# Relationships between symmetries depending on arbitrary functions and integrals of discrete equations

@article{Startsev2016RelationshipsBS,
title={Relationships between symmetries depending on arbitrary functions and integrals of discrete equations},
author={S Ya Startsev},
journal={Journal of Physics A: Mathematical and Theoretical},
year={2016},
volume={50}
}
• S. Startsev
• Published 7 November 2016
• Mathematics
• Journal of Physics A: Mathematical and Theoretical
The paper is devoted to the conjecture that an equation is Darboux integrable if and only if it possesses symmetries that depend on arbitrary functions. We note that the results of previous works together prove this conjecture for scalar partial differential equations of the form uxy=F(x,y,u,ux,uy). For autonomous semi-discrete and discrete analogues of these equations, we prove that the sequence of Laplace invariants is terminated by zero for an equation if this equation admits an operator…
In this paper, we consider symmetry drivers (i.e., operators that map arbitrary functions of one of independent variables into symmetries) and formal integrals (i.e., operators that map symmetries to
• Mathematics
• 2017
In this paper we construct a non-autonomous version of the Hietarinta equation [Hietarinta J., J. Phys. A: Math. Gen. 37 (2004), L67-L73] and study its integrability properties. We show that this
• S. Startsev
• Mathematics
Ufimskii Matematicheskii Zhurnal
• 2021
. The autonomous Hietarinta equation is a well-known example of the quad-graph discrete equation which is consistent around the cube. In a recent work, it was conjectured that this equation is
The paper is devoted to hyperbolic (generally speaking, non-Lagrangian and nonlinear) partial differential systems possessing a full set of differential operators that map any function of one

## References

SHOWING 1-10 OF 20 REFERENCES

• Mathematics
• 1999
The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The
• Mathematics
• 2010
A differential-difference equation with unknown t(n, x) depending on the continuous and discrete variables x and n is studied. We call an equation of such kind Darboux integrable if there exist two
• Mathematics
• 2015
In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial
• Mathematics
• 2012
We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this
The existence of a nontrivial integral and a sufficiently large set of symmetries with respect to one of the characteristics of a hyperbolic equation implies the existence of nontrivial integrals and
Darboux integrable difference equations on the quad-graph are completely described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A
• Mathematics
• 2008
We consider hyperbolic systems of equations that have full sets of integrals along both characteristics. The best known example of models of this type is given by two-dimensional open Toda chains.
• Mathematics
• 2012
We consider a discrete equation, defined on the two-dimensional square lattice, which is linearizable, namely, of the Burgers type and depends on a parameter �� . For any natural number �� we choose
• Mathematics
• 2001
This is a survey of the authors' results concerning non-linear hyperbolic equations of Liouville type. The definition is based on the condition that the chain of Laplace invariants of the linearized
A discrete analogue of the two-dimensional Toda molecule equation is obtained, which is expressed as follows \begin{aligned}