Conservation laws of the (2+1)-dimensional complex modified Korteweg-de Vries and Maxwell-Bloch equations

Abstract

: At present, the kmKdF equation is used in conjunction with the Maxwell-Bloch equations, and therefore they are
collectively called the equations of the complexly modified Korteweg-de Vries and Maxwell–Bloch (kmKdF and MB). In addition, these equations can be obtained by reconstructing the Hirota–Maxwell–Bloch (HMB) system of equations. The reductions of this equation are the nonlinear Schrodinger equation, the nonlinear Schrodinger-Mcwell-Bloch equation, and the Korteweg-de Vries–Maxwell-Bloch equation. These equations have been studied by different authors. The Lax pairs of these equations are presented. Using Lax pairs, Darboux transformations are constructed, namely, single transformations. Soliton solutions are obtained from different "seed" solutions using these Darboux transformations. Using the Darboux transformations one–fold, two–fold and N–fold the determinant representations of Darboux transformations are obtained. And also soliton solutions are built. 
In this paper, we obtain the conservation laws of the (2+1)-dimensional complex modified Korteweg-de Vries and MaxwellBloch equations through the Lax pair.