Rigorous analytical solutions for the multicomponent semi-discrete integrable systems

Construct rigorous analytical solutions for the twelve-component semi-discrete nonlinear integrable system on a quasi-one-dimensional lattice that couples the fields q_{+}, r_{+}, \bar{q}_{+}, \bar{r}_{+}, f_{+}, g_{+}, q_{-}, r_{-}, \bar{q}_{-}, \bar{r}_{-}, f_{-}, and g_{-}, and for the associated six-component semi-discrete system that couples w_{+}, \bar{w}_{+}, h_{+}, w_{-}, \bar{w}_{-}, and h_{-}.

Background

The paper introduces two new integrable semi-discrete systems: a twelve-component system on a quasi-one-dimensional lattice derived from a semi-discrete AKNS framework, and a reduced six-component system. Both exhibit rich coupling structures (linear and nonlinear) and admit symmetries, but explicit closed-form solutions are not provided.

The authors highlight that obtaining explicit solutions is a central objective for advancing both mathematical understanding and physical modeling of transport phenomena in quasi-one-dimensional structures. They point to Darboux–Bäcklund methods as likely tools, but emphasize that the existence of rigorous solutions remains an open task.