Darboux Integrability of Trapezoidal $H^{4}$ and $H^{6}$ Families of Lattice Equations II: General Solutions

Abstract

In this paper we construct the general solutions of two families of quad-equations, namely the trapezoidal $H^{4}$ equations and the $H^{6}$ equations. These solutions are obtained exploiting the properties of the first integrals in the Darboux sense, which were derived in [Gubbiotti G., Yamilov R.I., J. Phys. A: Math. Theor. 50 (2017), 345205, 26 pages, arXiv:1608.03506]. These first integrals are used to reduce the problem to the solution of some linear or linearizable non-autonomous ordinary difference equations which can be formally solved.