arXiv subjects

I. T. Habibullin

Publications and source records attributed to I. T. Habibullin.

At least 19 recordsLinked to original sources

arXiv2025

On the symmetry classification of integrable chains in 3D. Darboux-integrable reductions and their higher symmetries

This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations, called Toda-type chains, is that they admit finite-field reductions in the form of open chains with enhanced integrability. The paper results in a theorem stating that all known integrable Toda-type chains admit reductions in the form of an open chain of length three with a family of second-order evolutionary type symmetries. Apparently, this property of Toda-type chains can be used as an effective classification criterion when compiling lists of integrable differential-difference equations in 3D.

nlin.SI↗

arXiv2025

On the construction of solutions of the Davey--Stewartson I equation using an open Toda chain

An effective method for constructing explicit solutions to the Davey--Stewartson type integrable equations is discussed based on the use of a dressing chain. The application of the method is exemplified by the equation DS I, for which a new class of explicit solutions is constructed, containing freedom in two arbitrary functions. In this case the generalized Toda lattice corresponding to the simple Lie algebra $A_2$ is used as a dressing chain.

nlin.SI↗

arXiv2025

Construction of exact solutions of nonlinear PDE via dressing chain in 3D

The duality between a class of the Davey-Stewartson type coupled systems and a class of two-dimensional Toda type lattices is discussed. A new coupled system related to the recently found lattice is presented. A method for eliminating nonlocalities in coupled systems by virtue of special finite reductions of the lattices is suggested. An original algorithm for constructing explicit solutions of the coupled systems based on the finite reduction of the corresponding lattice is proposed. Some new solutions for coupled systems related to the Volterra lattice are presented as illustrative examples.

nlin.SI↗

arXiv2024

Higher symmetries of the lattices in 3D

It is known that there is a duality between the Davey--Stewartson type coupled systems and a class of integrable two--dimensional Toda type lattices. More precisely, the coupled systems are generalized symmetries for the lattices and the lattices can be interpreted as dressing chains for the systems. In our recent study we have found a novel lattice which apparently is not related to the known ones by Miura type transformation. In the article we described higher symmetries to this lattice and derived a new coupled system of the DS type.

nlin.SI↗

arXiv2024

Symmetries of Toda type 3D lattices

The duality between a class of the Davey-Stewartson type coupled systems and a class of two-dimensional Toda type lattices is discussed. For the recently found integrable lattice the hierarchy of symmetries is described. Second and third order symmetries are presented in explicit form. Corresponding coupled systems are given. An original method for constructing exact solutions to coupled systems is suggested based on the Darboux integrable reductions of the dressing chains. Some new solutions for coupled systems related to the Volterra lattice are presented as illustrative examples.

nlin.SI↗

arXiv2024

On integrable reductions of two-dimensional Toda-type lattices

The article considers lattices of the two-dimensional Toda type, which can be interpreted as dressing chains for spatially two-dimensional generalizations of equations of the class of nonlinear Schrödinger equations. The well-known example of this kind of generalization is the Davey-Stewartson equation. It turns out that the finite-field reductions of these lattices, obtained by imposing cutoff boundary conditions of an appropriate type, are Darboux integrable, i.e., they have complete sets of characteristic integrals. An algorithm for constructing complete sets of characteristic integrals of finite field systems using Lax pairs and Miura-type transformations is discussed.

nlin.SI↗

arXiv2023

Miura type transformations for integrable lattices in 3D

The article studies a class of integrable semidiscrete equations with one continuous and two discrete independent variables. Miura type transformations are obtained that relate the equations of the class. A new integrable chain of this type is found, for which the Lax pair is presented. Integrable in the sense of Darboux reductions of the chain are discussed, for small order reductions complete sets of integrals are constructed. Continuum limits for the chain are discussed. A method for finding particular solutions of chains based on integrable in sense of Darboux reductions is proposed. The effectiveness of the method is illustrated by an example.

nlin.SI↗

arXiv2021

Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds

The article discusses a new method for constructing algebro-geometric solutions of nonlinear integrable lattices, based on the concept of a generalized invariant manifold (GIM). In contrast to the finite-gap integration method, instead of the eigenfunctions of the Lax operators, we use a joint solution of the linearized equation and GIM. This makes it possible to derive Dubrovin type equations not only in the time variable $t$, but also in the spatial discrete variable $n$. We illustrate the efficiency of the method using the Ruijsenaars-Toda lattice as an example, for which we have derived a real and bounded particular solution in the form of a kink, a periodic solution expressed in terms of Jacobi elliptic functions and a solution expressed through Weierstrass $\wp$-function.

nlin.SI↗

arXiv2021

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie-Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.

nlin.SI↗

arXiv2021

Generalized invariant manifolds for integrable equations and their applications

In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to the nonlinear PDE, a much simpler, as a rule ordinary, differential equation, compatible with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this compatible ODE. Our generalization is that we look for an ordinary differential equation that is compatible not with the nonlinear partial differential equation itself, but with its linearization. Such a generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrability theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.

nlin.SI↗

arXiv2021

Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras. That allows one to derive effective integrability conditions in the form of differential equations on the right hand side of the equation under study. To test the hypothesis, we use known integrable equations from the class under consideration. In this article, we show that all known examples do have this property.

nlin.SI↗

arXiv2020

Invariant manifolds and separation of the variables for integrable chains

A notion of the generalized invariant manifold for a nonlinear integrable lattice is considered. Earlier it has been observed that this kind objects provide an effective tool for evaluating the recursion operators and Lax pairs. In this article we show with an example of the Volterra chain that the generalized invariant manifold can be used for constructing exact particular solutions as well. To this end we first find an invariant manifold depending on two constant parameters. Then we assume that ordinary difference equation defining the generalized invariant manifold has a solution polynomially depending on one of the spectral parameters and derive ordinary difference and differential equations, for the roots of the polynomials. Efficiency of the method is approved by some illustrative examples.

nlin.SI↗

arXiv2020

Integrability conditions for two-dimensional lattices

In the article some algebraic properties of nonlinear two-dimensional lattices of the form $u_{n,xy} = f(u_{n+1}, u_n, u_{n-1})$ are studied. The problem of exhaustive description of the integrable cases of this kind lattices remains open. By using the approach, developed and tested in our previous works we adopted the method of characteristic Lie-Rinehart algebras to this case. In the article we derived an effective integrability conditions for the lattice and proved that in the integrable case the function $f(u_{n+1}, u_n, u_{n-1})$ is a quasi-polynomial satisfying the following equation $\frac{\partial^2}{\partial u_{n+1}\partial u_{n-1}}f(u_{n+1}, u_n, u_{n-1})=Ce^{αu_n-{\frac{αm}{2}}u_{n+1}-{\frac{αk}{2}}u_{n-1}},$ where $C$ and $α$ are constant parameters and $k,\,m$ are nonnegative integers.

nlin.SI↗

arXiv2020

On a class of 2D integrable lattice equations

We develop a new approach to the classification of integrable equations of the form $$ u_{xy}=f(u, u_x, u_y, \triangle_z u \triangle_{\bar z}u, \triangle_{z\bar z}u), $$ where $\triangle_{ z}$ and $\triangle_{\bar z}$ are the forward/backward discrete derivatives. The following 2-step classification procedure is proposed: (1) First we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure which is Einstein-Weyl on every solution. (2) Secondly, to the candidate equations selected at the previous step we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions.

nlin.SI↗

arXiv2019

On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras

In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic equations of arbitrarily high order integrable in the Darboux sense. Darboux integrablity admits a remarkable algebraic interpretation: the Lie-Rinehart algebras related to both characteristic directions corresponding to the reduced system of the hyperbolic equations have to be of finite dimension. A classification algorithm based on the properties of the characteristic algebra is discussed. Some classification results are presented.

nlin.SI↗

arXiv2018

Algebraic Properties of Quasilinear Two-Dimensional Lattices connected with integrability

In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form $u_{n,xy}=α(u_{n+1} ,u_n,u_{n-1} )u_{n,x}u_{n,y} + β(u_{n+1},u_n,u_{n-1})u_{n,x}+γ(u_{n+1} ,u_n,u_{n-1} )u_{n,y}+δ(u_{n+1} ,u_n,u_{n-1})$. We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.

nlin.SI↗

arXiv2018

On the recursion operators for integrable equations

It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In the literature, we can find several methods for constructing recursion operators, some of them use the Hamiltonian approach and the others are based on the Lax representation of the equation. In the present article we discuss on an alternative method, suggested in \cite{HabKhaTMP18}, which is connected only with the first several generalized symmetries of the given equation. Efficiency of the method is illustrated with the examples of KdV, Krichever-Novikov and Kaup-Kupershmidt equations and discrete models.

nlin.SI↗

arXiv2017

On a direct algorithm for constructing recursion operators and Lax pairs for integrable models

We suggested an algorithm for searching the recursion operators for nonlinear integrable equations. It was observed that the recursion operator $R$ can be represented as a ratio of the form $R=L_1^{-1}L_2$ where the linear differential operators $L_1$ and $L_2$ are chosen in such a way that the ordinary differential equation $(L_2-λL_1)U=0$ is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter $λ\in \textbf{C}$. For constructing the operator $L_1$ we use the concept of the invariant manifold which is a generalization of the symmetry. Then for searching $L_2$ we take an auxiliary linear equation connected with the linearized equation by the Darboux transformation. Connection of the invariant manifold with the Lax pairs and the Dubrovin-Weierstrass equations is discussed.

nlin.SI↗