Breather and soliton solutions of semi-discrete negative order AKNS equation

In this paper, I propose the semi-discrete version of the negative order AKNS equation. I study the discrete Darboux and binary Darboux transformations in order to construct the multi-soliton solutions and also the binary Darboux matrix for the negative order AKNS equation. By the iteration of binary Darboux transformation, I obtain the quasi-Grammian solutions for the system. Further, I derive the explicit solutions for the binary Darboux matrix and then reduce them to the elementary Darboux transformation solutions and plot the dynamics of breather and soliton solutions.