The discrete mKdV equation revisited: a Riemann-Hilbert approach

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.

Rational solutions for three semi-discrete modified Korteweg–de Vries-type equations

“…Recently, soliton solutions and Jordan-block solutions for the equation (1.2) [21] was derived through the generalized Cauchy matrix approach [22]. For the sd-mKdV equation (1.3), many approaches, such as inverse scattering transform [23,24], Darboux transformation [25,26], bilinear approach [27], discrete Jacobi sub-equation method [28], algebro-geometric approach [29], Riemann-Hilbert approach [30] and Deift-Zhou nonlinear steepest descent method [31], have been developed to construct its exact solutions.…”