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A Comparison of the Undetermined Coefficient Method and the Adomian Decomposition Method for the Solutions of the Sasa-Satsuma Equation

Written By

Mir Asma

Reviewed: November 29th, 2021 Published: February 4th, 2022

DOI: 10.5772/intechopen.101817

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The Nonlinear Schrödinger Equation Edited by Nalan Antar

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The Nonlinear Schrödinger Equation [Working Title]

Dr. Nalan Antar and Prof. İlkay Bakırtaş

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Abstract

This chapter will talk about the mathematical as well as numerical aspects of the Sasa-Satsuma equation that is the extended nontrivial version of nonlinear Schrödinger’s equation. The exact solution will be found out by the undetermined coefficient method. After that, the Adomian decomposition method is secure numerical simulations of computed analytical solutions. The error plots are given to see the accuracy of the results.

Keywords

  • Sasa-Satsuma equation
  • solitons
  • Adomian decomposition method
  • undetermined coefficient method
  • telecommunication

1. Introduction

Solitons can be illustrated as special wave packets that are formed as a result of elegant balance among the fiber nonlinearity and dispersion. They have the ability to travel undistorted along trans-continental and trans-oceanic distances. Solitons are narrow pulses with immense peak power and exceptional properties. Mostly, pulses go through by spreading because of group velocity dispersion while propagating through optical fibers. Actually, solitons have the advantage of nonlinear effects that helps to overcome the broadening of pulse with the group velocity dispersion. When the corresponding dispersive effects and nonlinear effects are controlled and get the appropriate shape of the pulse. When these pulses balance compression and broadening and there is no change in the shape of the pulse or there are periodic changes in the shape of the pulse, this phenomenon is called soliton. Solitons are very advantageous for optical communication that they can overcome chromatic dispersion completely. In most Dense Wavelength Division Multiplexing (DWDM) systems, fiber losses are compensated periodically by using fiber amplifiers spaced 60-80 Km apart. Attenuation and higher powers are the indirect properties of solitons that are reimbursed by the optical amplifiers. When solitons and amplifiers are used together, they can assure the very high-bit rate, for the repeater-less data transmission for long distances. This combination can be responsible for the data transmission at a bit rate of 80 Gb/s for 10,000 km and it was testified in the laboratory. Hasegawa and Tappert in 1973, have discussed the possibilities of propagation of solitons through optical fibers and showed their remarkable stability by numerical computation [1]. Seven years later, Mollenauer, Stolon, and Gordon succeeded in observing soliton propagation experimentally [2].

In this chapter, Sasa-Satsuma equation (SSE) is going to be studied for the sake of optical solitons. SSE is the expansion of nonlinear Schrödinger equation (NLSE). In 1991, Narimasa Sasa and Junkichi Satsuma reported significant results that have a great impact in the field of nonlinear optics and the telecommunication industry [3]. Initially, Sasa and Satsuma displayed a nonlinear complex wave equation that was composed with the aid of inverse scattering transformation [4].

The Sasa-Satsuma equation with the linear temporal evolution is [5]:

iqt+aqnqxx+bq2q+iβ3qxxx+σq2xq+θq2qx=0E1

In (1), qxtis the dependent variable, xand tare independents variables and the subscripts serve as partial derivatives. The first term in (1) known as time evolution term, while ais the coefficient of nonlinear chromatic dispersion, bgives the self-phase modulation with kerr nonlinearity, β3is the coefficient of third-order dispersion, σand θare the coefficients of nonlinear dispersion. Finally, ngives the nonlinearity parameter of chromatic dispersion and n>0.

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2. Method of undetermined coefficients

Method of undetermined coefficients gives the spectrum of soliton solutions. In order to seek the soliton solution of SSE to (1) [6, 7, 8, 9, 10]:

qxt=PxtextE2

Where P(x,t) is the amplitude segment of the soliton. The phase component is

ϕxt=κx+ωt+θ0E3

Here, κ, ωand θ0are the soliton frequency, wave number and the phase constant respectively. By substituting (2) into (1) and equating real and imaginary parts leads to

aκ21+n2Pn+1+axPxt2n1+nPn1+a1+nPxtn+3β3κ2x2PxtPxt2σθκbPxt2+β3κ3+ω=0,E4

and

3x3Pxtβ3+21+n2Pxtn+2σ+θPxt23β3κ2xPxt+tPxt=0.E5

2.1 Solution of bright soliton

For the solution of bright soliton, the starting hypothesis is [7, 8, 9, 10]

Pxt=AD+coshτp,E6

Where

τ=BxvtE7

Ais known as amplitude, Bis the inverse width of the soliton, and Dis the parameter that relates to A and B. The p is an unknown parameter that can be found with the aid of balancing principle. By putting (6) into (4) gives:

A3κθ+bE3p+Aanp2n+2B2κ2+3B2β3κp2β3κ3ω+AB2p2aEp+n+11+n+1pa+3β3κp+1D+1D1ApB2Ep+22DApn+11/2+n+1pa+3κp+1/2β3B2Ep+1=0.E8

where E=D+coshτ.

With the balancing principle, the exponents of 3pand p+2gives p=1. By setting the coefficients of linearly independent function in (8) to zero that gives;

ω=n+12a+3β3κB2β3κ3aκ2E9
A=±κθ+bD1D+1n2+3n+2a+6β3κBκθ+b,E10
κθ+b>0,E11
D=n2+3n+2a+6β3κB2n+2n+1a6B2β3κ+κθ+bA2n2+3n+2a+6β3κB,E12
n2+3n+2a+6β3κB>0.E13

The solution of bright soliton of (1) is

qxt=AD+coshBxvteiκx+ωt+θ0,E14

Figure 1 represents the bright optical soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

Figure 1.

Bright soliton.

2.2 Solution of dark solitons

For the solution of dark soliton, the assumption is [7, 8, 9, 10]

Pxt=A+Btanhτp,E15

where Aand Bare free parameters and

τ=μxvt,E16

Here, pcan be found with the balancing principle. By substituting (15) into (4), the real part gives:

μ2AA+BBA+B4a1+n+1pn+1+3β3kp1pEp26ABAa1/2+n+1pn+1+2β3kp1/2A+Bμ2pEp1+1+2n+1pan+1+3β3k2p+1Aμ2pEp+1+1+n+1pan+1+3β3k1+pμ2pEp+2+B2+bE3p+18A21/3B2p21/3n+12a+β3kμ21/18B2β3k3+ak2+wEp=0E17

where E=A+Btanhτ

The value of pis similar to bright soliton and gives the value of coefficients of linearly independent function as zero that yields to the following relations of soliton parameters.

A=±B=±+ban2+3an+6β3k+2a+b,E18
+b>0,E19
an2+3an+6β3k+2>0,E20
μ=+ban2+3an+6β3k+2aan2+3an+6β3k+2E21
ω=ak2+4an2k3+8an+4a+12kE22

Hence, the solution of the dark soliton is given as:

qxt=A1±tanhμxvteiκx+ωt+θ0,E23

Figure 2 represents the dark soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

Figure 2.

Dark soliton.

2.3 Solution of singular solitons

For the solution of singular soliton, the starting assumption is [7, 8, 9, 10];

Pxt=AD+sinhτp,E24

Here, A, B, and Dare the free parameters with the unknown p. By putting (25) into (4) gives;

n+11+n+1pa+3β3p+1kCh2ApB2Ep+2AB2Shpan+3β3k+aEp+1+A3+bE3pAβ3k3+ak2+wEp=0,E25

where E=D+sinhτ.

By setting 3p=p+2, we get p=1and the free parameters are related as

A=±an2+3an+6β3k+2a+bB,E26
+b>0,E27
ω=an+12+3β3kB2β3k3ak2E28

Hence the solution of singular soliton of (1) is as:

qxt=AD+sinhBxvteiκx+ωt+θ0,E29

for designated parameters.

2.4 Solution of W-shaped solitons

For the solution of w-shaped soliton, the starting assumption is [7, 8, 9, 10];

Pxt=β+ρsechτp,E30
τ=μxvt,E31

Substituting (30) into (4) gives

4μ2an+11+2n+1p+3β3k2p+1Ep+1pβ+ρμ21+n+1pan+1+3β3kp1β2ρβEp22p1/2+n+1pan+1+3β3kp1/22β2+ρ2μ2βEp1+ρ2+bE3p+p2n+12a+3β3kμ2β3k3ak2wρ26μ2β2p2n+12a+3β3kEp1+n+1pan+1+3β3k1+ppμ2Ep+2=0,E32

where E=β+ρsechτ

By the aid of balancing principle, the value of p=1that gives parameters as;

β=±1/22ρ,E33
μ=±+ban2+3an+6kβ3+2aρ,E34
ρ=±an2+3an+6kβ3+2a+b,E35

and

w=6β2μ2an2μ2an2ρ2+12anμ2β22aμ2nρ2+18β2μ2kβ3+β3k3ρ23β3kμ2ρ2+6μ2aβ2+ak2ρ2μ2aρ2ρ2.E36

Therefore, W-shaped soliton solution is given by:

qxt=β+ρsechτeiκx+ωt+θ0,E37

Figure 3 represents the W-shaped optical soliton of SSE. The soliton solution appears with their corresponding constraints conditions.

Figure 3.

W-shaped soliton.

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3. Numerical investigation of soliton solutions

In this section, the Adomian decomposition method will be implemented. ADM has gained very much popularity in recent times in applied mathematics. This method is very robust, efficient, and effective to grasp a broad range of linear, nonlinear, ordinary or partial differential equations and linear or nonlinear integral equations. This method gives the fast convergence of the solution and has many symbolic advantages.

Geoge Adomian has introduced and developed this method and very well treated it in the literature. A very appreciable amount of work has been explored for the wide range of linear, nonlinear, ordinary differential equations, partial differential equations as well as integral equation [11].

3.1 Recapitulation of Adomian decomposition method

In this section, ADM is used to handle SSE numerically that show the broad spectrum analytically results. This method tackles the problem in a direct way that shows the accuracy of the exact solution of soliton solutions.

qxt=q1xt+iq2xtE38

By substituting (38) into (4) and breaking it down into real and imaginary parts, respectively

u2t+au12+u22xxnu1xx+u13+u1u22β3u2xxxσu1x2u2+u2x2u2θu12u2x2+u22u2x=0,E39
u1t+au12+u22xxnu2xx+u23+u12u2+β3u1xxx+σu1x2u1+u2x2u1+θu12u1x2+u22u1x=0,E40

The solution is decomposed into finite sums of components by decomposition method that is defined as;

uizt=n=0ui,nxt,E41

Here, i12. The components ui,n,n0and i=1,2will be found out recurrently. By using an operator form Lt=t, Eqs. (39) and (40) becomes

Ltu2xt+N2u1u2=0E42
Ltu1xt+N1u1u2=0,E43

Where

N2u1u2=au12+u22xxnu1xx+u13+u1u22β3u2xxxσu1x2u2+u2x2u2θu12u2x2+u22u2x=0,E44
N1u1u2=au12+u22xxnu2xx+u23+u12u2+β3u1xxx+σu1x2u1+u2x2u1+θu12u1x2+u22u1x=0,E45

By applying an inverse operator Lt1=0tdtto Eqs. (42) and (43), we get

u1xt=u1x0Lt1N2u1xtu2xtE46
u2xt=u2x0+Lt1N1u1xtu2xt,E47

where u1x0=Reux0and u2z0=Imqz0.

u1,0xt=u1x0u2,0xt=u2x0u1,k+1xt=Lt1N2,ku2,k+1xt=Lt1N1,k,E48
An=1n!dndηnNjn=0ηnu1,nxtn=0ηnu2,n(xt).E49
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4. Numerical simulations

4.1 Bright solitons

To depict the ability, reliability and the accuracy of the ADM for Sasa-Satsuma equation for bright solitons where, a= 110, b= 4100, β3= 1100, σ= 110, θ= 173200, θ0= 0, ω= 311800, and κ= 12. The results and the profile of bright soliton shown in Table below. Figures 414, present the plots of exact and approximate solution with their error plots respectively by varying the values of tand x.

Figure 4.

The graph of analytical and numerical solution with absolute error att=0anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 5.

The graph of analytical and numerical solution with absolute error att=0.1anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 6.

The graph of analytical and numerical solution with absolute error att=0.2anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 7.

The graph of analytical and numerical solution with absolute error att=0.3anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 8.

The graph of analytical and numerical solution with absolute error atx=0anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 9.

The graph of analytical and numerical solution with absolute error atx=0.1anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 10.

The graph of analytical and numerical solution with absolute error atx=0.2anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 11.

The graph of analytical and numerical solution with absolute error atx=0.3anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 12.

The graph of analytical and numerical solution with absolute error att=1anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 13.

The graph of analytical and numerical solution with absolute error att=2anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 14.

The graph of analytical and numerical solution with absolute error att=3anda=110,b=4100,β3=1100,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

tqeqa
08.×1017
0.18.×1017
0.28×1017
0.38.×1017
10.10
20.2
30.2

xqeqa
09.×108
0.19.×108
0.27.×108
0.35.×108

4.2 Dark solitons

In order to depict ADM, we acknowledge the Sasa-Satsuma equation in investigated in detail where, a= 110, b= 12, β3= 15, σ= 110, θ= 173200, θ0=0, ω= 311800, and κ= 12. The results and the profile of dark soliton shown in Table below. Figures 1522, present the plots of exact and approximate solution with their error plots respectively by varying the values of tand x.

Figure 15.

The graph of analytical and numerical solution with absolute error att=0anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 16.

The graph of analytical and numerical solution with absolute error att=0.1anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 17.

The graph of analytical and numerical solution with absolute error att=0.2anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 18.

The graph of analytical and numerical solution with absolute error att=0.3anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 19.

The graph of analytical and numerical solution with absolute error atx=0.1anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 20.

The graph of analytical and numerical solution with absolute error atx=0.2anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 21.

The graph of analytical and numerical solution with absolute error atx=0.3anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 22.

The graph of analytical and numerical solution with absolute error att=1anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

tqeqa
01.5×108
0.11.×1016
0.21.×1016
0.31.×1016
11.×1016
28.×1017
46.×1017

xqeqa
0.14.×1019
0.21.5×1018
0.33.×1018

4.3 Wshaped solitons

In order to depict ADM, we acknowledge the Sasa-Satsuma equation in investigated in detail where, a= 4100, b= 12, β3= 4100, σ= 110, ρ= 310, θ= 173200, θ0= 0, ω= 311800, and κ= 12. The results and the profile of W-shaped soliton shown in Table below. Figures 2334, present the plots of exact and approximate solution with their error plots respectively by varying the values of tand x.

Figure 23.

The graph of analytical and numerical solution with absolute error att=2anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 24.

The graph of analytical and numerical solution with absolute error att=4anda=110,b=12,β3=15,σ=110,θ=173200,θ0=0,ω=311800,κ=12.

Figure 25.

The graph of analytical and numerical solution with absolute error att=0anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 26.

The graph of analytical and numerical solution with absolute error att=0.1anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 27.

The graph of analytical and numerical solution with absolute error att=0.2anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 28.

The graph of analytical and numerical solution with absolute error att=0.3anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 29.

The graph of analytical and numerical solution with absolute error atx=0.1anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 30.

The graph of analytical and numerical solution with absolute error atx=0.2anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 31.

The graph of analytical and numerical solution with absolute error atx=0.3anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 32.

The graph of analytical and numerical solution with absolute error att=2anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 33.

The graph of analytical and numerical solution with absolute error att=3anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

Figure 34.

The graph of analytical and numerical solution with absolute error att=5anda=4100,b=12,β3=4100,σ=110,ρ=310,θ=173200,θ0=0,ω=311800,κ=12.

tqeqa
01.6×1011
0.11.6×1011
0.21.6×1011
0.31.6×1011
23.×1018
32.×1018
53.×1018

xqeqa
0.18.×1019
0.28.×1019
0.38.×1019

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5. Conclusion

In this chapter, SSE has been discussed. First, the undetermined coefficient method has been used. This method secured bright, dark, singular, and W-shaped solitons solutions. The method has given a spectrum of solitons. After that, the Adomian decomposition method has been used for the numerical simulations. This is a very powerful method that has given rapid convergence. Along with, error plots have also been given to witness the accuracy of the exact solution. The graphs have also shown the comparison of exact and absolute solution.

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Conflict of interest

“The authors declare no conflict of interest”.

References

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Written By

Mir Asma

Reviewed: November 29th, 2021 Published: February 4th, 2022