arXiv subjects

A. R. Khakimova

Publications and source records attributed to A. R. Khakimova.

12 recordsLinked to original sources

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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arXiv2016

On a method for constructing the Lax pairs for nonlinear integrable equations

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 pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.

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