# Numerical Solution of Dispersive Optical Solitons with Schrödinger-Hirota Equation by Improved Adomian Decomposition Method

Authors: H. O. Bakodah; M. A. Banaja; A. A. Alshaery; A. A. Al Qarni
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we present new numerical results for the dispersive optical soliton solutions of the nonlinear Schrödinger-Hirota equation. The spatio-temporal dispersion term is included, in addition to group velocity dispersion Kerr law of nonlinearity are studied. A general recursive numerical scheme for the equation is devised via the Improved Adomian Decomposition Method (IADM) and further sought for some analytical results for validation. The scheme is shown to be efficient and possessed high level of accuracy as demonstrated.


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## Body

## 1. Introduction

Nonlinear Schrodinger equations (NLSE) play a significant role in the fiber optics, optical soliton, prorogation of pulses in metamaterials and network engineering applications among others [1, 2]. Thus, different forms of NLSE exist and serve for a variety of purposes mostly in communication and networking engineering including, for example, the trans-continental and trans-oceanic data transfer requiring phase modulation [3] and beyond; see [4–16] for different mathematical studies on these equations. Furthermore, due to their immense application, many analytical methods and few numerical methods have been proposed in the past decades. Such methods include the Backlund transformation method [17], Hirota's direct method [18, 19], tanh-sech method [20, 21], extended tanh method [22, 23], sine-cosine method [24], and homogeneous balance method [25].However, we will propose in this paper a numerical method for solving a class of NLSE called the nonlinear Schrodinger-Hirota equation. The method will heavily depend on the Improved Adomian Decomposition Method (IADM) [26–31]. The IADM is an efficient numerical method for functional and integral solutions based on the Adomian decomposition method [32]. Further, a recent analytical study on the nonlinear Schrodinger-Hirota equation will be sought for to validate the proposed scheme. This form of solution is being reported for the first time in this paper.


## 2. Governing Equation

The nonlinear Schrodinger equation used in modeling propagation of solitons through optical fibers with Third-Order Dispersion (TOD) is given by(1)iut+12uxx+u2u=-iλuxxx,where u=u(x,t) is a complex-valued function of x (space) and t (time); λ is the coefficient of TOD. The first term on the left hand side is the linear temporal evolution, the second term is the group velocity dispersion term, and the third term accounts for Kerr law nonlinearity. Also when the group velocity is low, the TOD is justified.We now introduced the Lie symmetry concept to be able to study (1). With q=qx,t, let (2)q=u-3iλux+2u∫-∞xuξ2dξ,which transforms (1) to(3)iqt+12qxx+q2q+iλqxxx+6q2qx=0.In (3), higher order terms are neglected [11, 33, 34] with Kerr law nonlinearity that models transmission of dispersive optical solitons through nonlinear fibres. We express (3) with general coefficients as (4)iqt+aqxx+cq2q+iγqxxx+σq2qx=0,where physically σ represents nonlinear dispersion. Further, (4) happens to be ill-posed due to the group velocity dispersion term. However, with the addition of the spatiotemporal dispersion (STD) term, (4) possesses a well-posedness [35, 36] given by (5)iqt+aqxx+bqxt+cq2q+iγqxxx+σq2qx=0,where the coefficient b represents STD. Finally, in presence of perturbation terms (5) becomes [37–39] (6)iqt+aqxx+bqxt+cq2q+iγqxxx+σq2qx=iαqx+iλq2qx+ivq2xq.However, we consider an analytical 1-soliton solution of (6) as special case presented by Al Qarni et al. [40] for the numerical simulation sake in Section 4 given by (7)qx,t=Asech⁡Bx-vtei-klx+ωlt+θ,where k is the frequencies of the solitons, ω is the wave number, θ is the phase velocity, and A and B are the amplitude and width B given, respectively, by(8)A=2ω+ak-ωkb+ak2+γk3a-bvω-3γk,(9)B=2ω+ak-ωkb+ak2+γk3c+σk-λk.The constraint conditions, for the existence of bright solitons from (7)-(9), are as follows: (10)a-bvω-3γk2ω+ak-ωkb+ak2+γk3>0,(11)c+σk-λk2ω+ak-ωkb+ak2+γk3>0.


## 3. Numerical Method

In [26–31], authors introduced the IADM to convert a special case of the complex-valued system into a real-valued system by splitting qx,t as (12)qx,t=u1+iu2,where (uk,k=1,2) are real functions. By substituting equation (12) into (6) we obtain the following system:(13)u1t+au2xx+bu2xt+cu12+u22u2+γu1xxx+σu12+u22u1x=αu1x+λu12+u22u1x-vu1x2+u2x2u1,(14)u2t-au1xx-bu1xt-cu12+u22u1+γu2xxx+σu12+u22u2x=αu2x+λu12+u22u2x+vu1x2+u2x2u2,where(15)u1x,0=qx,0R,u2x,0=qx,0I.The decomposition method [32] decomposes the solution into infinite sums of components defined by(16)u1x,t=∑n=0∞u1nx,t,(17)u2x,t=∑n=0∞u2nx,t,where the components u1n,u2n,n≥0 will be determined recursively. In an operator form with Lt=∂/∂t, (14) and (15) become(18)Ltu1+bu2x+au2xx+cu12+u22u2+γu1xxx+σu12+u22u1x=αu1x+λu12+u22u1x-vu1x2+u2x2u1,(19)Ltu2-b1u1x-au1xx-cu12+u22u1+γu2xxx+σu12+u22u2x=αu2x+λu12+u22u2x+vu1x2+u2x2u2.Taking the inverse operator Lt-1 to both sides of (18) and (19) gives (20)u1x,t=u1x,0-bu2xx,t-u2xx,0-Lt-1au2xx-Lt-1γu1xxx+Lt-1αu1x+Lt-1A1,(21)u2x,t=u2x,0+bu1xx,t-u1xx,0+Lt-1au1xx-Lt-1γu2xxx+Lt-1αu2x+Lt-1A2.Assuming that the nonlinear terms in (20) and (21) are represented by the series,(22)A1=-cu12+u22u2-σu12+u22u1x+λu12+u22u1x+vu1x2+u2x2u1,(23)A2=cu12+u22u1-σu12+u22u2x+λu12+u22u2x+vu1x2+u2x2u2,A1n,…,A2n,…. are the so-called Adomian polynomials that can be constructed for all forms of nonlinearity according to specific algorithms set by Adomian [32]. Substituting the nonlinear terms into (22) and (23) and the solution from (16) and (17) into (20) and (21) gives(24)∑n=0∞u1nx,t=u1x,0-b∑n=0∞u2nx,tx+bu2nx,0x-Lt-1a∑n=0∞u2nx,txx-Lt-1γ∑n=0∞u1nx,txxx+Lt-1α∑n=0∞u1nx,tx+Lt-1∑n=0∞A1n,(25)∑n=0∞u2nx,t=u2x,0+b∑n=0∞u1nx,tx-bu1nx,0x+Lt-1a∑n=0∞u1nx,txx-Lt-1γ∑n=0∞u2nx,txxx+Lt-1α∑n=0∞u2nx,tx+Lt-1∑n=0∞A2n.Following the decomposition analysis, the following recursive relations are introduced:(26)u1,0x,t=u1x,0+bu2x,0x,(27)u2,0x,t=u2x,0-bu1x,0x,(28)u1,k+1x,t=-bu2,kx,tx-Lt-1au2,kx,txx-Lt-1γu1,kx,txxx+Lt-1αu1,kx,tx+Lt-1A1,m,(29)u2,k+1x,t=bu1,kx,tx+Lt-1au1,kx,txx+Lt-1γu2nx,txxx+Lt-1αu2nx,tx+Lt-1A2,m.Thus, we determine u1 and u2 as follows: (30)u1=u1,0+u1,1+u1,2+⋯u2=u2,0+u2,1+u2,2+⋯and the overall approximate solution for (6) is obtained by substituting the above into (12) coupled to (26)-(29) to get (31)qx,t=u1,0+u1,1+u1,2+⋯+iu2,0+u2,1+u2,2+⋯.


## 4. Numerical Results

In this section, we consider three different cases for the Schrödinger-Hirota equation with spatiotemporal dispersion given in (6) to illustrate the application of the IADM scheme we presented in the above section. We also consider the bright soliton solution given in (7)-(11) for numerical simulation with the following fixed parameters: ω=1, k1=k=0.1, θ=0, c=1, a=0.5. We also represent the graphs for absolute value for each of exact solution qe and approximate solution qa.Example 1.
We consider Schrödinger-Hirota equation with spatiotemporal dispersion equation (6) with b=0.0 and the following three special cases.Case 1.
Letγ=0.0.Case 2.
Letγ=0.6.Case 3.
Letγ=1.0.The results and the profiles of this example are presented in Table1 and Figure 1.Table 1
The absolute error for Example1 when x=20.
t b = 0 γ = 0 γ = 0.6 γ = 1 0.0 0.0 0.0 0.0 0.1 0.000010261373 0.00003209069 0.00006378300 0.2 0.000021764973 0.00006266170 0.00012599913 0.3 0.000034488427 0.00009173715 0.00018667002 0.4 0.000048407918 0.00011934200 0.00024581792 0.5 0.000063498224 0.00014550173 0.00030346539Figure 1
Comparison of the exact and approximate solution for Example1 for -20≤x≤20.Example 2.
We consider Schrödinger-Hirota equation with spatiotemporal dispersion equation (6) with b=0.1 and the following three special cases.Case 1.
Letγ=0.0.Case 2.
Letγ=0.6.Case 3.
Letγ=1.0.The result and the profiles of this example are presented in Table2 and Figure 2.Table 2
The absolute error for Example2 when x=20.
t b = 0.1 γ = 0 γ = 0.6 γ = 1 0.0 5.47003×10−7 7.5968×10−7 8.7637×10−7 0.1 0.000013623790 0.00003154594 0.00006498593 0.2 0.000028096531 0.00006216823 0.00012912151 0.3 0.000043940472 0.00009113126 0.00019155154 0.4 0.000061129371 0.00011845995 0.00025229764 0.5 0.000079635465 0.00014417984 0.00031138194Figure 2
Comparison of the exact and approximate solution for Example2 for -20≤x≤20.Example 3.
We consider Schrödinger-Hirota equation with spatiotemporal dispersion equation (6) with b=0.6 and the following three special cases.Case 1.
Letγ=0.0.Case 2.
Letγ=0.6.Case 3.
Letγ=1.0.The result and the profiles of this example are presented in Table3 and Figure 3.Table 3
The absolute error for Example3 when x=20.
t b = 0.6 γ = 0 γ = 0.6 γ = 1 0.0 0.00014373490 0.00017576127 0.00019563219 0.1 0.00014417980 0.00011583001 0.00009275551 0.2 0.00014701256 0.00005817085 0.00000758921 0.3 0.00015221500 0.00000366844 0.00010541473 0.4 0.00015976708 0.00004859351 0.00020073458 0.5 0.00016964674 0.00009833252 0.00029356299Figure 3
Comparison of the exact and approximate solution for Example3 for -20≤x≤20.


## 5. Conclusion

In this work, an Improved Adomian Decomposition Method (IADM) has been proposed to solve the nonlinear Schrodinger-Hirota equation in presence of several Hamiltonian type perturbation terms. The obtained results possess high precision and converged to the exact solution with less computational efforts. It was noticed that a high accuracy of results is obtained when is the coefficient of 3ODγ=0 and within the range 0>γ, b<1 (b is the coefficient of spatio-temporal dispersion term) as commented also in [40]. It is also clear from Figures 1–3 that the complete correspondence of the two solutions is indeed remarkable. Thus, the IADM is an efficient method for nonlinear Schrodinger equations since it shows a high level of accuracy with lesser computational efforts as compared to other numerical methods. This topic is still open for further researches for different values of parameters and other types of soliton solutions.



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