On a series of Darboux integrable discrete equations on the square lattice

Abstract

We present a series of Darboux integrable discrete equations on the square lattice. Equations of the series are numbered with natural numbers $M$. All the equations have a first integral of the first order 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 $M$. In the cases $M=1,2,3$ we show that those equations are integrable in quadratures. More precisely, we construct their general solutions in terms of the discrete integrals. We also construct a modified series of Darboux integrable discrete equations which have in different directions the first integrals of the orders $2$ and $3M-1$, where $M$ is the equation number in series. Both first integrals are unobvious in this case.

11 pages

References in corpus (3)

• On Non-Point Invertible Transformations of Difference and Differential-Difference Equations
• Examples of Darboux Integrable Discrete Equations Possessing First Integrals of an Arbitrarily High Minimal Order
• Integrable Discrete Nonautonomous Quad-equations as Bäcklund Auto-transformations for Known Volterra and Toda Type Semidiscrete Equations