The Darboux Transform and some Integrable cases of the q-Riccati Equation

TL;DR

This paper employs the q-Darboux transform to derive general solutions for q-difference Riccati equations and related Schrödinger equations, expanding the toolkit for solving integrable q-difference equations.

Contribution

It introduces a method to obtain general solutions of q-difference Riccati equations using the q-Darboux transform, a novel approach in this context.

Findings

Derived the general solution of q-difference Riccati equations.

Constructed solutions for a large class of q-difference Schrödinger equations.

Demonstrated the effectiveness of the q-Darboux transform in integrable systems.

Abstract

Using the q-version of the Darboux transform we obtain the general solution of q-difference Riccati equation from a special one by the action of one-parameter group. This allows us to construct the solutions for the latge class of q-difference Riccati equations as well as q-difference Schrodinger equations