of semi-discrete Toda-type equations using characteristic integrals in this paper, probably it is new. For all seven models, associated linear systems (Lax pairs) are given.

nlin.SI↗

arXiv2025

On a class of exact solutions of the Ishimori equation

In this paper, a class of particular solutions of the Ishimori equation is found. This equation is known as the spatially two-dimensional version of the Heisenberg equation, which has important applications in the theory of ferromagnets. It is shown that the two-dimensional Toda-type lattice found earlier by Ferapontov, Shabat and Yamilov is a dressing chain for this equation. Using the integrable reductions of the dressing chain, the authors found an essentially new class of solutions to the Ishimori equation.

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arXiv2020

Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations

In the article differential-difference (semi-discrete) lattices of hyperbolic type are investigated from the integrability viewpoint. More precisely we concentrate on a method for constructing generalized symmetries. This kind integrable lattices admit two hierarchies of generalized symmetries corresponding to the discrete and continuous independent variables $n$ and $x$. Symmetries corresponding to the direction of $n$ are constructed in a more or less standard way while when constructing symmetries of the other form we meet a problem of solving a functional equation. We have shown that to handle with this equation one can effectively use the concept of characteristic Lie-Rinehart algebras of semi-discrete models. Based on this observation, we have proposed a classification method for integrable semi-discrete lattices. One of the interesting results of this work is a new example of an integrable equation, which is a semi-discrete analogue of the Tzizeica equation. Such examples were not previously known.

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arXiv2015

Peculiar symmetry structure of some known discrete nonautonomous equations

We study the generalized symmetry structure of three known discrete nonautonomous equations. One of them is the semidiscrete dressing chain of Shabat. Two others are completely discrete equations defined on the square lattice. The first one is a discrete analogue of the dressing chain introduced by Levi and Yamilov. The second one is a nonautonomous generalization of the potential discrete KdV equation or, in other words, the H1 equation of the well-known Adler-Bobenko-Suris list. We demonstrate that these equations have generalized symmetries in both directions if and only if their coefficients, depending on the discrete variables, are periodic. The order of the simplest generalized symmetry in at least one direction depends on the period and may be arbitrarily high. We substantiate this picture by some theorems in the case of small periods. In case of an arbitrarily large period, we show that it is possible to construct two hierarchies of generalized symmetries and conservation laws. The same picture should take place in case of any nonautonomous equation of the Adler-Bobenko-Suris list.

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arXiv2012

Affine Lie algebras, Lax pairs and integrable discrete and continuous systems

A consistent set of six integrable discrete and continuous dynamical systems are suggested corresponding to arbitrary affine Lie algebra. The set contains a system of partial differential equations which can be treated as a version of generalized Toda lattice while semi-discrete systems in the set define the Backlund transform for this Toda lattice and the fully discrete representative of the set can be obtained as a superposition of such kind Backlund transforms. Four linear Zakharov-Shabat type systems taken pairwise realize Lax pairs for these six dynamical systems.

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arXiv2011

Method for searching higher symmetries for quad graph equations

Generalized symmetry integrability test for discrete equations on the square lattice is studied. Integrability conditions are discussed. A method for searching higher symmetries (including non-autonomous ones) for quad graph equations is suggested based on characteristic vector fields.

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arXiv2011

Classification of integrable discrete Klein-Gordon models

The Lie algebraic integrability test is applied to the problem of classification of integrable Klein-Gordon type equations on quad-graphs. The list of equations passing the test is presented containing several well-known integrable models. A new integrable example is found, its higher symmetry is presented.

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arXiv2010

An algebraic method of classification of S-integrable discrete models

A method of classification of integrable equations on quad-graphs is discussed based on algebraic ideas. We assign a Lie ring to the equation and study the function describing the dimensions of linear spaces spanned by multiple commutators of the ring generators. For the generic case this function grows exponentially. Examples show that for integrable equations it grows slower. We propose a classification scheme based on this observation.

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