# 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} }

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…

## 4 Citations

### Symmetry Drivers and Formal Integrals of Hyperbolic Systems

- Mathematics
- 2020

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

- 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…

### On Darboux non-integrability of Hietarinta equation

- MathematicsUfimskii 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

- Mathematics
- 2017

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…

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