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

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

2009-07-22

arxiv.org/abs/0907.3785v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

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 $x$, $\{t(n\pm k,x)\}_{k=-\infty}^\infty$, ${\frac{d^k}{dx^k}t(n,x)}_{k=1}^\infty$, 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: $Dp(n)=p(n+1)$. 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 $f(u,v,w)=w+g(u,v)$.

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