On series of Darboux integrable discrete equations on square lattice

Abstract

We present a series of Darboux integrable discrete equations on a square lattice. The equations in the series are numbered by natural numbers $M$. All the equations possess a first order first integral in one of directions of the two-dimensional lattice. The minimal order of a first integral in the other direction is equal to $3M$ for an equation with the number $M$. First integrals in the second direction are defined by a simple general formula depending on the number $M$. In the cases $M=1,2,3$ we show that the equations are integrable by quadrature. More precisely, we construct their general solutions in terms of the discrete integrals. We also construct a modified series of Darboux integrable discrete equations having the first integrals of the minimal orders $2$ and $3M-1$ in different directions, where $M$ is the equation number in series. Both first integrals are not obvious in this case. A few similar series of integrable equations were known before; however, they were of Burgers or sine-Gordon type. A similar series of the continuous hyperbolic type equations was discussed by A.V. Zhiber and V.V. Sokolov in 2001.