Semi discrete Konno–Oono system: Integrable lattice formulations, Bäcklund transformation, and Liouville-type reduction

This study presents three integrable semi discrete versions of the Konno–Oono system of equations constructed through Type-I, Type-II, and Type-III Lax discretizations. Each formulation preserves integrability through a discrete zero curvature condition and admits a Darboux transformation that produces exact soliton solutions. The semi-discrete equations are also interpreted as a Bäcklund transformation connecting neighboring lattice sites, where a constant seed leads to a one dimensional Liouville type equation in time. This reduction yields analytical pulse and kink solutions that provide direct insight into the elementary soliton generation mechanism. The Darboux transformation is then employed to obtain one- and two-soliton structures and to analyze their elastic interactions on the lattice. Numerical simulations demonstrate stable propagation and confirm that the soliton collisions are perfectly elastic, consistent with the analytical asymptotic analysis. The continuum limit recovers the classical Konno–Oono equations, confirming consistency between discrete and continuous models. The combination of the Bäcklund transformation, Liouville-type reduction, and Darboux transformation provides a unified algebraic and numerical framework for exploring integrable lattice soliton dynamics.