Geodesic
Necessary conditions of Darboux integrability for differential-difference equations of a~special kind
Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 78-82

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This work dwells upon chains of differential equations of the form $\varphi(x,u_{i+1},(u_{i+1})_x)=\psi(x,u_i,(u_i)_x)$, where $u$ depends on the discrete variable $i$ and the continuous variable $x$, and the functions $\varphi(x,y,z)$, $\psi(x,y,z)$ and $x$ are functionally-independent. We demonstrate that necessary Darboux integrability conditions for chains of the above form can be easily derived from already known results. These conditions are not sufficient but may be useful for classification of Darboux-integrable differential-difference equations. As an auxiliary result, we also prove a proposition about structure of symmetries for differential-difference equations of a more general form.
Keywords: Darboux integrability, differential-difference equations. 
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S. Ya. Startsev. Necessary conditions of Darboux integrability for differential-difference equations of a~special kind. Ufa mathematical journal, Tome 3 (2011) no. 1, pp. 78-82. http://geodesic.mathdoc.fr/item/UFA_2011_3_1_a6/