On the Classification of Darboux Integrable Chains

Abstract

We study differential-difference equation of the form $t_x(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$ (called an $n$-integral), both of a finite number of variables $x$, $t(n)$, $t(n\pm 1)$, $t(n\pm 2)$, $...$, $t_x(n)$, $t_{xx}(n)$, $...$, such that $D_{x}F=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that gives an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial $x$-integrals is given in the case when the function $f$ is of the special form $f(x,y,z)=z+d(x,y)$.

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Cited by in corpus (9)

• Characteristic Lie Algebra and Classification of Semi-Discrete Models
• Method for searching higher symmetries for quad graph equations
• Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$
• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• On Darboux Integrable Semi-Discrete Chains
• Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations
• Classification of integrable discrete Klein-Gordon models
• Discretization of Liouville type nonautonomous equations preserving integrals
• On existence of an x-integral for a semi-discrete chain of hyperbolic type