New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach

doi:10.1016/j.ijleo.2018.08.007

Optik

Volume 175, December 2018, Pages 328-333

https://doi.org/10.1016/j.ijleo.2018.08.007 Get rights and content

Highlights

•
New optical solitary wave solutions of Fokas-Lenells equation (FLE) were constructed.
•
The solutions of the FLE were found using the generalized exponential function method.
•
The physical meaning of the geometrical structures for some of these solutions is discussed.

Abstract

A variety of new optical waves solutions of the Fokas-Lenells equation in presence of perturbation terms is investigated. A new approach is used, namely the generalized exponential function method. The physical meaning of the geometrical structures for some of these solutions is discussed for different choices of the free parameters present in the solutions. It is shown that the proposed methodology provides powerful mathematical tools for obtaining the exact traveling wave solutions of different nonlinear evolution equations.

Introduction

In the last years, the investigation of the complex waves propagation described by a certain type of nonlinear evolution equations (NLEE) has drawn the attention of the research community. The NLEE describing the composition and dynamical behavior of these waves in plasma physics, optical communications, laser technology, signal processing, and others, represent a significant challenge [1], [2], [3], [4], [5]. Some models with higher order and power-law nonlinearity are implemented to depict the optical solitons propagations in optical fibers [6], [7]. The most appropriate way to comprehend the dynamics of these models is to find their exact solutions. The explicit solutions of these equations, if available, facilitate the verification of numerical researchers and aid in studying the stability analysis.

Different approaches are used in literature for calculating the exact solutions for the NLEE. Among these methods; the improved fractional sub-equation method [8], [9], Kudryashov method and its extended form [10], [11], [12], [13], the unified method and its generalized scheme [14], [15], [16], [17], [18], [19], the homotopy perturbation method [20], [21], and the new extended trial equation method [22], [23].

In this paper, we study the Fokas-Lenells equation (FLE) in presence of perturbation terms [24], [25] using the generalized exponential rational function (GER) method [26], that is a novel extended approach of the exponential rational function method [27], [28].

The FLE is given by: $i ψ_{t} + a_{1} ψ_{xx} + a_{2} ψ_{xt} | ψ |^{2} (b ψ + i σ ψ_{x}) = {α ψ_{x} + λ {(| ψ |^{2} ψ)}_{x} + μ {(| ψ |^{2})}_{x} ψ} .$ In (1), ψ(x, t) represents a complex field envelope, and x and t are spatial and temporal variables, respectively. Here, the first term represents the linear evolution of the pulses in nonlinear optical fibers, while the coefficient a₁ is the spatiotemporal dispersion (STD) and a₂ is the group velocity dispersion (GVD). Then the fourth term introduces the cubic nonlinear term, while the fifth term accounts for dispersion. On the righthand side of (1), the coefficient of α is the inter-modal dispersion (IMD), while λ is the self-steepening perturbation term and finally μ is the nonlinear dispersion (ND) coefficient.

Based on these ideas this paper is organized as follows. The main aspects of the GER method are presented in Section 2. The discussion and application of this method to the FLE is discussed in Section 3. The physical meaning for some of the obtained solutions is presented in Section 4. Finally, the main conclusions outline are in Section 5.

Section snippets

Description of the method

In this section, we present the main steps used with the GER method for finding traveling wave solutions of NLEE. We suppose that a given NLEE for u(x, t) to be in the form $N (u, u_{x}, u_{t}, u_{xx}, \dots) = 0 .$ Using the transformations u = u(ξ) and ξ = x − ν t, we reduce Eq. (2) to the following nonlinear ordinary differential equation (NODE) $N (u, u^{'}, u^{″}, \dots) = 0,$ where ν is a constant and $u^{'} = \frac{d u}{d ξ}$ .

Step 1

The key of this method is to suppose that Eq. (3) has the formal solution

u (ξ) = A_{0} + \sum_{k = 1}^{N} A_{k} {Φ (ξ)}^{k} + \sum_{k = 1}^{N} B_{k} {Φ (ξ)}^{- k},

where

Φ (ξ) = p_{1} e^{q_{1} ξ} + p

Application of the GER method

To solve Eq. (1), first we need to apply the traveling wave transformation $ψ (x, t) = u (ξ) e^{i η (x, t)}, ξ = x - ν t,$ where $η (x, t) = - κ x + ω t + θ .$ Here, ν is the velocity of the soliton, κ is the frequency while ω is the soliton wave number and θ is the phase constant to be determined. Applying Eq. (6) into Eq. (1) we obtain the following pair of equations of real and imaginary components, respectively as $a_{1} - a_{2} ν u^{″} - (α + ω + a_{1} κ^{2} - a_{2} κ ω) u + (b - κ λ + κ σ) u^{3} = 0,$ and $(ν + α + 2 a_{1} κ - a_{2} (ν κ + ω) + (3 λ + 2 μ - σ) u^{2}) u^{'} = 0 .$ From (9), the velocity of the soliton is

Physical explanation

In this part, we introduce the physical interpretation for some of the complex wave solutions of the FLE.

Fig. 1, Fig. 2 depict the 3D and 2D charts of the absolute, real and imaginary parts of ψ_i(x, t), i = 1, 5.

Fig. (1) represents the complex wave solution of the FLE given by Eq. (12) with the parameters a₁ = 0.1, a₂ = 0.25, b =−0.2, ω = 0.3, α = 0.3, λ = 0.2, σ =−0.1, κ = 0.5, and θ = 0. We observe that in Fig. (1)(a) and (b) the absolute value of ψ₁(x, t) is a singular periodic wave. The

Conclusion

In this paper, we derived several types of complex solitary wave solutions for the FLE. These new formulae are obtained by mean of the GER method. The GER method not only has the advantage of giving a unified formulation to obtain exact traveling wave solutions for NLEE, but also provide a guideline to classify the types of these solutions. The physical interpretation of the obtained expressions was discussed for different choices of the parameters that occur in the solutions. Furthermore, the

Conflicting interests

The authors declare that there is no conflict of interest.

Ethical standard

The authors state that this research complies with ethical standards and it does not involve either human participants or animals.

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View full text

New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach

Highlights

Abstract

Introduction

Section snippets

Description of the method

Application of the GER method

Physical explanation

Conclusion

Conflicting interests

Ethical standard

Appl. Math. Comput.

Appl. Math. Comput.

Comput. Math. Appl.

Appl. Math. Comput.

Appl. Math. Lett.

Optik

Optik

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Opt. Quant. Electron.

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Math. Methods Appl. Sci.

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Pramana J. Phys.

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Commun. Theor. Phys.

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