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

## TL;DR

This paper introduces a series of Darboux integrable discrete equations on the square lattice, detailing their integrability properties, first integrals, and explicit solutions for specific cases, along with a modified series with different integral orders.

## Contribution

It constructs a new series of Darboux integrable discrete equations with explicit integrals and solutions, and introduces a modified series with different integral structures.

## Key findings

- Equations have a first integral of first order in one direction.
- Minimal order of the other direction's first integral is 3M.
- Explicit solutions are constructed for M=1, 2, 3.

## 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.