Karakteristik lie cebiri ve yarı-ayrık modellerin sınıflandırılması

In this thesis, we studied a differential-difference equation of thefollowing formt_{x}(n+1,x)=f(t(n,x),t(n+1,x),t_x(n,x)), (1)where the unknown t=t(n,x) is a function of two independentvariables: discrete n and continuous x. The equation (1) iscalled a Darboux integrable equation if it admits nontrivial x-and n-integrals. A function F(x,t,t_{\pm 1},t_{\pm 2},...)is called an x-integral if D_xF=0, where D_x is the operatorof total differentiation with respect to x. A functionI(x,t,t_x,t_{xx},...) is called an n-integral if DI=I,where D is the shift operator: Dh(n)=h(n+1).In this work, we introduced the notion of characteristic Lie algebra for semi-discretehyperbolic type equations. We used characteristic Lie algebra as atool to classify Darboux integrability chains and finally gave thecomplete list of Darboux integrable equations in the case when thefunction f in the equation (1) is of the special form f=t_x(n,x)+d(t(n,x),t(n+1,x)).