Analysis of the solitary wave solutions of the negative order modified Korteweg –de Vries equation with a self-consistent source

In this work, the initial value problem for the negative order modified Korteweg–de Vries equation (nmKdV) with a self-consistent source was analyzed. The inverse scattering transform method for obtaining evolution equations of scattering data of the Dirac operator, which potential is the solution of the considered problem was implemented. For the first time, the real matrix triplet ( A , B , C ) method was applied to construct a multisoliton solution of nmKdV equation with a self-consistent source. Furthermore, the wave phenomena of solitons were demonstrated by varying the normalization conditions. • The considered operator has no spectral singularities and all eigenvalues are simple. • The Jost functions admit an analytical continuation to the upper-half plane. • The potential is reconstructed through the scattering data of the Dirac system. • The soliton solution arise in the case of reflectionless coefficient.