Gaussons: optical solitons with log-law nonlinearity by Laplace–Adomian decomposition method

1 Introduction

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 [10]. 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

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 [27]. The LADM was established and used for the first time by Khuri [28] for solving differential equations.

We consider the general form of nonlinear partial differential equations with the initial condition in the following equation:

where L _t u = u _t , R is a linear operator that includes partial derivatives with respect to x, N is a nonlinear operator, and the function h is a source term, which must be independent of u.

Solving for L _t u(x, t) and applying Laplace transform on both sides of equation (13), we obtain

Using the known property of the Laplace transform with respect to the first derivative, that is, ℒ { u t ( x , t ) } = s ℒ { u ( x , t ) } − u ( x , 0 ) , the expression (14) is equivalent to

In the homogeneous case, because ℒ is a linear operator, we have

Now, considering the initial condition given in equation (13), u(x, t) are obtained easily by applying the inverse Laplace transform ℒ − 1 on both sides of equation (16)

The ADM decomposes u(x, t) as a series with components u _n (x, t) and N(u) as a series with components A _n (u ₀, u ₁,…,u _n ), namely,

In equation (19), { A n } n = 0 ∞ is the so-called Adomian polynomials sequence established in ref. [29], which are calculated recursively by

Substituting (18) and (19) into equation (17) gives rise to

From equation (21), each of the components of the series (18) is obtained by means of the recursion scheme, for each n = 0, 1, 2,…

The convergence of this series has been studied in ref. [30]. Finally, for numerical purposes, the N-term approximate

can be used to approximate the exact solution. The exact solution is obtained by adding all the terms, that is,

Now we will implement the algorithm provided by LADM for the solution of NLSE with log-law nonlinearity.

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

Table 1 and Figures 1–3.

Table 1

Gaussons without detuning term

Cases	a	b	A	B	κ	ν	ω	N	|Max error|
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

Figure 1

Gausson without detuning term: case 1: numerically computed profile (left) and absolute error (right).

Figure 2

Gausson without detuning term: case 2: numerically computed profile (left) and absolute error (right).

Figure 3

Gausson without detuning term: case 3: numerically computed profile (left) and absolute error (right).

4.2 Gaussons with detuning term

Table 2 and Figures 4–6.

Table 2

Gaussons with detuning term

Cases	a	b	A	C	D	κ	ω	f(x)	N	|Max error|
1	0.5	1.0	2.5	0.50	1.0	1.00	0.33	1	12	4.0 × 10−10
2	0.3	0.2	3.0	4.25	0.57	0.5	0.16	2	12	6.0 × 10−10
3	0.2	0.3	4.0	5.00	0.86	0.3	0.51	3	12	2.0 × 10−9

Figure 4

Gausson with detuning term: case 1: numerically computed profile (left) and absolute error (right).

Figure 5

Gausson with detuning term: case 2: numerically computed profile (left) and absolute error (right).

Figure 6

Gausson with detuning term: case 3: numerically computed profile (left) and absolute error (right).

5 Conclusions

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.