During the last four decades, optical solitons have been the object of intensive theoretical and experimental studies. ^{[ 1 – 4 ]} This is because that optical soliton can be treated as a natural data bit in ultrahigh-speed optical telecommunication systems. ^{[ 5 , 6 ]} The all-optical communication link will play the major role in optical fiber transmission systems such as all optical networks, optical communication, and optical logic devices. There are many nonlinear dynamic models in different branches of science which have soliton-like solutions. Based on the balance of the group velocity dispersion and the self-phase modulation, an optical soliton can be formed in fiber. The nonlinear Schrödinger (NLS) equation was reported theoretically by Hasegawa and Tappert ^{[ 1 , 2 ]} to describe self-focusing and self-phase modulation of light waves ^{[ 7 – 14 ]} in nonlinear optics. The propagation of optical solitons ^{[ 15 – 18 ]} are generated by the NLS equation. ^{[ 19 – 21 ]} In recent years, the femtosecond optical pulse transmission technology in optical fibers has a rapid development and potential prospect. As early as 1987, the higher-order NLS equation was derived by using the multiple-scale method, ^{[ 11 ]} which is the transmission evolution equation of femtosecond optical pulses in optical fibers. The higher-order NLS equation includes the terms caused by third-order dispersion, ^{[ 22 ]} self-steepening, ^{[ 23 ]} self-frequency shift effects, ^{[ 24 ]} and so on. However, it is more difficult to deal with both in physics and mathematics because the higher-order NLS equation is a higher-order nonlinear partial differential equation. Many authors have obtained dark and bright solitary wave solutions ^{[ 25 – 29 ]} and W-shaped solitary wave solutions ^{[ 30 ]} of a higher-order equation by using various methods, such as the Hirota algorithm, ^{[ 31 ]} traveling wave transformation, ^{[ 32 ]} and Darboux transformation. ^{[ 33 – 35 ]} Some types of exact N -soliton solutions ^{[ 36 – 41 ]} are reported for the higher-order nonlinear equation. In this paper, we will investigate Kuznetsov–Ma soliton and Akhmediev breather solutions of the higher-order NLS equation. In terms of Darboux transformation we obtain the analytical soliton solutions and breather solutions on a continuous wave (cw) background. The higher-order effects on the breather solution are discussed in detail.

The higher-order NLS equation ^{[ 42 ]} takes the form

Modulation instability (MI) ^{[ 48 ]} as a parametric process exists by the interplay between dispersive and nonlinearity effects, in which a continuous or quasi-continuous wave experience a modulation of its amplitude or phase in the presence of weak perturbation. ^{[ 49 ]} Therefore, it may act as a precursor for the formation of bright solitons. Conversely, the stable dark solitons will exist if the absence of MI is under the constant intensity background. The phenomenon of MI has been discussed and studied in many physical systems. The MI process of the steady-state solutions can be discussed by introducing the following perturbed solution:

The higher-order NLS equation can be presented by the condition of compatibility of the Lax equation as

In the above equations the overbar stands for the complex conjugate. It is easy to verify that equation ( 1 ) can be recovered under the compatibility condition ∂ B / ∂ t − ∂ L / ∂z + [ B , L ] = 0. It is well known that the bright solitary wave solutions exist for Eq. ( 1 ), which is given by the vanishing boundary condition. In addition, the bright soliton solutions can be obtained on the non-vanishing cw background. To this propose, we take the initial “seed” solution as q = A _c e ^{i( k _c z + ω _c t )} , which is a cw solution and satisfies the nonlinear dispersion relation

Following the standard procedure of Darboux transformation, ^{[ 50 – 52 ]} we obtain the soliton solution of Eq. ( 1 ) on a cw background corresponding to eigenvalue λ ₁ = ω _s +i A _s . This solution takes the following form:

Therefore, equation ( 6 ) denotes a soliton solution embedded in a cw background. With the increase of A _s , the cw background is gradually localized and forms Akhmediev breather ^{[ 53 , 54 ]} due to the interaction of soliton and cw background.

3.1. Kuznetsov–Ma soliton

When A _c ≠ 0, ω _c = −2 ω _s , and , the solution in Eq. ( 6 ) reduces to

where Θ ₁ = − 2 ζ ₁ t − 2 ζ ₁ (2 ω _s − Fα − Gγ ) z + t ₀ , Φ ₁ = −2 ζ ₁ ( A _s − 12 ω _s A _s α − Eγ ) z + φ ₀ , with , , , and . From Eq. ( 8 ) we see that the main characteristic properties of a bright soliton is aperiodic in t coordinate and periodic in x coordinate, which is usually called the Kuznetsov–Ma soliton. The propagation of the Kuznetsov–Ma soliton is depicted in Fig. 1 on a cw background with the higher-order effects. Also, it is easy to verify that equation ( 8 ) can reduce to the special form q ₁ = e ^iδ (2 iζ ₁ /cosh Θ ₁ − A _c ), at any position z ₀ = (2 nπ )/[2 ζ ₁ (12 ω _s A _s α + Eγ − A _s )], n = 0, ±1, ±2, …. This result shows that this solution can be generated by coherently adding in quadrature a bright soliton to a cw background. Furthermore, from Eq. ( 8 ) we can find that the soliton intensity | q ₁ | ² has a maximum value . However, it has a minimum value , at cosh Θ ₁ = 2 A _s / A _c sin Φ ₁ − A _c sin Φ ₁ / A _s .

Fig. 1.

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	Fig. 1. Evolution of Kuznetsov–Ma soliton solution in Eq. ( 8 ). Parameters are as follows: A c = 0.3, A s = 0.5, κ c = ω c = 0.2, t 0 = 6, φ 0 = 0, α = 0.05, and γ = 0.025.

From Fig. 2(a) , one can find that the maximum and minimum intensities evolve along the propagation direction. The location of maximum and minimum intensity in the time–space plane is shown in Fig. 2(b) . It is clear that the narrower the soliton, the sharper the peak and the deeper the two dips at the wings of the soliton. This phenomenon indicates the characteristic breather behavior of the Kuznetsov–Ma soliton. It is easy to find that the peak position of the soliton is moved by the presence of the amplitude A _c . With careful analysis we can find the soliton’s peak position is moved from its initial value t _s = ( F′α + G′γ −2 ω _s ) z + t ₀ /2 A _s to the value t _st = ( Fα + Gγ −2 ω _s ) z + t ₀ /2 ζ ₁ with and . Thus, the amount of shift for the soliton’s peak position is given by

Fig. 2.

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	Fig. 2. (a) Evolution of the maximum intensity (red curve), the minimum intensity (blue curve), and the intensity of cw background (solid curve) of the solution in Eq. ( 8 ); (b) location of the maximum intensity (solid curve) and the minimum intensity (dotted curve) in the time-propagation-distance plane. The parameters are the same as in Fig. 1 .

Figure 3 describes the location of the peak of the soliton under two different situations ( A _c ≠ 0 and A _c = 0). From Eq. ( 9 ) we can find that the higher-order terms give the shift and , respectively. For the NLS equation ( α = γ = 0), the value of shift of the soliton’s peak position is t ₀ /(2 ζ ₁ ) − t ₀ /(2 A _s ). Compared with the NLS equation we can see that the main feature of the solution has no change except for a move of the soliton’s peak position. This phenomenon shows that the injection of a cw pulse can be used for parametric control of soliton timing. ^{[ 55 ]}

Fig. 3.

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	Fig. 3. Evolution of Akhmediev breather in Eq. ( 10 ) with the soliton’s peak position of the one-soliton solution q 1 in Eq. ( 7 ) (red line) and the Kuznetsov–Ma soliton solution ( 8 ) (blue line). The parameters are the same as in Fig. 1 .

3.2. Akhmediev breather

When A _c ≠ 0, ω _c = − 2 ω _s , and , the solution in Eq. ( 6 ) reduces to the form

where

with Θ ₁ = (− A _s + (12 ω _s A _s ) α + Eγ ) N ₁ z + t ₀ , and Φ ₁ = N ₁ t + (2 ω _s − Fα − Gγ ) N ₁ z + φ ₀ , with . The solution of Eq. ( 10 ) is aperiodic in the z direction and periodic in the t direction, which is known as Akhmediev breather. Therefore, the solution of Eq. ( 10 ) can be used to describe the modulation instability process. ^{[ 50 , 51 ]} Figure shows the higher-order effects on the modulation instability process. We can observe that the main features of solitons have no change under different values of z except for the time shift.

Fig. 4.

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	Fig. 4. Evolution of the soliton’s peak position shift of the single soliton in Eq. ( 10 ) with and without the higher-order effects at (a) z = 0.5, (b) z = 2.5, (c) z = 4.2, and (d) z = 7.2, respectively. The parameters are α = 0.05, γ = 0.025 (blue curves); and α = γ = 0 (red curve), A s = 0.5, A c = 0.9. The other parameters are the same as in Fig. 1 .

In summary, we have exactly solved the higher-order integrable NLS equation which describes the propagation of ultrashort optical pulse in optical fibers. We have discussed the modulation instability process in detail and obtained the modulation instability conditions for a higher-order nonlinear equation. Based on the Lax pair and Darboux transformation method, bright solitary soliton solutions on a continuous wave background have been presented. With the different parameters, we give the Kuznetsov–Ma soliton and Akhmediev breather solutions of the higher-order nonlinear Schrödinger equation. The propagation characteristics of those solutions on a cw background have been analyzed in detail. When the amplitude of cw light exceeds the half of the soliton amplitude, this one-soliton solution describes the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses. However, the amplitude of cw light is less than half of the soliton amplitude, the one-soliton solution describes the propagation of a bright pulse on the cw background in the presence of higher-order effects, and the soliton’s peak position is shifted owing to the presence of a nonvanishing background field amplitude.