The study of optical solitons with log-law nonlinearity, also known as optical Gaussons, has gained popularity during the last decade. Several analytical results have been reported [1,2,3,4,5,6,7,8,9]. In this context, soliton perturbation theory, quasi-stationary optical Gaussons, and birefringent fibers with dense wavelength division multiplexing (DWDM) technology have been addressed. Recently, shifting gears, the interest on the nonlinear Schrödinger’s equation (NLSE) with log-law nonlinearity, has emerged in the direction of numerical studies . The travelling wave solution technology for finding the partial differential equations (PDEs) was developed by various studies [11,12,13,14,15,16,17,18]. This paper addresses optical Gaussons using the Laplace–Adomian decomposition method (LADM). This is a modified version of the popular Adomian decomposition method (ADM) that has gained extreme popularity during the last decade or so. ADM has been successfully implemented to a wide variety of nonlinear evolution equations, and several impressive numerical results have been reported. In this work, the LADM scheme will be first derived and discussed in detail and subsequently implemented to NLSE with log-law nonlinearity. The model will be studied both in the presence and in the absence of the detuning term. The numerical simulations all appear with their respective error analyses, and these are all depicted in respective tables and figures.
2 Governing equation and optical Gaussons
2.1 The model
Although the usual norm is to study solitons in a Kerr nonlinear medium, there are some advantages to study log-law nonlinearity over Kerr law nonlinearity. One immediate advantage is that solitons from Kerr law nonlinearity produce radiation that is not present for NLSE in a log-law medium .
2.2 Optical Gaussons
2.2.1 Case f(x) = 0
Now, choosing the phase as
The velocity of the Gausson is obtained from the frequency and coefficients of the model (1) and is given by
Observing equation (3), the constraint condition that guarantees the existence of Gaussons is given as follows:
Finally, in the present case, the optical Gaussons solution of the NLSE with log-law nonlinearity is given by
2.2.2 Case f(x) = n
From the phase component, the wavenumber ω is given by
Finally, equation (10) prompts the constraint condition as follows:
3 Analysis of the methodology
The Adomian method combined with the Laplace transform LADM is a decomposition method that gives us solutions for nonlinear differential equations in terms of a convergent series . The LADM was established and used for the first time by Khuri  for solving differential equations.
We consider the general form of nonlinear partial differential equations with the initial condition in the following equation:
Solving for Ltu(x, t) and applying Laplace transform on both sides of equation (13), we obtain
The convergence of this series has been studied in ref. . Finally, for numerical purposes, the N-term approximate
3.1 Gaussons solution of the NLSE with log-law nonlinearity through LADM
Using (20), the first Adomian’s Polynomials of N(u) are as given as follows:
From equation (22), we achieve the series solutions as follows:
4 Numerical simulations
In this section, some examples are provided to show the reliability and the efficiency of the proposed method in solving nonlinear differential equations used in the modeling of dynamics in quantum optics of the type (1). Our computations are performed by MATHEMATICA software.
4.1 Gaussons without detuning term
Gaussons without detuning term
|1||0.5||1.0||2.0||1.00||0.5||−0.5||0.26||12||1.0 × 10−10|
|2||0.3||2.0||2.5||1.82||1.0||−0.6||1.36||12||1.5 × 10−10|
|3||0.2||3.0||3.0||2.73||1.0||−0.4||3.39||12||1.5 × 10−9|
This paper successfully studied optical Gaussons that emerged from NLSE with log-law nonlinearity by the aid of LADM. The results of this paper are being reported for the first time. The error analysis proved that the results appear with grand success and are truly impressive. The results of this paper thus stand on a strong footing to move further along. Later, this scheme will be implemented to address the model further along. They are the application of the scheme to study Gaussons in birefringent fibers, DWDM systems, as well as optical couplers, photonic crystal fiber (PCF), metamaterials, and other variety of optoelectronic devices. Moreover, additional numerical schemes will be implemented to address the model in such optoelectronic devices; one such scheme is the variational iteration method. Such studies are underway. This is just the tip of the iceberg.
This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia, under Grant No. (KEP-65-130-38). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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