The discrete mKdV equation revisited: a Riemann-Hilbert approach

TL;DR

This paper revisits the discrete mKdV equation using a Riemann-Hilbert approach, deriving solutions for plus and minus types through symmetry conditions and gauge transformations.

Contribution

It introduces new symmetry conditions and a Riemann-Hilbert framework to solve the discrete mKdV equation, providing explicit solutions for both types.

Findings

Derived matrix Riemann-Hilbert problem for discrete mKdV

Obtained complex and real solutions for plus type

Connected solutions of minus type via gauge transformation

Abstract

We study the plus and minus type discrete mKdV equation. Some different symmetry conditions associated with two Lax pairs are introduced to derive the matrix Riemann-Hilbert problem with zero. By virtue of regularization of the Riemann-Hilbert problem, we obtain the complex and real solution to the plus type discrete mKdV equation respectively. Under the gauge transformation between the plus and minus type, the solutions of minus type can be obtained in terms of the given plus ones.