1. Introduction

Research on exact solutions of nonlinear systems has reached cruising speed these days. The best known solitons solutions have found their applications in several research areas including optical fibers, plasm, biology, and quantum physics, just to name a few. Thus, optical solitons did not remain anonymous for a long time because of their direct implications in trans-continental and trans-oceanic data transport [¹-¹⁰]. It is important to remember that thanks to the advent of optical solitons, the field of communication has experienced an expansion through high speed data transfer, and guaranteed protection. Thereafter, many theoretical and experimental results have been followed with the mathematical tools to handle them. In this way, exact optical solitons resulting from the nonlinear Schrodinger equations with different¨ nonlinearities have been reported [¹⁰-¹³]. Some of the analytical methods that facilitated the success of these results are expansion methods including modified exp(−ψ(ξ)) function, (G’/G), trial-error, auxiliary equation, rational function, Riccati-Bernoulli sub-ODE , the sine-Gordon expansion, exp(−(((ξ))-expansion, see [²-²³].

The present work, aims to unearth exact optical solitons to PNLSE [¹⁴,¹⁵]:

ψ(x,t) is complex wave profile, t represents temporal variable and x is spatial variable. α ₁ is the STD, α₂ the coefficient of group velocity dispersion (GVD), while α ₃ is the cubic nonlinearity which will be combined with the dispersion terms in order to build optical solitons. However, β is the self-steepening coefficient.

Section 2 concern the traveling-wave solution to transform the PNLSE into to integrable form of nonlinear ordinary differential equation. In Secs. 3 and 4, we apply two integration schemes to look optical solitons and we establish also the existence criteria of the obtained results. In Sec. 5 we discussed the modulation instability of of the obtained results. The last part of the paper is devoted to conclusion. To determine optical solitons to Eq. (1), we will use sinh-Gordon and exp(−Φ(ξ))-expansion techniques with the traveling-wave hypothesis.

2. Sinh-Gordon expansion method

Following expression is adopted as in [¹⁹-²¹], the detail of the method can be obtained in [¹⁹]. By principle of balance from (6) between ϕ _ξξ and ϕ ³, N = 1.

and α₀, α₁ and B₁ are parameters to be determined.

To build some special Jacobian elliptic function solution, we used the following expression

m represents modulus of Jacobian functions [¹⁹-²¹]. Now, substituting Eq. (15) along with others derivatives into Eq. (6) and taking into consideration (16a-16b), it is recovered after some computations:

• cosh³(w(ξ)):

• cosh²(w(ξ)):

• sinh(w(ξ))cosh²(w(ξ)):

• cosh(w(ξ)):

• sinh(w(ξ))cosh(w(ξ)):

• Constant:

• sinh(w(ξ)):

With help of Maple 14, we solve Eqs. (17-23), and it is recovered the following solution families

• Family 1:

• Family 2:

• Family 3:

• Family 4:

3. exp(−Φ(ξ))-expansion method

Let

in which α_i are unknown parameters, while Φ(ξ) satisfies [²²,²³] the ODE:

Using homogeneous principle of balance between ϕ_ξξ and ϕ³ in (6), it is obtained N = 1. Hence,

Substituting Eqs. (41) and (42) into (6) leads to a polynomial in (exp(−Φ(ξ))). By MAPLE software and letting coefficients of (exp(−Φ(ξ))) to 0:

• Set 1: A0=A0,A1=A1,κ=1a1,ω=2a1

• Set 2:

From set 1, it is obtained five families of solutions: (1.1) If λ² − 4µ > 0, and µ 6= 0, it is acquired

(1.2) If λ² − 4µ < 0, and µ 6= 0, it is obtained

(1.3) If λ² − 4µ < 0, and µ0 and λ 6= 0, it is gained

(1.4) If λ² − 4µ = 0, and µ 6= 0 and λ 6= 0, it is gained

(1.5) If λ² − 4µ = 0, and µ = 0 and λ = 0, we have

From set 2, it is obtained five general solutions as follows

(2.1) If λ² − 4µ > 0, and µ 6= 0, it is acquired

(2.2) If λ² − 4µ < 0, and µ 6= 0, it is obtained

(2.3) If λ² − 4µ < 0, and µ0 and λ 6= 0, it is gained

(2.4) If λ² − 4µ = 0, and µ 6= 0 and λ 6= 0, it is gained

(2.5) If λ² − 4µ = 0, and µ = 0 and λ = 0, we have

Where ξ₀ is a constant of integration.

4. Modulation analysis

This section of the paper will address the modulation analysis of the unstable solutions. It is well established that bright and dark solitons are stable; thus, it is necessary to verify the regime of the instability of the rest of solutions. On this fact, it will be used the following solutions of Eq. (1).

while P₀ is the incident power. To seek the perturbation, we use Eq. (53) into Eq. (1)

Hence, Eq. (54) maybe solved in the frequency domain. To do so, it will be considered form of α(x,t) as:

Inserting Eq. (55) into Eq. (54) gives the coefficient matrix of b₁ and b₂,

The dispersion relation is obtained when a solution of Eq. (56) exists. Spreading out determinant leads to

For α²₁K² 6= 1, it is appear the modulation instability (MI) ,

and the gain spectrum G(K) = 2Im(Ω) and it is revealed

Figures 1 and 2 plot the analytical 2D and 3D free-chirp bright solutions for |ψ ₁| at at α₁ = 1, α₃ = 1.14,ω = 0.324, κ = 0.2,v = 0.25. We note that when the self-steepening is absent (β = 0), it is obtained bright and dark optical solitons with normal-GVD without any deformation of waveform. However, Kerr nonlinearity, STD and low GVD lead to dark solitons to the proposed model of NLSE. In addition, the obtained bright solitons depend on the GVD, STD, and Kerr nonlinearity parameters. Also, it is observed that the (MI) gain spectrum depends on the STD, Kerr nonlinearity and self-steepening parameters; Figures 3a) and 3b) stress the (MI) gain spectrum with the effect of self-steepening. More and more the presence of SPM in the MI gain instability will accentuate the instability zone and the MI gain will grow exponentially. The formation of soliton due the fighting between nonlinear and dispersion terms of the nonlinear Schrödinger equation will be impacted during the propagation in long distance because of the act of SPM and the signal received will be highly perturbed.

FIGURE 1 Spatiotemporal plot evolution 3-D of the chirp-free bright of |ψ ₁(x,t)|² of (12) at α₁ = 1, α₃ = 1.14,ω = 0.324,κ = 0.2, v = 0.25. 

FIGURE 2 Spatiotemporal plot evolution 2-D of the chirp-free bright of |ψ ₁(x,t)|² of (12) at α₁ = 1,α₃ = 1.14,ω = 0.324,κ = 0.2, v = 0.25. 

FIGURE 3 The modulation instability gain spectrum a) in the absence of self-steepening β = 0 and b) with the effect of the self-steepening β = 0.007 at α ₁ = 1,α ₃ = 0.007,P₀ = 100. 

5. Conclusion

This work studies modulation analysis and optical solitons of PNLSE having STD, perturbation and some nonlinearity terms. We employ the sinh-Gordon technique and exp(−Φ(ξ))-technique to handle the obtained ordinary differential equation (ODE). Obtained results are bright and dark optical solitons and trigonometric functions. Compare our findings with some previous works [¹⁴,¹⁵], some additional are obtained by adopting the famous exp(−Φ(ξ))-expansion method. In addition, the STD and the cubic nonlinearity terms have contributed to generation og bright and dark solitons. Thus, in order to verify the instability regime, we used the modulation instability (MI), to establish the gain spectrum. From there, two areas emerge, where one is stable and the second unstable. The model will take into account modulations of self-phase and cross-phase in a future work.