Darboux integrability of trapezoidal $H^{4}$ and $H^{6}$ families of lattice equations I: First integrals

TL;DR

This paper proves the Darboux integrability of trapezoidal H^4 and H^6 lattice equations, indicating their linearizability and potential for explicit solutions, based on algebraic entropy and first integrals.

Contribution

It establishes the Darboux integrability of specific lattice equations, providing new insights into their structure and solution methods.

Findings

Trapezoidal H^4 and H^6 equations are Darboux integrable.

These equations are linearizable.

First integrals can be used to derive general solutions.

Abstract

In this paper we prove that the trapezoidal $H^{4}$ and the $H^{6}$ families of quad-equations are Darboux integrable systems. This result sheds light on the fact that such equations are linearizable as it was proved using the Algebraic Entropy test [G. Gubbiotti, C. Scimiterna and D. Levi, Algebraic entropy, symmetries and linearization for quad equations consistent on the cube, \emph{J. Nonlinear Math. Phys.}, 23(4):507543, 2016]. We conclude with some suggestions on how first integrals can be used to obtain general solutions.