On Darboux Integrable Semi-Discrete Chains

Abstract

Differential-difference equation $\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables 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: $Dp(n)=p(n+1)$. It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for general solution to Darboux integrable chains is discussed and for a class of chains such solutions are found.

19 pages

References in corpus (5)

• Discrete analogues of the Liouville equation
• A survey of Hirota's difference equations
• Characteristic Lie Algebra and Classification of Semi-Discrete Models
• On the Classification of Darboux Integrable Chains
• Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$

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• Darboux integrable discrete equations possessing an autonomous first-order integral
• On relationships between symmetries depending on arbitrary functions and integrals of discrete equations
• An algebraic criterion of the Darboux integrability of differential-difference equations and systems
• On existence of an x-integral for a semi-discrete chain of hyperbolic type