Soliton and Breather Solutions for the Mixed Nonlinear Schrödinger Equation via N-Fold Darboux Transformation

Under investigation in this paper is the mixed nonlinear Schrödinger equation, which describes the propagation of the subpicosecond or femtosecond optical pulse in a monomodal optical fiber. The Darboux transformation is constructed and Ntimes iterative potential formula is presented. Two-soliton and breather solutions are derived on the vanishing and two types of nonvanishing backgrounds: the continuous wave(cw) background and constant background, respectively. The dynamic features of the solitons and breathers are discussed via analytic solutions and graphical illustration.


Introduction
Investigations on the nonlinear Schrödinger equation have attracted certain interest in nonlinear optics [1][2][3][4][5][6].In this paper, we mainly investigate the following mixed nonlinear Schrödinger equation (MNLS) [7]: where  denotes the slowly varying complex pulse envelope,  and  are the partial derivatives with respect to the longitudinal distance and retarded time,  = ±1 corresponds to the focusing and defocusing nonlinear Schrödinger (NLS) equation,  is a real parameter describing the measure of the derivative cubic nonlinearity, and Γ denotes the loss or gain coefficient.Equation (1) usually governs the subpicosecond or femtosecond optical pulse propagation in a monomodal optical fiber.As a complete integrability model, ( 1) is equivalent to the following Wadati-Konno-Ichikawa (WKI) spectral system [7,8]: where Ψ = ( 1 ,  2 )  (  denotes the transpose of a matrix).One can deduce (1) directly from the zero curvature equation   −   +  −  = 0.
During the past decades, [8][9][10] have derived some soliton solutions via the dressing method, the Hirota bilinear method and the technique of determinant calculation, respectively.However, to our knowledge, the soliton and breather solutions of (1) have not been generated through fold Darboux transformation and the interaction characters between two breathers have not been analysed.The aim of this paper is mainly to construct -fold Darboux transformation (DT) in Section 2, derive two-soliton, one-and two-breather solutions via the obtained DT, and discuss dynamic features of those solitons and breathers in Section 3. Finally, our conclusions will be addressed in Section 4.
Theorem 1.The solution  for (1) is mapped into the new solution   under the Darboux transformation: 3. Soliton and Breather Solutions for (1) In this section, we will construct soliton solutions for (1).
Case 1. Considering the vanishing background of  = 0, we can generate elastic interactions and bound states between two solitons for (1) as shown in Figure 1.
Figure 1(a) depicts the interactions of two solitons on the vanishing background.One can find that main features of the interaction are that the shapes, amplitudes, and pulse widths all remain invariant except for slightly visible phase shifts, so the interaction is elastic.Figure 1(b) shows that when suitable parameters are chosen, two bound solitons with the same amplitude propagate in parallel without any effect on each other even if the propagation distance grows long enough.From Figure 1, we can conclude that the parallel bound solitons will form when increases, the elastic interactions between two solitons will happen.
Case 2. In the case of the cw background as  =   exp ( + ), we can derive the nonlinear dispersion relation  = (4 2  − 8 2 + 3 2   + 16Γ)/16 for (1).By the method of separation of variables and the superposition principle, we derive where with For simplicity, setting  4 =  1 and through direct computations, we can generate that with  = 2( 2 1 − 1)/3.Substituting (27) into (18) and taking Re( 1 ) = Im( 1 ),  = 0, we can obtain Akhmedievbreather solutions on the nonvanishing background for (1) as shown in Figure 2(a).One can observe that the main feature is propagation of the Akhmediev breather which is periodic in the space coordinate and aperiodic in the time coordinate.Therefore, it is considered as a modulation instability (MI) process in which a cw beam becomes unstable [15].MI was predicted to occur in optical fibers and was experimentally observed [16].
Iterating the DT again, we can obtain the two-breather solution for (1) as shown in Figures 2(b), 3(a), and 3(b).One can observe that the interactions between those breathers are also elastic.Through adjusting the value of  1 and  2 , we can control the directions of those breathers as shown in those figures.
One can observe from Figure 4(a) that the breathers time periodically propagate on the constant backgrounds; that is, they are the Ma-breathers.In addition, as  1 +  2 approaches zero, the Ma-breather will become the one-soliton solution.Since  2 1 +  2 2 = −4Γ and Γ denotes the gain or loss coefficient in in (1), we can conclude that the absence of the gain or loss term can become breathers into one-soliton solutions.
From Figure 4(b), one can find that the Akhmediev breather is periodic in the space coordinate and aperiodic in the time coordinate.Generally, the time-aperiodic solution can be regarded as a homoclinic or separatrix trajectory in the infinite-dimension phase space of the solutions for (1) with periodic boundary conditions in space.Through numerical simulation, one can gain the facts that are in Figure 4(b) as follows: (1) the periods are in inverse proportion to the value of 16 + , so the group velocities of the Akhmediev breathers are dependent on parameters ; (2) parameters   can affect the amplitudes.

Conclusions
Our attention has been focused on (1) which describes the propagation of the subpicosecond or femtosecond optical pulse in a monomodal optical fiber.With symbolic computation, we have constructed -fold Darboux transformation and derived two-soliton and breather solutions on the vanishing and two types of nonvanishing backgrounds, respectively.In addition, some figures have been plotted to display the dynamic characteristics of those solitons.