# On some classes of semi-discrete darboux integrable equations

In this thesis we consider Darboux integrable semi-discrete hyperbolic equations of the form $t_{1x} = f(t,t_{1}, t_{x}), \frac{\partial f}{\partial t_{x}} \neq 0.$ We use the notion of characteristic Lie ring for a classification problem based on dimensions of characteristic $x$- and $n$-rings. Let $A = (a_{ij})_{N\times N}$ be a $N\times N$ matrix. We also consider semi-discrete hyperbolic equations of exponential type $u_{1,x}^{i} - u_{x}^{i} = e^{\sum a_{ij}^{+}u_{1}^{j} + \sum a_{ij}^{-}u^{j}}, i,j = 1,2,\dots,N.$ We find the conditions on $a_{ij}$'s so that the above equation is Darboux integrable when $N=2$.