The q-difference 2D Toda lattice, the q-difference sine-Gordon equation and classifications of solutions

In this paper, we have developed Cauchy matrix approach to construct the q-difference two-dimensional Toda lattice (q-2DTL) and q-difference sine-Gordon (q-sG) equation, and explore their integrability such as Lax pair and explicit solutions. By leveraging specific dispersion relations pertaining to r and s of the Sylvester equation \(KM + ML = rs^\top \), we establish the q-2DTL and derive its Lax pair. We also clarify the connection of the \(\tau \) function of the q-2DTL with Cauchy matrix approach. Besides, explicit solutions of the q-2DTL are formulated and classified by comprehensively investigating its underlying systems of linear q-difference equations. As typical examples, the dynamical behaviors of both soliton solutions and a double-pole solution are simulated numerically. Under the assumption \(K = L\), we demonstrate how to reduce the q-sG equation from the q-2DTL both by Cauchy matrix approach and by 2-periodic reductions. Besides, the bilinear representation for the q-sG equation is reported for the first time. Furthermore, rich solutions such as kink solutions and breathers are explicitly presented and graphically illustrated for the q-sG equation.