RUS  ENG
Full version
JOURNALS // Ufimskii Matematicheskii Zhurnal // Archive

Ufimsk. Mat. Zh., 2010 Volume 2, Issue 4, Pages 39–51 (Mi ufa70)

This article is cited in 2 papers

Characteristic Lie algebra and Darboux integrable discrete chains

N. A. Zheltukhinaa, A. U. Sakievab, I. T. Habibullinb

a Bilkent University, Bilkent, Turkey
b Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences, Ufa, Russia

Abstract: Differential-difference equation
$$ \frac d{dx}t(n+1,x)=f\left(x,t(n,x),t(n+1,x),\frac d{dx}t(n,x)\right) $$
with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is considered. An equation is said to be Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator $Dp(n)=p(n+1)$. It is proved that an equation is Darboux integrable if and only if its characteristic Lie algebras are finite-dimensional in both directions. The structure of integrals is described. Characteristic algebras for a certain class of integrable equations are described.

Keywords: integrable chains, classification, $x$-integral, $n$-integral, characteristic Lie algebra, integrability conditions.

UDC: 517.957

Received: 01.07.2010



Bibliographic databases:


© Steklov Math. Inst. of RAS, 2024