Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$

Abstract

We study differential-difference equation of the form $$\frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equation of such kind is called Darboux integrable, if there exist two functions $F$ and $I$ of a finite number of arguments $x$, $\{t(n\pm k,x){\}}_{k=-\infty }^{\infty }$, ${\frac{d^k}{d{x}^k}t(n,x)}_{k=1}^{\infty }$, 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: $Dp(n)=p(n+1)$. Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function $f$ is of the special form $f(u,v,w)=w+g(u,v)$.

References in corpus (4)

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Cited by in corpus (9)

• Method for searching higher symmetries for quad graph equations
• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• On Darboux Integrable Semi-Discrete Chains
• Discretization of hyperbolic type Darboux integrable equations preserving integrability
• 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
• An algebraic criterion of the Darboux integrability of differential-difference equations and systems