Original research articleOptical solitons with differential group delay and four-wave mixing using two integration procedures
Introduction
The phenomenon of birefringence or double refraction in an optical fiber is inevitable. This comes with the splitting of pulses into two that leads to differential group delay. The cumulative effect of this delay implies birefringence. Most of the mathematical analysis for birefringence are carried out without the effect of four-wave mixing (4WM). However, there are quite a few results that are reported with the inclusion of 4WM effect in birefringent fibers for parabolic law and Kerr law nonlinearity [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. This paper studies the effect of birefringence in optical fibers with 4WM effect in parabolic and Kerr law nonlinear fibers. The trial equation methodology and the modified simple equation approach are applied to retrieve dark, bright with singular soliton type solutions for the model of study. With the inclusion of 4WM effect, phase-matching condition is implemented to permit integrability of the governing model. Not only the effect of spatio-temporal dispersion (STD) but also group velocity dispersion (GVD) is included. The remainder of the paper will yield the particulars of the subtraction of soliton solutions.
Section snippets
A quick brush-up of trial equation method
The fundamental stages of this method are enumerated by following steps:
Step 1: Let's consider a nonlinear evolution equation (NLEE)Eq. (1) transforms to ordinary differential equation (ODE)by help of the wave variables q(x, t) = Q(ζ), ζ = x − vt, where Q = Q(ζ) stands for an dependent function when Δ implies to a polynomial which includes Q with its derivatives.
Step 2: In what follows, we consider the following auxiliary first order ODE
A quick glance at modified simple equation method
Let us consider a NLEE
The main steps of this method are enumerated by following lines:
Step 1: We consider the following wave variable transformation:
where c is velocity which must be found. By using this conversion, Eq. (79) decreases to:
where Δ is a ODE and represents a polynomial including not only the dependent function Q(ζ) but also its derivatives.
Step-2: Eq. (81) permits the formal solution:
where δl are
Conclusions
This paper recovered dark, bright as well as singular optical soliton solutions to coupled NLSE in birefringent fibers which have been considered for parabolic and Kerr laws of nonlinearity with the effect of 4WM terms included. Phase-matching condition is the key ingredient to the integrability of the model. The modified simple equation scheme and the trial equation method gave way to these results that are truly inciting to first form the groundwork of the models with equations other than
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