Darboux integrable discrete equations possessing an autonomous first-order integral

Abstract

All Darboux integrable difference equations on the quad-graph are described in the case of the equations that possess autonomous first-order integrals in one of the characteristics. A generalization of the discrete Liouville equation is obtained from a subclass of these equation via a non-point transformation. The general proposition on the symmetry structure of the quad-graph equations is proved as an auxiliary result.

References in corpus (6)

• Generalized symmetry classification of discrete equations of a class depending on twelve parameters
• Method for searching higher symmetries for quad graph equations
• On Darboux Integrable Semi-Discrete Chains
• Discretization of hyperbolic type Darboux integrable equations preserving integrability
• On Non-Point Invertible Transformations of Difference and Differential-Difference Equations
• Linearizability of Nonlinear Equations on a Quad-Graph by a Point, Two Points and Generalized Hopf-Cole Transformations

Cited by in corpus (3)

• Darboux integrability of trapezoidal $H^4$ and $H^6$ families of lattice equations I: First integrals
• On relationships between symmetries depending on arbitrary functions and integrals of discrete equations
• On Darboux non-integrability of the Hietarinta equation