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

  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},
  • 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… 

Symmetry Drivers and Formal Integrals of Hyperbolic Systems

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

Reconstructing a Lattice Equation: a Non-Autonomous Approach to the Hietarinta Equation

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

On Darboux non-integrability of Hietarinta equation

  • 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

Formal Integrals and Noether Operators of Nonlinear Hyperbolic Partial Differential Systems Admitting a Rich Set of Symmetries

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



Discrete analogues of the Liouville equation

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

On Darboux-integrable semi-discrete chains

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

On Partial Differential and Difference Equations with Symmetries Depending on Arbitrary Functions

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

Generalized symmetry classification of discrete equations of a class depending on twelve parameters

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

Differential substitutions of the miura transformation type

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 discrete equations possessing an autonomous first-order integral

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

Symmetries of nonlinear hyperbolic systems of the Toda chain type

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.

Examples of Darboux integrable discrete equations possessing first integrals of an arbitrarily high minimal order

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

Exactly integrable hyperbolic equations of Liouville type

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

Discrete Two-Dimensional Toda Molecule Equation

A discrete analogue of the two-dimensional Toda molecule equation is obtained, which is expressed as follows \begin{aligned}