arXiv subjects

M. N. Kuznetsova

Publications and source records attributed to M. N. Kuznetsova.

4 recordsLinked to original sources

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.

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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.

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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.

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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.

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