New optical solitary wave solutions of Fokas-Lenells equation in presence of perturbation terms by a novel approach
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: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 formUsing the transformations u = u(ξ) and ξ = x − ν t, we reduce Eq. (2) to the following nonlinear ordinary differential equation (NODE)where ν is a constant and . Step 1 The key of this method is to suppose that Eq. (3) has the formal solution
Application of the GER method
To solve Eq. (1), first we need to apply the traveling wave transformationwhereHere, 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 asandFrom (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 ψi(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.
References (28)
- et al.
Extended trial equation method to generalized nonlinear partial differential equations
Appl. Math. Comput.
(2013) - et al.
Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations
Appl. Math. Comput.
(2011) On multi-soliton solutions for the (2+1)-dimensional breaking soliton equation with variable coefficients in a graded-index waveguide
Comput. Math. Appl.
(2018)- et al.
An efficient algorithm to construct multi-soliton rational solutions of the (2+1)-dimensional KdV equation with variable coefficients
Appl. Math. Comput.
(2018) - et al.
The first-integral method applied to the Eckhaus equation
Appl. Math. Lett.
(2012) On complex wave solutions governed by the 2D Ginzburg-Landau equation with variable coefficients
Optik
(2018)- et al.
Optical soliton perturbation with full nonlinearity for Fokas-Lenells equation
Optik
(2018) - et al.
On nonautonomous complex wave solutions described by the coupled Schrödinger-Boussinesq equation with variable-coefficients
Opt. Quant. Electron.
(2018) - et al.
Complex acoustic gravity wave behaviors to some mathematical models arising in fluid dynamics and nonlinear dispersive media
Opt. Quant. Electron.
(2018) - et al.
Dynamic of DNA's possible impact on its damage
Math. Methods Appl. Sci.
(2016)
Topological and non-topological soliton solutions to some time-fractional differential equations
Pramana J. Phys.
Abundant soliton solutions of the resonant nonlinear Schrödinger equation with time-dependent coefficients by ITEM and He's semi-inverse method
Opt. Quant. Electron.
A new trial equation method and its applications
Commun. Theor. Phys.
Exact solutions for Fitzhugh-Nagumo model of nerve excitation via Kudryashov method
Opt. Quant. Electron.
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