Breathers and Soliton Solutions for a Generalization of the Nonlinear Schrödinger Equation

A generalized nonlinear Schrödinger equation, which describes the propagation of the femtosecond pulse in single mode optical silica fiber, is analytically investigated. By virtue of the Darboux transformation method, some new soliton solutions are generated: the bright one-soliton solution on the zero background, the dark one-soliton solution on the continuous wave background, the Akhmediev breather which delineates the modulation instability process, and the breather evolving periodically along the straight line with a certain angle of x-axis and t-axis. Those results might be useful in the study of the femtosecond pulse in single mode optical silica fiber.


Introduction
Investigations on the dynamic features of solitons have attracted certain interest in nonlinear optics [1][2][3][4].Optical solitons have been regarded as a candidate for the optical communication networks [5][6][7][8].On the basis of the balance between the group velocity dispersion and self-phase modulation [9,10], the propagation of optical soliton is usually governed by the nonlinear Schrödinger (NLS) equation [11][12][13][14]: However, when optical pulses are shorter, the NLS equation becomes inadequate, and it is necessary to include additional terms [6,7].For example, in single mode optical silica fiber, in order to describe the propagation of femtosecond pulse, the higher order asymptotic terms should be retained [15]; to understand such phenomena, we consider the following generalization of the NLS equation [16]: Analogous to the circumstance that the Camassa-Holm equation provides a better approximation of the KdV equation [15], (2) is related to the NLS equation, provided that one retains terms of the next asymptotic order.Under the transformation  =   , (2) can be converted into the following equation [16,17]: where  denotes the complex field envelope and the subscripts  and  are the longitudinal distance and retarded time, respectively.In recent years, some results have been obtained for (1): (1) Reference [15] has analyzed the dynamic features of the rogue wave solutions; (2) Reference [16] has analyzed the conservation laws, bi-Hamiltonian structure, Lax pair, and initial-value problem; (3) Reference [17] has derived some soliton solutions by using the bilinear method.The aim of this paper is mainly to derive some new soliton solutions for (3) using the Darboux transformation (DT) method and analyze the dynamic features of soliton solutions.This paper will be organized as follows.In Section 2, we will give the Lax pair and construct the DT for (3).In Section 3, we will obtain bright one-soliton, dark onesoliton, and breather solutions and analyze the dynamic features of soliton solutions by using some figures.Finally, our conclusions will be addressed in Section 4.

Lax Pair and Darboux Transformation
Employing the Ablowitz-Kaup-Newell-Segur formalism [18], [15,16] has given the Lax pair associated with (3) as where Ψ = ( 1 ,  2 )  ( denotes the transpose of a matrix), and the matrices  and  have the form where  is a spectral parameter and Through direct computations, it can be verified that the zero curvature equation   −   +  −  = 0 exactly gives rise to (3).
Through direct computation, we can obtain So, if  is a seed solution of (3),   is also a solution of (3).

One and Breather Solutions (3)
In this section, we will apply the DT constructed to obtain one and breather solutions for (3).Now we take the nonzero continuous wave (cw) solution  =  exp ( + ) as the initial seed for (3), where , , and  are all real parameters.Equation (3) requires that the frequency  satisfies the nonlinear dispersion relation: Solving (4a) and (4b) and setting  1 =  1 exp ( + ),  2 =  2 , one can obtain where Through tedious computations, one can arrive at where and  1 ,  2 ,  3 , and  4 are complex constants satisfying: with with  1 ,  2 ,  1 ,  1 ,  2 , and  2 as real numbers, and we can derive Now, substituting ( 14) into (10), we can obtain the solution for (1) as where with Next according to different values of those parameters in solution ( 19), we will analyse the novel properties of solitons.(3).When  = 0, that is to say, the initial seed for (3) is zero, solution (19) reduces to onesoliton solution as

One-Soliton Solutions for
with Mathematical Problems in Engineering So, the amplitude and envelope velocity will increase when the value of  1 becomes bigger.As shown in Figures 1, the amplitude is higher and the envelope velocity is bigger in Figure 1(a) than in Figure 1(b).

Breather and Dark
One-Soliton Solutions for (1).When  ̸ = 0, the nonzero initial seed  =  exp ( + ) describes the nonvanishing boundary conditions.For simplicity, taking  1 =  2 ,  = 2/ 2 , we have the following relations: With the previous conclusions, solution (19) can be converted into: where with ) , Figure 2 display the propagation characteristics of solitons via solutions (26).Figures 2(a) and 2(b) depict the dynamic features of breathers; as shown in Figure 2(a), the main feature is the propagation of the breather that is periodic in the space coordinate and aperiodic in the time coordinate; that is to say, we can obtain the Akhmediev breather [23] via solutions (26) under suitable parameters chosen.In addition, the Akhmediev breather can be regarded as a modulation instability process.

Conclusions
Our main attention has been focused on (3), which can describe the propagation of femtosecond pulse in single mode optical silica fiber.By using the Darboux transformation method, we have obtained (1) bright one soliton on the zero background; (2) two types of breathers: the Akhmediev soliton which delineates the modulation instability process and the breather evolving periodically along the straight line with a certain angle of -axis and -axis; (3) the dark onesoliton solution on the continuous wave background.

Figure 2 (
b) portrays the propagation of the breather evolving periodically along the straight line with a certain angle of -axis and -axis.Figure2(c) describes the dynamic feature of the dark one-soliton solutions via solutions (26) on the continuous wave background, which is different with Figures 1 via Solutions(22) on the zero background.