19 citations · 66 across the 18 of their papers we have counts for
18 papers
Discrete integrable equations with three independent variables
Ismagil T. Habibullin, Aigul R. Khakimova
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 Hirot…
On the construction of solutions of the Davey--Stewartson I equation using an open Toda chain
I. T. Habibullin, A. R. Khakimova
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…
Higher symmetries of the lattices in 3D
I. T. Habibullin, A. R. Khakimova
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 sy…
Construction of exact solutions of nonlinear PDE via dressing chain in 3D
I. T. Habibullin, A. R. Khakimova
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 recentl…
Reduction of the Laplace sequence and sine-Gordon type equations
K I Faizulina, A R Khakimova
In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously pr…
On the classification of nonlinear integrable three-dimensional chains by means of characteristic Lie algebras
I T Habibullin, A R Khakimova
The article continues the work on the description of integrable nonlinear chains with three independent variables of the following form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n}, u^{j}_…
Laplace transformations and sine-Gordon type integrable PDE
I T Habibullin, K I Faizulina, A R Khakimova
It is well known that the Laplace cascade method is an effective tool for constructing solutions to linear equations of hyperbolic type, as well as nonlinear equations of the Liouv…
Miura type transformations for integrable lattices in 3D
I. T. Habibullin, A. R. Khakimova, A. U. Sakieva
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…
Construction of exact solutions to the Ruijsenaars-Toda lattice via generalized invariant manifolds
I. T. Habibullin, A. R. Khakimova, A. O. Smirnov
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…
Generalized invariant manifolds for integrable equations and their applications
I. T. Habibullin, A. R. Khakimova, A. O. Smirnov
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 kn…
Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
I. T. Habibullin, A. R. Khakimova
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 discret…
Invariant manifolds and separation of the variables for integrable chains
I. T. Habibullin, A. R. Khakimova
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 e…
Integrable boundary conditions for the Hirota-Miwa equation and Lie algebras
Ismagil Habibullin, Aigul Khakimova
Systems of discrete equations on a quadrilateral graph related to the series of the affine Lie algebras are studied. The systems are derived from the Hirota-Miwa equati…
Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras
I T Habibullin, A R Khakimova
The article deals with the problem of the integrable discretization of the well-known Drinfeld-Sokolov hierarchies related to the Kac-Moody algebras. A class of discrete exponentia…
On the recursion operators for integrable equations
I. T. Habibullin, A. R. Khakimova
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 symme…
Invariant manifolds for the hyperbolic type integrable equations and their applications
Ismagil Habibullin, Aigul Khakimova
We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semi-discrete variant). First, we linearize the equation around its arbitrary solution . Th…
On a method for constructing the Lax pairs for integrable models via quadratic ansatz
Ismagil Habibullin, Aigul Khakimova
A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold to the linearization of the given equation. E…
On a method for constructing the Lax pairs for nonlinear integrable equations
I. T. Habibullin, A. R. Khakimova, M. N. Poptsova
We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax…