2008/06/19 by I. T. Habibullin, Ismagil Habibullin, Natalya Zheltukhina +1 · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Numerical methods for differential equations #nlin.SI
paper · pdf · doi:10.1063/1.2992950
arxiv created 2008/06/19 · openalex publication_date 2008/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a 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±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx 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 give 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).