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The nonlocal nonlinear Schrödinger and Maxwell – Bloch equation

In this paper, the nonlocal nonlinear Schrödinger and Maxwell – Bloch equations are is introduced. A particular case of this system, namely the Schrödinger equation, is integrable by the inverse scattering method as shown in the work of M. Ablowitz and Z. Musslimani. Following their idea, we prove the integrability of the nonlocal nonlinear Schrödinger and Maxwell – Bloch equation using its Lax pairs. Also the Darboux transformations are constructed, and soliton solutions are obtained from different "seed" solutions using them. One–fold, two–fold and N–fold determinant representations are obtained by this transformation. Moreover, soliton and solitons–like solutions, such as dynamic and topological soliton, periodic, domain walls, kink, lamp, bright and dark solitons, bright and dark rogue waves, bright and dark positons, etc., of this equation are built. In future papers, we will investigate the conservation laws of the nonlocal nonlinear Schrödinger and Maxwell – Bloch equation using the Lax pair.


Introduction
It is known that the nature of nonlinear real systems is one of the basic notions in modern science. Nonlinearity is the property which is applied in almost all fields of science. A nonlinear phenomenon is usually modeled in terms of nonlinear ordinary and/or differential equations in particular cases. Most of such nonlinear differential equations (NDE) are completely integrable. That is they accept some classes of interesting exact solutions, such as solitons, dromions, destructive waves, semilaritons, king, etc. They draw both mathematical and physical interest. Investigations of solitons, positons, brakes, dromions, "destructive waves" have become one of the interesting and highly active areas for research in modern science and technology during the last several decades. In particular, many of the completely integrable NDEs have already been established and investigated [1][2][3][4][5][6][7][8].
Among such integrable nonlinear systems, the Schrödinger and Maxwell -Bloch equations have a crucial role. The Schrödinger and Maxwell -Bloch equations describe solitons in fibers with resonance and erbium systems and have a (1+1) -dimension [9]. Using the Darboux transformation, the (1+1)-dimensional Schrödinger and Maxwell -Bloch equations were analyzed in [9], where soliton and periodic solutions were constructed from various "seeds".
Recently, a (2+1) -dimensional Schrödinger and Maxwell -Bloch equation was introduced in [10]. In this section, our aim is to built a Darboux transformation for (2+1) -dimensional Schrödinger and Maxwell -Bloch equation and to obtain soliton solutions. It where  denotes complex conjugation and p q, are complex functions of the real variables x and t . The equations (1)-(2) are called the nonlocal nonlinear Schrödinger equations and they were introduced by M. Ablowitz and Z. Musslimani in [10]. The nonlocal nonlinear Schrödinger equation is obtained from a new and simple reduction of the well-known AKNS system. It admits a Lax pair and an infinite number of conservation laws. The IST for decaying data is developed and a one breathing soliton solution is construced. The IST requires different scattering data symmetries than the classical NLS equation. A nonlocal NLS hierarchy and novel nonlocal Painleve´ type equations are also obtained. Now consider equations (1) -(5) so called Schrödinger and Maxwell -Bloch equation. The system of equations (1) -(5) admits a Lax pair representation and possesses an infinite number of conservation laws; hence, it is integrable. Using the inverse scattering method, corresponding to rapidly decaying initial data, one can linearize the equation and аштв solutions to the system of equations (1)-(5) including pure soliton solutions. Some of the essential properties of the nonlocal NLS equation are derived from the classical NLS equation where the nonlocal nonlinear term . Infact, we note that the system of equations (1) -(2) and the classical NLS share the symmetry that when and a complex conjugate is taken, then the equation stays invariant [16].
Corresponding Lax representation for the nonlocal nonlinear Schrödinger -Maxwell -Bloch equation (1) -(5) can be written as follows is the Pauli matrix and So the Lax representation of the integrable nonlocal nonlinear Schrödinger -Maxwel l-Bloch equation is given by A and B. It can be seen that from the compatibility condition from tx xt    we can obtain the zero curvature equation of the system of equations (6) and (7).
In the next section, one-fold Darboux transformation of the nonlocal nonlinear Schrödinger -Maxwell-Bloch equation will be obtained.

The one-fold Darboux transformation for the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation
Using the Darboux transformation for the AKNS system [11], consider the transformation of the linear system of equations (6)-(7) The function   1  is assumed to satisfy The relation between is the same as the relation between A  . In order to make the system (9) invariant under the transformation (8), T must satisfy the following The relation between can be derived from these equations, which is actually the Darboux transformation of the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation (1)-(5).
From the above identities, after simplifications and comparisons of the coefficients from equation (11), it follows that ] , [ : Finally, from (13)-(15) we obtain solutions [1] 12 Similarly, comparing the coefficients of i  of the two sides of the equation (12) gives us These solutions (16) and (22) In the following section we give the determinant representation of the Darboux transformation for the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation.

Conclusion
In this paper, we have obtained the DT for the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation. Using the derived DT, some exact solutions including, the one-soliton solution are obtained. The determinant representations are given for one-fold, two-fold and n-fold DT for the onlocal nonlinear Schrödinger and Maxwell -Bloch equation. Using obtained results, one can also find the n-solitons, breathers and rogue wave solutions of the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation. It is interesting to note that the rogue wave solutions of nonlinear equations are currently one of the most active topics in nonlinear physics and mathematics. The application of the obtained solutions in physics is an interesting subject. In particular, we hope that the presented solutions may be used in experiments or optical fibre communication. Also we will study some important generalizations of the nonlocal nonlinear Schrödinger and Maxwell -Bloch equation in future.