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

Abstract

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.

1. Introduction

Investigations on the dynamic features of solitons have attracted certain interest in nonlinear optics [1–4]. Optical solitons have been regarded as a candidate for the optical communication networks [5–8]. 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 [11–14]: 𝑖𝐸_𝜁+12𝐸_𝜏𝜏+|𝐸|²𝐸=0. (1) 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]: 𝑖𝑢_𝑡−𝑢_𝑥𝑡+𝑢_𝑥𝑥+|𝑢|²𝑢+𝑖|𝑢|²𝑢_𝑥=0. (2) 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 𝑞=𝑒^𝑖𝑥𝑢, (2) can be converted into the following equation [16, 17]: 𝑖𝑞_𝑥𝑡−𝑖𝑞_𝑥𝑥+2𝑞_𝑥−𝑞²𝑞_𝑥+𝑖𝑞=0, (3) where 𝑞 denotes the complex field envelope and the subscripts 𝑥 and 𝑡 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 one-soliton, and breather solutions and analyze the dynamic features of soliton solutions by using some figures. Finally, our conclusions will be addressed in Section 4.

2. Lax Pair and Darboux Transformation

Employing the Ablowitz-Kaup-Newell-Segur formalism [18], [15, 16] has given the Lax pair associated with (3) as Ψ_𝑥Ψ=𝑈Ψ,_𝑡=𝑉Ψ, (4a)(4b)where Ψ=(𝜓₁,𝜓₂)^𝑇 (𝑇 denotes the transpose of a matrix), and the matrices 𝑈 and 𝑉 have the form 𝑈=𝜆²𝑈₂+𝜆𝑈₁,𝑉=𝜆²𝑉₂+𝜆𝑉₁+𝑉₀+1𝜆𝑉₋₁+1𝜆²𝑉₋₂, (5a)(5b)where 𝜆 is a spectral parameter and 𝑈₂=−𝑖00𝑖,𝑈₁=0𝑞_𝑥𝑞^∗_𝑥0,𝑉₂=𝑈₂,𝑉₁=𝑈₁,𝑉₋₂=14𝑈₂,𝑉₀=𝑖𝑖−2𝑞²0𝑖0−𝑖+2𝑞²,𝑉₋₁=0𝑖2𝑞−𝑖2𝑞^∗0. (6)

Through direct computations, it can be verified that the zero curvature equation 𝑈_𝑡−𝑉_𝑥+𝑈𝑉−𝑉𝑈=0 exactly gives rise to (3).

Next, based on Lax pair (4a) and (4b), we will give the DT [19–22] formalism for (3). Define Ψ^′=𝜆𝐷₁+𝐷₀+1𝜆𝐼Ψ, (7) where 𝐼 denotes the identity matrix and 𝐷₁=𝑠₁00𝑠₂,𝐷₀=0𝑠₃𝑠₄0, (8) with 𝑠₁=−(1/𝜆)𝜙₁²+1/𝜆^∗𝜙₂²𝜆𝜙₁²−𝜆^∗𝜙₂²,𝑠₂=−(1/𝜆)𝜙₂²+1/𝜆^∗𝜙₁²𝜆𝜙₂²−𝜆^∗𝜙₁²,𝑠₃=𝜆^∗/𝜆−𝜆/𝜆^∗𝜙₁𝜙^∗₂𝜆𝜙₁²−𝜆^∗𝜙₂²,𝑠₄=𝜆^∗/𝜆−𝜆/𝜆^∗𝜙₂𝜙^∗₁𝜆𝜙₂²−𝜆^∗𝜙₁². (9) One can verify that if (𝜙₁,𝜙₂)^𝑇 is the solution of Lax pair (4a) and (4b) with 𝜆, then (𝜙^∗₂,𝜙^∗₁)^𝑇 is also the solution of Lax pair (4a) and (4b) corresponding to 𝜆^∗.

Through direct computation, we can obtain 𝑞^′𝜆=𝑞+^∗2−𝜆²𝜙₁𝜙^∗₂𝜆𝜆^∗𝜆𝜙₁²−𝜆^∗𝜙₂². (10)

So, if 𝑞 is a seed solution of (3), 𝑞^′ is also a solution of (3).

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 𝑞=𝑎exp𝑖(𝐴𝑡+𝐵𝑥) as the initial seed for (3), where 𝑎, 𝐴, and 𝐵 are all real parameters. Equation (3) requires that the frequency 𝐴 satisfies the nonlinear dispersion relation: 𝐴=2−𝑎²1+𝐵+𝐵. (11)

Solving (4a) and (4b) and setting 𝜙₁=𝑓₁exp𝑖(𝐴𝑡+𝐵𝑥),  𝜙₂=𝑓₂, one can obtain 𝑓_1𝑥+𝑖𝐵𝑓₁=−𝑖𝜆²𝑓₁+𝑖𝑎𝐵𝜆𝑓₂,𝑓_2𝑥=−𝑖𝑎𝐵𝜆𝑓₁+𝑖𝜆²𝑓₂,𝑓_1𝑡+𝑖𝐴𝑓₁=𝛼𝑓₁+𝛽𝑓₂,𝑓_2𝑡=−𝛽𝑓₁−𝛼𝑓₂, (12) where 𝛼=−𝑖𝜆²𝑖+𝑖−2𝑎²−𝑖4𝜆²𝑖,𝛽=𝑖𝑎𝐵𝜆+2𝜆𝑎. (13)

Through tedious computations, one can arrive at 𝜙₁=𝑐₁exp𝜃₁+𝑐₂exp𝜃₂(),𝜙exp𝑖𝐴𝑡+𝐵𝑥₂=𝑐₃exp𝜃₁+𝑐₄exp𝜃₂, (14) where 𝜃₁=𝑥2−𝑖𝐵+𝜁₁+𝑖𝜂₁+𝑡2−𝑖𝐴+𝜁₂+𝑖𝜂₂,𝜃₂=𝑥2−𝑖𝐵−𝜁₁−𝑖𝜂₁+𝑡2−𝑖𝐴−𝜁₂−𝑖𝜂₂, (15) and 𝑐₁,  𝑐₂,  𝑐₃, and 𝑐₄ are complex constants satisfying: 𝑐₂=(𝐿+𝑖𝑀)𝑐₄,𝑐₃=(𝐿+𝑖𝑀)𝑐₁, (16) with 1𝜆=2𝜆₁+𝑖𝜆₂,𝜁₁+𝑖𝜂₁=−𝐵²−4𝜆²𝐵+𝜆⁴−𝑎²𝐵²𝜆²,𝜁₂+𝑖𝜂₂=−𝐴²−4𝛽²−𝛼²+𝑖𝐴𝛼, (17) with 𝜆₁,  𝜆₂,  𝜁₁,  𝜂₁,  𝜁₂, and 𝜂₂ as real numbers, and we can derive 𝐿=−2𝜁₁𝜆₂+𝜆₁𝜆²₁+𝜆²₂+2𝐵+2𝜂₁2𝑎𝐵𝜆²₁+𝜆²₂,𝑀=2𝜁₁𝜆₁+𝜆₂𝜆²₁+𝜆²₂−2𝐵−2𝜂₁2𝑎𝐵𝜆²₁+𝜆²₂. (18)

Now, substituting (14) into (10), we can obtain the solution for (1) as 𝑞=𝑎−4𝑖𝜆₁𝜆₂𝐺𝜆²₁+𝜆²₂𝐹exp𝑖(𝐴𝑡+𝐵𝑥), (19) where 𝐺=2𝐿coshΘ₁−2𝑖𝑀sinhΘ₁+𝐿²+𝑀²+1cosΘ₂+𝑖1−𝐿²−𝑀²sinΘ₂,𝐹=1−𝐿²−𝑀²𝜆₁sinhΘ₁+𝑖𝐿²+𝑀²+1𝜆₂coshΘ₁+2𝜆₁𝑀sinΘ₂+2𝑖𝜆₂𝐿cosΘ₂, (20) with Θ₁=𝜁₁𝑥+𝜁₂𝑡,Θ₂=𝜂₁𝑥+𝜂₂𝑡. (21)

Next according to different values of those parameters in solution (19), we will analyse the novel properties of solitons.

3.1. One-Soliton Solutions for (3)

When 𝑎=0, that is to say, the initial seed for (3) is zero, solution (19) reduces to one-soliton solution as 𝜆𝑞=²−𝜆^∗2𝜆𝜆^∗2sech𝜒₁+𝜒^∗₁exp𝜒₁−𝜒^∗₁, (22) with 𝜒₁=−𝑖𝜆²𝑖𝑥+𝑖−4𝜆²−𝑖𝜆²𝜒𝑡,_∗1=𝑖𝜆^∗2𝑖𝑥+−𝑖+4𝜆^∗2−𝑖𝜆^∗2𝑡. (23) Solution (22) represents a bright soliton whose dynamic features are delineated in Figures 1. Through symbolic computation, we can conclude the following physical quantities for solution (22): the maximum amplitude |(𝜆²−𝜆^∗2)/𝜆𝜆^∗2|, the width 1/|𝜆²−𝜆^∗2|, the envelope velocity (1/4𝜆^∗2−1/4𝜆²−𝜆²−𝜆^∗2)/(𝜆²−𝜆^∗2), and the energy of the one-soliton solution 𝐸₁=^+∞_−∞𝑞(𝑥,𝑡)²−𝑞(±∞,𝑡)²𝑑𝑥=128𝜆₁𝜆₂𝜆²₁+𝜆²₂³. (24) So, the amplitude and envelope velocity will increase when the value of 𝜆₁ 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).

(a)

(b)

Figure 1 

Evolution of the one-soliton solution of (22). The parameters are (a) 𝜆1=2,  𝜆2=4; (b) 𝜆1=1,  𝜆2=4.

3.2. Breather and Dark One-Soliton Solutions for (1)

When 𝑎≠0, the nonzero initial seed 𝑞=𝑎exp𝑖(𝐴𝑡+𝐵𝑥) describes the nonvanishing boundary conditions. For simplicity, taking 𝜆₁=𝜆₂,  𝐵=2/𝑎², we have the following relations:2𝐴=2+𝑎²−𝑎²2,𝜁₁=𝜆²₂,𝜂₁=2𝑎²,𝜁₂1=−𝜆²₂−𝜆²₂,𝜂₂=𝑎²2−𝑎²21,𝐿=𝜆₂𝑎1,𝑀=−𝜆₂𝑎+𝜆₂𝑎2. (25a)(25b)With the previous conclusions, solution (19) can be converted into: 𝑞=𝑎−4𝑖𝜆₁𝜆₂𝐺^′𝜆²₁+𝜆²₂𝐹^′2exp𝑖2+𝑎²−𝑎²22𝑡+𝑎²𝑥, (26) where 𝐺^′=𝐷₁coshΘ^′₁+𝐷₂sinhΘ^′₁+𝐷₃cosΘ^′₂+𝐷₄sinΘ^′₂,𝐹=𝐷₅coshΘ^′₁+𝐷₆sinhΘ^′₁+𝐷₇cosΘ^′₂+𝐷₈sinΘ^′₂, (27) with Θ^′₁=𝜆²₂𝑥−𝜆²₂+1𝜆²₂Θ𝑡,_′2=𝑎²2−2𝑎²2𝑡+𝑎²𝐷𝑥,₁=2𝑎𝜆₂,𝐷₂=𝑖𝑎𝜆₂−𝐷₁,𝐷₃=2𝑎²𝜆²₂+𝑎²𝜆²₂4,𝐷₄=𝑖2−𝐷₃,𝐷₅2=𝑖𝑎²𝜆₂+𝑎²𝜆³₂4,𝐷₆=2𝜆₂+𝑖𝐷₅,𝐷₇=2𝑖𝑎,𝐷₈=𝑎𝜆²₂+𝑖𝐷₇. (28) 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. Figure 2(b) portrays the propagation of the breather evolving periodically along the straight line with a certain angle of 𝑥-axis and 𝑡-axis. Figure 2(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.

(a)

(b)

(c)

Figure 2 

Evolution of the soliton solutions of (26). The parameters are (a) 𝑎=1, 𝜆2=0.5; (b) 𝑎=−2,  𝜆2=1.1; (c) 𝑎=1,  𝜆2=3.

4. 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 𝑥-axis and 𝑡-axis; (3) the dark one-soliton solution on the continuous wave background.

Acknowledgments

The express their sincere thanks to each member of their discussion group for their suggestions. This work has been supported by the National Natural Science Foundation of China under Grant nos. 11172194 and 61250011 and by the Natural Science Foundation of Shanxi Province under Grant no. 2012011004-3.