Riemann–Hilbert approach and the soliton solutions of the discrete mKdV equations

Abstract

In this paper, we present the inverse scattering transform of the discrete mKdV equation by the Riemann–Hilbert approach. By its Lax pair, we construct the Jost solution and the reflection coefficients. With these, we assume that there are higher-order zeros for the scattering coefficient a(λ), and construct the corresponding Riemann–Hilbert (RH) problem. In this vein, by the RH problem and the reconstruction formula, we obtain the multiple-pole solutions for the discrete mKdV equations. Compared with simple-pole solutions, multiple-pole solutions possess more complicated profiles.