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

Abstract

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 𝑁-times 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.

1. Introduction

Investigations on the nonlinear Schrödinger equation have attracted certain interest in nonlinear optics [1–6]. In this paper, we mainly investigate the following mixed nonlinear Schrödinger equation (MNLS) [7]: 𝑖𝑢_𝑧+12𝑢_𝑡𝑡1+𝜎4|𝑢|²𝑢−𝜎3𝑖𝛿|𝑢|16²𝑢_𝑡+Γ𝑢=0, (1) where 𝑢 denotes the slowly varying complex pulse envelope, 𝑧 and 𝑡 are the partial derivatives with respect to the longitudinal distance and retarded time, 𝜎=±1 corresponds to the focusing and defocusing nonlinear Schrödinger (NLS) equation, 𝛿 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]:Ψ_𝑡𝜆=𝑈Ψ=2𝑖²−1𝑖3𝛿2𝑖𝜆𝑢2𝜆𝑢^∗2𝑖𝜆²−13𝛿Ψ,Ψ_𝑧−=𝑉Ψ=4𝑖9𝛿²𝜆²−1²+𝑖8𝜆²|𝑢|²Γ+𝑖2−𝑖𝜆𝜆²−13𝛿𝑢+3𝑖|𝑢|32𝛿𝜆²𝜆𝑢−4𝑢_𝑡−𝑖𝜆𝜆²−1𝑢3𝛿^∗+3𝑖|𝑢|32𝛿𝜆²𝑢^∗+𝜆4𝑢^∗_𝑡4𝑖9𝛿²𝜆²−1²−𝑖8𝜆²|𝑢|²Γ−𝑖2Ψ, (2a)(2b)where Ψ=(𝜓₁,𝜓₂)^′ (′ denotes the transpose of a matrix). One can deduce (1) directly from the zero curvature equation 𝑈_𝑧−𝑉_𝑡+𝑈𝑉−𝑉𝑈=0.

During the past decades, [8–10] 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 𝑁-fold Darboux transformation and the interaction characters between two breathers have not been analysed. The aim of this paper is mainly to construct 𝑁-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.

2. 𝑁-Fold Darboux Transformation

Taking the transformation as Ψ^′Ψ=()=𝑇Ψ=𝐴𝐵𝐶𝐷−1^𝑁+^𝑁_𝑛=1𝑎_2𝑛𝜆^𝑁2𝑛_𝑛=1𝑏_2𝑛−1𝜆_𝑁^2𝑛−1_𝑛=1𝑐_2𝑛−1𝜆^2𝑛−1(−1)^𝑁+^𝑁_𝑛=1𝑑_2𝑛𝜆^2𝑛Ψ, (3) we require that Ψ^′ must satisfyΨ^′_𝑡=𝑈^′Ψ^′=𝑇_𝑡+𝑇𝑈𝑇⁻¹Ψ^′,Ψ^′_𝑧=𝑉^′Ψ^′=𝑇_𝑧+𝑇𝑉𝑇⁻¹Ψ^′, (4a)(4b)where 𝑈^′, 𝑉^′ have the same form as 𝑈, 𝑉. From the rules of the Darboux transformation [11–14], we know that 𝑇|_𝜆=𝜆1Ψ=0, 𝑎_2𝑛, 𝑏_2𝑛−1, 𝑐_2𝑛−1, and 𝑑_2𝑛 can be calculated by the linear algebraic system: ^𝑁_𝑛=1𝑎_2𝑛𝜆_𝑗+𝛼_𝑗𝑏_2𝑛−1𝜆_𝑗^2𝑛−1=(−1)^𝑁,_𝑁𝑛=1𝑐_2𝑛−1+𝛼_𝑗𝑑_2𝑛𝜆_𝑗𝜆_𝑗^2𝑛−1=(−1)^𝑁+1𝛼_𝑗, (5) with 𝛼_𝑗=𝜙₂𝜆_𝑗−𝛾_𝑗𝜓₂𝜆_𝑗𝜙₁𝜆_𝑗−𝛾_𝑗𝜓₁𝜆_𝑗,1≤𝑗≤2𝑁, (6) and 𝜙=(𝜙₁,𝜙₂)^′, 𝜓=(𝜓₁,𝜓₂)^′ as two basic solutions of the spectral system (2a)-(2b) and 𝜆_𝑗 and 𝛾_𝑗(𝜆_𝑘≠𝜆_𝑗, 𝛾_𝑘≠𝛾_𝑗, as 𝑘≠𝑗) as parameters suitably chosen such that the determinant of coefficients for (5) is nonzero. Thus, 𝑎_2𝑛, 𝑏_2𝑛−1, 𝑐_2𝑛−1, and 𝑑_2𝑛 are uniquely determined by (5). Equations (3) and (5) show that det𝑇(𝜆) is a 2𝑁 polynomial of 𝜆, and det𝑇𝜆_𝑗=𝐴𝜆_𝑗𝐷𝜆_𝑗−𝐵𝜆_𝑗𝐶𝜆_𝑗,𝐴𝜆_𝑗=−𝛼_𝑗𝐵𝜆_𝑗,𝐶𝜆_𝑗=−𝛼_𝑗𝐷𝜆_𝑗. (7) Therefore, one can derive that det𝑇(𝜆)=∏^2𝑁_𝑗=1(𝜆−𝜆_𝑗)=0, so 𝜆_𝑗(𝑗=1,2,…) are 2𝑁 roots for the determinant of matrix 𝑇(𝜆).

Meanwhile, defining 𝑇_𝑡+𝑇𝑈𝑇^∗=𝑓₁₁(𝜆)𝑓₁₂(𝜆)𝑓₂₁(𝜆)𝑓₂₂(𝜆),𝑇_𝑧+𝑇𝑉𝑇^∗=𝑔₁₁(𝜆)𝑔₁₂(𝜆)𝑔₂₁(𝜆)𝑔₂₂(𝜆), (8) one can deduce that 𝑓₁₁(𝜆) and 𝑓₂₂(𝜆) are (4𝑁+2), 𝑓₁₂(𝜆) and 𝑓₂₁(𝜆) are 4𝑁+1, 𝑔₁₁(𝜆) and 𝑔₂₂(𝜆) are (4𝑁+4), 𝑔₁₂(𝜆) and 𝑔₂₁(𝜆) are 4𝑁+3th-order polynomials of 𝜆, and𝛼_𝑗𝑡=𝑖2𝜆_𝑗𝑢^∗−4𝑖𝜆²_𝑗−1𝛼3𝛿^𝑗−𝑖2𝜆_𝑗𝑢𝛼²_𝑗,𝛼_𝑗𝑧=𝑐𝜆_𝑗−2𝑎𝜆_𝑗𝛼_𝑗−𝑏𝜆_𝑗𝛼²_𝑗, (9a)(9b)where 𝑎𝜆_𝑗=−4𝑖9𝛿²𝜆²_𝑗−1²+𝑖8𝜆²_𝑗|𝑢|²,𝑏𝜆_𝑗=−𝑖𝜆_𝑗𝜆²_𝑗−13𝛿𝑢+3𝑖32𝛿𝜆_𝑗|𝑢|²𝜆𝑢−^𝑗4𝑢_𝑥,𝑐𝜆_𝑗=−𝑖𝜆_𝑗𝜆²_𝑗−1𝑢3𝛿^∗+3𝑖32𝛿𝜆_𝑗|𝑢|²𝑢^∗+𝜆_𝑗4𝑢^∗_𝑥. (10) By virtue of the above conclusions, one can deduce that all 𝜆_𝑗(1⩽𝑗⩽2𝑁) are roots of 𝑓_𝑘𝑗(𝜆), 𝑔_𝑘𝑗(𝜆)(𝑘,𝑗=1,2). Noting (7), one can arrive at det𝑀∣𝑓_𝑘𝑗(𝜆),det𝑀∣𝑔_𝑘𝑗(𝜆)𝑘,𝑗=1,2. (11) According to 𝑇𝑇^∗=det|𝑇|𝐸, one can transform (8) into 𝑇_𝑡+𝑇𝑈=𝑃(𝜆)𝑇,𝑇_𝑧+𝑇𝑉=𝑄(𝜆)𝑇, (12) where𝑃(𝜆)=𝑝⁽²⁾₁₁𝜆²+𝑝⁽¹⁾₁₁𝜆+𝑝⁽⁰⁾₁₁𝑝⁽¹⁾₁₂𝜆+𝑝⁽⁰⁾₁₂𝑝⁽¹⁾₂₁𝜆+𝑝⁽⁰⁾₂₁𝑝⁽²⁾₂₂𝜆²+𝑝⁽¹⁾₂₂𝜆+𝑝⁽⁰⁾₂₂,𝑄(𝜆)=^4𝑗=0𝑞^(𝑗)₁₁𝜆^𝑗3_𝑗=0𝑞^(𝑗)₁₂𝜆^𝑗_3𝑗=0𝑞^(𝑗)₂₁𝜆^𝑗4_𝑗=0𝑞^(𝑗)₂₂𝜆^𝑗. (13a)(13b)Comparing the coefficients of 𝜆^2𝑁+4, 𝜆^2𝑁+3, 𝜆^2𝑁+2, 𝜆^2𝑁+1, and 𝜆^2𝑁 in (12), one can obtain𝑝⁽²⁾₁₁=−𝑝⁽⁰⁾₁₁=−𝑝⁽²⁾₂₂=𝑝⁽⁰⁾₂₂=2𝑖,𝑝3𝛿₍₁₎₁₁=𝑝⁽⁰⁾₁₂=𝑝⁽⁰⁾₂₁=𝑝⁽¹⁾₂₂=0,𝑝⁽¹⁾₁₂=𝑖2𝑢𝑎_2𝑁𝑑_2𝑁−8𝑏3𝛿𝑑^2𝑁−1_2𝑁=𝑖2𝑢^′,𝑝⁽¹⁾₂₁=𝑖2𝑢^∗𝑑_2𝑁𝑎_2𝑁+8𝑐3𝛿𝑎^2𝑁−1_2𝑁=𝑖2𝑢^′∗,𝑞⁽⁴⁾₂₂=𝑞⁽⁰⁾₂₂=−𝑞⁽⁴⁾₁₁=−𝑞⁽⁰⁾₁₁=4𝑖9𝛿²,𝑞⁽³⁾₁₁=𝑞⁽¹⁾₁₁=𝑞⁽²⁾₁₂=𝑞⁽⁰⁾₁₂=𝑞⁽²⁾₂₁=𝑞⁽⁰⁾₂₁=𝑞⁽³⁾₂₂=𝑞⁽¹⁾₂₂=0,𝑞⁽³⁾₁₂𝑖=−𝑢𝑎3𝛿𝑑^2𝑁_2𝑁−8𝑏3𝛿𝑑^2𝑁−1_2𝑁𝑖=−𝑢3𝛿^′,𝑞⁽³⁾₂₁𝑖=−𝑢3𝛿^∗𝑑_2𝑁𝑎_2𝑁+8𝑐3𝛿𝑎^2𝑁−1_2𝑁𝑖=−𝑢3𝛿^′∗,𝑞⁽²⁾₁₁=8𝑖9𝛿²−𝑖𝑏3𝛿𝑎^2𝑁−1_2𝑁𝑢^∗+𝑖8𝑢𝑢^∗+𝑞⁽³⁾₁₂𝑐_2𝑁−1𝑎_2𝑁=8𝑖9𝛿²+𝑖8𝑢^′𝑢^′∗,𝑞⁽²⁾₂₂=−8𝑖9𝛿²+𝑖𝑐3𝛿𝑎^2𝑁−1_2𝑁𝑢^∗−𝑖8𝑢𝑢^∗−𝑞⁽³⁾₂₁𝑏_2𝑁−1𝑎_2𝑁=−8𝑖9𝛿²−𝑖8𝑢^′𝑢^′∗,𝑞⁽¹⁾₁₂=1𝑑_2𝑁−8𝑖9𝛿²𝑏_2𝑁−1+8𝑖9𝛿²𝑏_2𝑁−3−𝑞⁽²⁾₁₁𝑏_2𝑁−1−𝑞⁽³⁾₁₂𝑑_2𝑁−2+𝑖𝑢𝑎3𝛿^2𝑁−𝑖𝑢𝑎3𝛿^2𝑁−2−𝑖|𝑢|²8𝑏_2𝑁−1+3𝑖𝛿𝑎32^2𝑁|𝑢|²𝑢𝑢−^𝑡4𝑎_2𝑁,𝑞⁽¹⁾₂₁=1𝑎_2𝑁8𝑖9𝛿²𝑐_2𝑁−1−8𝑖9𝛿²𝑐_2𝑁−3−𝑞⁽²⁾₂₂𝑐_2𝑁−1−𝑞⁽³⁾₂₁𝑎_2𝑁−2+𝑖𝑢^∗𝑑3𝛿^2𝑁−𝑖𝑢^∗𝑑3𝛿^2𝑁−2+𝑖|𝑢|²8𝑐_2𝑁−1+3𝑖𝛿𝑑32^2𝑁|𝑢|²𝑢^∗+𝑢^∗_𝑡4𝑑_2𝑁. (14a)(14b)(14c)(14d)(14e)(14f)(14g)(14h)(14i)(14j)Substituting 𝑢^′=𝑢(𝑎_2𝑁/𝑑_2𝑁)−(8/3𝛿)(𝑏_2𝑁−1/𝑑_2𝑁), 𝑢^′∗=𝑢^∗(𝑑_2𝑁/𝑎_2𝑁)+(8/3𝛿)(𝑐_2𝑁−1/𝑎_2𝑁) into (14i) and (14j), one can calculate that 𝑞⁽¹⁾₁₂=𝑖𝑢^′+3𝛿3𝑖𝛿𝑢32^′2𝑢^′−14𝑢^′_𝑥,𝑞⁽¹⁾₂₁=𝑖𝑢^′∗+3𝛿3𝑖𝛿𝑢32^′2𝑢^′∗−14𝑢_𝑥^′∗. (15) Summarying the above conclusions, one can arrive at the following theorm.

Theorem 1. The solution 𝑞 for (1) is mapped into the new solution 𝑞^′ under the Darboux transformation: 𝑢^′𝑎=𝑢^2𝑁𝑑_2𝑁−8𝑏3𝛿𝑑^2𝑁−1_2𝑁,𝑢^′∗=𝑢^∗𝑑_2𝑁𝑎_2𝑁+8𝑐3𝛿𝑎^2𝑁−1_2𝑁. (16)

3. Soliton and Breather Solutions for (1)

In this section, we will construct soliton solutions for (1). Choosing 𝜙(𝜆)=(𝜙₁(𝜆),𝜙₂(𝜆))^′ and 𝜓(𝜆)=(−𝜓^∗₂(𝜆^∗),𝜓^∗₁(𝜆^∗))^′ as two solutions of the spectral system (2a)-(2b), one can derive that 𝜆_2𝑗=𝜆^∗_2𝑗−1 and 𝛾_2𝑗=𝛾^∗_2𝑗−1, so 𝛼_2𝑗−1=[𝜙₂(𝜆_2𝑗−1)−𝛾_2𝑗−1𝜓₂(𝜆_2𝑗−1)]/[𝜙₁(𝜆_2𝑗−1)−𝛾_2𝑗−1𝜓₁(𝜆_2𝑗−1)].

For 𝑁=1, from (5), one can obtain that 𝑎₂=𝜆₁𝜙₂²+𝜆^∗₁𝜙₁²𝜆₁²𝜆₁𝜙₁²+𝜆^∗₁𝜙₂²,𝑏₁=𝜆₁^∗2−𝜆²₁𝜙₁𝜙^∗₂𝜆₁²𝜆₁𝜙₁²+𝜆^∗₁𝜙₂², (17) and 𝑑₂=𝑎^∗₂, 𝑐₁=−𝑏^∗₁. Then from (16), one can obtain 𝑢⁽¹⁾Δ=𝑢^∗Δ²+83𝛿Δ²𝜆²₁−𝜆₁^∗2𝜙₁𝜙^∗₂Δ^∗, (18) with Δ=𝜆₁|𝜙₁|²+𝜆^∗₁|𝜙₂|².

For 𝑁=2, from 𝑑₄=𝑎^∗₄, 𝑐₃=−𝑏^∗₃, and (5), one can obtain that²_𝑛=1𝑎_2𝑛𝜆_2𝑗−1+𝛼_2𝑗−1𝑏_2𝑛−1𝜆^2𝑛−1_2𝑗−1=1,_2𝑛=1𝑎_2𝑛𝜆^∗_2𝑗−1𝛼^∗_2𝑗−1−𝑏_2𝑛−1𝜆^{∗(2𝑛−1)}_2𝑗−1=−𝛼_2𝑗−1, (19a)(19b)with 𝑗=1,2, so (19a)-(19b) result: 𝑎₄=Ω_𝑎4Ω,𝑏₃=Ω_𝑏3Ω, (20) where 𝜆Ω=²¹𝜆₁𝛼₁𝜆⁴₁𝛼₁𝜆³₁𝜆²₃𝜆₃𝛼₃𝜆⁴₃𝛼₃𝜆³₃𝜆₁^∗2𝛼^∗₁−𝜆^∗₁𝜆₁^∗4𝛼^∗₁−𝜆₁^∗3𝜆₁^∗2𝛼^∗₃−𝜆^∗₃𝜆₃^∗4𝛼^∗₃−𝜆₃^∗3. (21)Ω_𝑎4 and Ω_𝑏3 are produced from Ω by replacing its third and fourth columns with (1,1,−𝛼^∗₁,−𝛼^∗₃), respectively. Then the two-soliton solutions for (1) can be derived as 𝑢⁽²⁾𝑎=𝑢⁴𝑑₄−8𝑏3𝛿𝑑³₄. (22)

Then we we mainly discuss soliton solutions from three different seeds.

Case 1. Considering the vanishing background of 𝑢=0, we can generate elastic interactions and bound states between two solitons for (1) as shown in Figure 1.

(a)

(b)

Figure 1 

Evolution of the two-soliton solutions of (1). Parameters are (a) 𝜆1=2+2𝑖, 𝜆2=3+2𝑖, 𝛿=2.5, Γ=0.1 and (b) 𝜆1=1+2𝑖, 𝜆2=1.0001−2𝑖, 𝛿=2.5, Γ=0.1.

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 |𝜆₁|→|𝜆₂|, while if the value of ||𝜆₁|−|𝜆₂|| increases, the elastic interactions between two solitons will happen.

Case 2. In the case of the cw background as 𝑢=𝑢_𝑐exp𝑖(𝜅𝑡+𝜔𝑧), we can derive the nonlinear dispersion relation 𝜔=(4𝑢²_𝑐−8𝜅²+3𝑢²_𝑐𝜅𝛿+16Γ)/16 for (1). By the method of separation of variables and the superposition principle, we derive 𝜙₁=𝑐₁𝑒^𝜃1+𝑐₂𝑒^𝜃2exp𝑖(𝜅𝑡+𝜔𝑧),𝜙₂=𝑐₃𝑒^𝜃1+𝑐₄𝑒^𝜃2, (23) where 𝜃₁=𝑡21−𝑖𝜅−3𝛿𝐵²−𝐴²+𝑧21−𝑖𝜔−3𝛿𝐵²−𝐴²Θ,𝜃₂=𝑡21−𝑖𝜅+3𝛿𝐵²−𝐴²+𝑧21−𝑖𝜔+3𝛿𝐵²−𝐴²Θ, (24) with 𝐴=4𝜆²₁−1,𝐵=3𝑖𝑢_𝑐𝛿𝜆₁,Θ=9𝛿²𝑢²_𝑐−32𝜆²₁−1.48𝛿 (25) For simplicity, setting 𝑐₄=𝑐₁ and through direct computations, we can generate that 𝑐₃=𝜃_1𝑡+𝑖𝜅−𝛾(𝑖/2)𝜆₁𝑢_𝑐𝑐₁,𝑐₂=(𝑖/2)𝜆₁𝑢_𝑐𝜃_1𝑡𝑐+𝑖𝜅−𝛾⁴, (26) with 𝛾=2𝑖(𝜆²₁−1)/3𝛿. Substituting (27) into (18) and taking Re(𝜆₁)=Im(𝜆₁), 𝜅=0, we can obtain Akhmediev-breather 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 𝜆₁ and 𝜆₂, we can control the directions of those breathers as shown in those figures.

(a)

(b)

Figure 2 

Evolution of the breather solutions of (1). Parameters are (a) 𝑢𝑐=1, 𝜅=0, 𝛿=3, 𝜆1=2.05+2.05𝑖, Γ=0 and (b) 𝑢𝑐=1, 𝜅=0, 𝜆1=2.05+2.05𝑖, 𝜆2=2+2𝑖, 𝛿=3, Γ=0.

(a)

(b)

Figure 3 

Evolution of the breather solutions of (1). Parameters are (a) 𝑢𝑐=1, 𝜅=0, 𝜆1=2.05+1.05𝑖, 𝜆2=2+2𝑖, 𝛿=3, Γ=0 and (b) 𝑢𝑐=1, 𝜅=0, 𝜆1=1.5+𝑖, 𝜆2=3+1.1𝑖, 𝛿=3, Γ=0.

Case 3. Considering the constant background of 𝑢=𝜅₁+𝑖𝜅₂, (1) gives the relation 𝜅²₁+𝜅²₂=−4Γ. Defining 𝜆=𝐴_𝑐+𝑖𝜅_𝑐, and for simplicity setting 𝜅_𝑐=8(𝐴²_𝑐−1)−9Γ𝛿², we can obtain 𝜙₁=𝑐₁𝑒^𝜗+𝑐₂𝑒^−𝜗,𝜙₂=𝑐₃𝑒^𝜗+𝑐₄𝑒^−𝜗, (27) where 𝜗=2𝑡𝜉₁+𝑖𝜂₁+𝑧𝜉₁+𝑖𝜂₁8𝜆²−1+9Γ𝛿²36𝛿²,𝑐₃=𝜗_𝑡−(2𝑖/3𝛿)𝜆²−1(𝑖/2)𝜆𝜅₁+𝑖𝜅₂𝑐₁,𝑐₂=(𝑖/2)𝜆𝜅₁+𝑖𝜅₂−𝜗_𝑡−(2𝑖/3𝛿)𝜆²𝑐−1⁴, (28) with 𝜉₁+𝑖𝜂₁=𝛽16+𝛽,𝛽=9Γ𝐴²_𝑐. (29) Owing to 𝛽 which is always negative, we discuss two different cases according to the sign of (16+𝛽).(A)When (16+𝛽)>0, the value of 𝛽(16+𝛽) will be negative; then we can arrive at 𝜉₁=0, 𝜂₁=−𝛽(16+𝛽). Substituting these conclusions and (27) into (18), we can derive the Ma-breather solution for (1) as shown in Figure 4(a).(B)When (16+𝛽)<0, the value of 𝛽(16+𝛽) will be positive; then we can arrive at 𝜉₁=𝛽(16+𝛽), 𝜂₁=0. Substituting these conclusions and (27) into (18), we can derive the Akhmediev-breather solution for (1) as shown in Figure 4(b).
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 𝜅₁+𝑖𝜅₂ approaches zero, the Ma-breather will become the one-soliton solution. Since 𝜅²₁+𝜅²₂=−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+𝛽, so the group velocities of the Akhmediev breathers are dependent on parameters 𝛽;(2)parameters 𝐴_𝑐 can affect the amplitudes.

(a)

(b)

Figure 4 

Evolution of the breather solutions of (1). Parameters are (a) 𝐴𝑐=0.5, 𝜅1=1, 𝜅2=1, 𝛿=1 and (b) 𝐴𝑐=0.05, 𝜅1=1, 𝜅2=1, 𝛿=1.

4. 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 𝑁-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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express sincere thanks to each member of their discussion group for their suggestions. This work has been supported by the Special Funds of the National Natural Science Foundation of China under Grant nos. 11347165 and 11172194, by the Natural Science Foundation of Shanxi Province under Grant no. 2012021011-3, and by the Science and Technology Innovation Fund of Taiyuan University of Technology.