arXiv subjects

I T Habibullin

Publications and source records attributed to I T Habibullin.

3 recordsLinked to original sources

arXiv2023

On the classification of nonlinear integrable three-dimensional chains by means of characteristic Lie algebras

The article continues the work on the description of integrable nonlinear chains with three independent variables of the following form $u^j_{n+1,x}=u^j_{n,x}+f(u^{j+1}_{n}, u^{j}_n,u^j_{n+1 },u^{j-1}_{n+1})$ by the presence of a hierarchy of reductions integrable in the sense of Darboux, started in (1). The classification algorithm is based on the well-known fact that the characteristic algebras of Darboux integrable systems have a finite dimension. In this paper, we used the characteristic algebra in the direction $x$, whose structure for a given class of models is determined by some polynomial $P(λ)$, whose degree does not exceed three for known examples. The article assumes that $P(λ)=λ^2$, in this case the classification problem is reduced to finding eight unknown functions of one variable. In the paper, a rather narrow class of candidates for integrability is obtained, among which there is a new example of an integrable chain.

nlin.SI↗

arXiv2023

Laplace transformations and sine-Gordon type integrable PDE

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 Liouville type. The connection between the Laplace method and soliton equations of hyperbolic type remains less studied. The article shows that the Laplace cascade also has important applications in the theory of hyperbolic equations of the soliton type. Laplace's method provides a simple way to construct such fundamental objects related to integrability theory as the recursion operator, the Lax pair and Dubrovin-type equations, allowing one to find algebro-geometric solutions. As an application of this approach, previously unknown recursion operators and Lax pairs are found for two nonlinear integrable equations of the sine-Gordon type.

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arXiv2019

Discrete exponential type systems on a quad graph, corresponding to the affine Lie algebras $A^{(1)}_{N-1}$

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 exponential systems connected with the Cartan matrices has been suggested earlier in \cite{GHY} which coincide with the corresponding Drinfeld-Sokolov systems in the continuum limit. It was conjectured that the systems in this class are all integrable and the conjecture has been approved by numerous examples. In the present article we study those systems from this class which are related to the algebras $A^{(1)}_{N-1}$. We found the Lax pair for arbitrary $N$, briefly discussed the possibility of using the method of formal diagonalization of Lax operators for describing a series of local conservation laws and illustrated the technique using the example of $N=3$. Higher symmetries of the system $A^{(1)}_{N-1}$ are presented in both characteristic directions. Found recursion operator for $N=3$. It is interesting to note that this operator is not weakly nonlocal.

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