On series of Darboux integrable discrete equations on 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 . 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 3 for an equation with the number .In the cases = 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 3 −1, where is the equation number in series. Both first integrals are unobvious in this case.

Modified series of integrable discrete equations on a square lattice with a non-standard symmetry structure

“…где a n = b + cn как в (6). При N = 2, β N = −1 высшая симметрия второго порядка, соответствующая (7), имеет вид:…”

“…Тогда уравнение (37) легко записывается в терминах Y n,m . При N = 2, β N = −1 высшая симметрия второго порядка, соответствующая (7), имеет вид: При N = 3 высшая симметрия третьего порядка, соответствующая (8), имеет вид:…”

“…Аналогичные серии неавтономных дискретных уравнений рассматривались в [4]. Серии дискретных уравнений, интегрируемых по Дарбу, и уравнений типа Бюргерса изучались в [5,7].…”

“…где a n = b n +cn как в(7). Очевидно, что при c = 0 она является Мёбиус-инвариантной относительно w n,m как и (37).…”

An algebraic criterion of the Darboux integrability of differential-difference equations and systems

“…On the other hand side, the concept of Darboux integrability is widespread and important in itself; therefore, its generalization to differential-difference models is of independent interest (see [8], [9], [10], [11], [12]). Fundamental ideas in the study of the problem of constructing in closed form general solution of partial differential equations of hyperbolic type go back to the classical works of Laplace, Liouville, Lie, Darboux, Goursat, Vessio, and others.…”