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

In this paper, we consider the (2+1)-dimensional complex modified Korteweg-de Vries and Maxwell-Bloch (cmKdVMB) equations. Lax pairs of cmKdVMB equations are presented. Using the Lax pair, we construct a Darboux transformation and namely one-fold transformations. The soliton solutions are obtained from the different “seeds” by using this Darboux transformation.


Introduction
The theory of nonlinear partial differential equations has attracted a lot of attention among researcher and is fundamentally linked to several basic developments in this area of soliton theory. It is well known that nonlinear equations such as the Korteweg-de Vries (KdV) equation, modified Korteweg-de Vries (mKdV) equation and the nonlinear Schrodinger (NLS) equation are the most typical and well-studied integrable evolution equations which describe nonlinear wave phenomena for a range of dispersive physical systems. In the study of nonlinear waves their solutions play an important role [1][2][3][4][5][6][7][8][9][10][11][12]. One of the generalizations of the mKdV equation is complex mKdV (cmKdV) equation which is one of the well-known and completely integrable equations in soliton theory. It possesses all the basic characters of integrable models. From a physical point of view cmKdV equation has been derived for, e.g. the fundamental evolution of nonlinear lattice, fluid dynamics, plasma physics, ultra-short pulses in nonlinear optics, nonlinear transmission lines and so on [13][14].
At the present time, the cmKdV equation is used in pair with Maxwell-Bloch system of equations, and so it called as complex modified Korteweg-de Vries and Maxwell-Bloch (cmKdVMB) equations. Moreover, this equation can be received by the reduction of the Hirota-Maxwell-Bloch (HMB) system of equations. In (1+1)-dimensions cmKdVMB equations were studied in [15][16][17] by the reduction of HMB system of equations.
In this work, we consider (2+1)-dimensional cmKdVMB equations which were suggested in [18]. Here we use the method of Darboux transformation. It has been proved to be an efficient way to find the different solutions from "seed" solutions this transformation allows to construct non-trivial analytical solutions of nonlinear partial differential equations [19][20][21][22][23].
The paper is organized as follows. The (2+1)-dimensional cmKdVMB equations we present in Section 2. In Section 3, we construct the DT for the (2+1)-dimensional cmKdVMB equations. In Section 4, using the constructed one-fold DT, the one-soliton solutions of the (2+1)-dimensional cmKdVMB equations is given.
In particular, we give in detail the one-fold DT and briefly the n-fold DT.
In order to satisfy the constraints of M and B [1] −1 as mentioned above, we first notes that  λ 1 ; t, x, y) ) , where So the matrix M has the form ) .
Finally we can write the one-fold DT for the (2+1)-dimensional cmKdVMB equations as: v [1] = v + 4(m 12 q * y + m * 12 q y + 2im 11 m 11y − 2im * 12 m 12y ), At last, we note that the expressions of m ij can be rewritten in the determinant form as where . (58)