Optical solitons for complex Ginzburg–Landau model with Kerr, quadratic–cubic and parabolic law nonlinearities in nonlinear optics by the exp($-\Phi(\zeta))$ expansion method

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https://doi.org/10.1007/s12043-020-02000-0

Optical solitons for complex Ginzburg–Landau model with Kerr, quadratic–cubic and parabolic law nonlinearities in nonlinear optics by the exp (−(ζ)) expansion method

M K ELBOREE

Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt E-mail: mm_kalf@yahoo.com; mohammed.Elboree@sci.svu.edu.eg

MS received 7 July 2019; revised 8 November 2019; accepted 1 July 2020

Abstract. The optical solitons for the complex Ginzburg–Landau model with Kerr law, quadratic–cubic law and parabolic law are obtained via the exp(−(ζ ))expansion method. Many abundant solutions such as complex dark- singular, complex periodic-singular and plane-wave solutions are derived for this model. These complex solutions are useful for understanding the physical properties for this model. Figures are presented for these solutions to show the dynamics for these waves.

Keywords. Complex Ginzburg–Landau model; exp(−(ζ ))expansion method; complex wave solutions.

PACS Nos 02.30.Jr, 02.30.Hq; 04.20.Jb; 42.25.−p

1. Introduction

The optical soliton is a pulse travelling without dis- tortion due to dispersion or other effects [1,2]. The

‘soliton’ in optics is used to point out any optical field that does not change through propagation because of the delicate balance between nonlinear and linear effects in the medium. New optical devices are in various stages of development such as soliton information processing, basic research in nonlinear optical phenomena. Nonlinear Schrödinger- type solitons are optical solitons that are observed in non-resonant nonlinear silica glass fibres. There are many models such as nonlinear Schrödinger’s equation, Sasa–Satsuma equation, Schrödinger–Hirota equation, Biswas–Milovic equation, Manakov equation, Ginzburg–Landau (GL) model etc. [3–5] which describe the dynamics of soliton propagation through optical fibres for long distances.

In §2, we give a brief description of the method used, and in §3, we derive optical solitons including complex dark-singular, complex periodic-singular and plane-wave solutions for complex Ginzburg–Landau (CGL) model with Kerr law, quadratic–cubic law and parabolic law. In §3, we present the numerical simula- tions via some figures. Physical interpretations of the

obtained solutions are given in §4. Finally, conclusions are given in §5.

2. Description of the exp(−(ζ ))expansion method

We assume that the given nonlinear partial differential equations (NLEEs) is as follows:

F(, t, x, xt, x x, ...)=0, (1) where F is a polynomial in unknown function(x,t) and its derivatives. The steps for this method are as follows:

Step1. Introduce the travelling wave solutions to (1) as follows:

(x,t)=U(ζ ), ζ =kx −ct, (2) wherek is the wave number andc is the wave speed.

Inserting (2) into (1) yields an ordinary differential equation forU(ζ ). We can convert eq. (1) into an ordinary differential equation (ODE) as follows:

F(U,−cU,U,U, . . .)=0, (3) whereindicates the derivative with respect toζ. 0123456789().: V,-vol

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The analytical solutions for eq. (3) according to the exp(−(ζ ))expansion method, can be expressed as follows:

U(ζ )= N i=0

aiexp(−(ζ ))ⁱ, (4)

whereai = 0 are real constants to be determined later and(ζ )satisfies the following ODE:

(ζ )=exp(−(ζ ))+βexp((ζ ))+δ, (5) whereβ, δare constants.

Equation (1) has the following solutions:

Case1: Ifβ =0 andδ²−4β >0, then (ζ )

=ln

⎛

⎜⎜

⎝−

δ²−4βtanh √

δ²−4β

2 (C+ζ )

+δ 2β

⎞

⎟⎟

⎠. (6) Case2: Ifβ =0 andδ²−4β <0, then

(ζ )=ln

⎛

⎜⎜

⎝

4β−δ²tan √

4β−δ²

2 (C+ζ )

−δ 2β

⎞

⎟⎟

⎠. (7) Case3: Ifβ =0, δ=0 andδ²−4β >0, then

(ζ )

= −ln

δ

cosh(δ(C+ζ ))+sinh(δ(C+ζ ))−1

. (8) Case4: Ifβ =0, δ =0 andδ²−4β =0, then

(ζ )=ln

−2(δ ( ζ + C)+2) δ²(C+ζ )

. (9)

Case5: Ifβ =0, δ=0 andδ²−4β =0, then

(ζ )=ln(ζ +C), (10)

whereC is an integration constant.

3. Complex solutions for the CGL model We consider the CGL model [6]

it+α1x x+α2F(||²)− α3

||²^∗(2||²(||²)x x

−((||²)x)²)−α4=0, (11)

where x is the non-dimensional distance along the fibre, t is in dimensionless form, αi,i = 1,2 are the group of velocity dispersion and nonlinearity parameters, α2 = 1,−1 correspond to the anomalous and normal group velocity dispersions in the active core and αj, j = 3,4 arise from the perturbation effects. The functionF(||²)is a real-valued algebraic function and is k-times continuously differentiable [7]. This model describes the wave profile showing in different physical systems.

To apply the exp(−(ζ ))expansion method, we use the travelling wave transformation

(x,t)=U(ζ )eⁱ^η, (12)

whereζ = x −ct andη = −kx +ωt +λ,c, η,k, ω andλare the velocity, phase component, wave number, frequency and phase constant for the wave solutions for eq. (11). Inserting eq. (12) into eq. (11), we can obtain the real and imaginary parts for eq. (11) as follows:

AU_ζζ −BU+α2F(ψ)U =0, (13) where A=α1−4α3andB =ω+k²α1+α4.

−U_ζ(2kα1+c)=0. (14) From the imaginary part of eq. (14), we find that

C = −2kα1. (15)

Now, we shall study the three forms of nonlinear fibres for the CGL model, Kerr law, quadratic–cubic law and parabolic law nonlinearities, in the following subsec- tions.

3.1 Kerr law nonlinearity

According to Kerr law nonlinearity F(s) = s which arises in nonlinear fibre optics [8], eq. (13) converts to

AU_ζζ −BU+α2U³ =0. (16) BalancingU³andU_ζζ in eq. (16) givesN =1, then eq.

(4) becomes

U(ζ )=a0+a1exp(−(ζ )). (17) Substituting eq. (17) into eq. (16) using eq. (5) and collecting all terms of the same power of exp(−(ζ )), we obtain the systems of parametersA,B, α2,a0, β, δand a1. Solving it we get

B = A(4β−δ²)

2 , α2 = −2A

a₁², a1= 2a0

δ , (18) where A, β, δanda0 are arbitrary constants.

From eqs (18) and (17) we get the family of complex wave solutions according to eqs (6)–(10) as follows: If β = 0 and δ² −4β > 0, the complex dark-singular

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optical soliton solutions for eq. (11) can be obtained as follows:

(x,t)=

δ²−4βtanh √

δ²−4β

2 (x−ct+C)

+δ²−4β

a1e⁻ⁱ⁽^kx^−ω^t^−λ)

2

δ²−4βtanh √

δ²−4β

2 (x−ct +C)

+δ

. (19)

Ifβ =0 andδ²−4β <0, the complex periodic-singular solutions for eq. (11) can be obtained as follows:

(x,t)=

4β−δ²tan √

4β−δ²

2 (x−ct+C)

−δ²+4β

a₁e⁻ⁱ⁽^kx^−ω^t^−λ)

2

4β−δ²tan √

4β−δ²

2 (x−ct +C)

−δ

. (20)

Ifβ = 0, δ = 0 andδ²−4β > 0, the complex bright singular soliton solutions for eq. (11) can be obtained as follows:

(x,t)= (cosh(δ(x−ct+C))+sinh(δ(x−ct+C))+1)a1δe⁻ⁱ⁽^kx^−ω^t^−λ)

2(cosh(δ(x−ct+C))+sinh(δ(x −ct+C))−1) . (21) If β = 0, δ = 0 and δ² −4β = 0, the complex

plane-wave solutions for eq. (11) can be obtained in the following form:

(x,t)= δa₁eⁱ⁽⁻^kx^+ω^t^+λ)

(x−ct+C) δ+2. (22) If β = 0, δ = 0 and δ² − 4β = 0, the complex plane-wave solutions for eq. (11) can be obtained in the following form:

(x,t)= (δ (x−ct +C)+2)b1eⁱ^(−kx+ω^t^+λ) 2(x−ct+C) . (23)

Figure 1. Dark-singular solution when β = 0 and δ² − 4β >0.

Figure1shows the absolute dark-singular optical soliton solution (19) for (11) with suitable choice of parameters,

Figure2shows the absolute periodic-singular solution (20) for (11) with suitable choice of parameters. Figure 3shows the absolute bright-singular solution (21) and figures 4 and5 show the absolute plane-wave optical solutions (22) and (23) for (11) with suitable choice of parameters.

3.2 Quadratic–cubic law nonlinearity

According to quadratic–cubic law nonlinearityF(s)= b1s+b2

√swhich appears in [7], eq. (13) becomes AU_ζζ −BU+α2U²(b1U +b2)=0. (24)

Figure 2. Periodic-singular solution whenβ = 0 andδ²− 4β <0.

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BalancingU³ andU_ζζ in eq. (24) gives N = 1. Then eq. (4) converts to (17).

Substituting (24) into (23) using (5) and collecting all terms of the same power of exp(−(ζ )), we obtain systems of parameters A, B, β, δ, α2, b1, b2, a0 and a1. Solving it we obtain

A= −1

2a12α2b1, B = −1

2a12α2b1δ²+2a0α2a1b1δ−2a02α2b1, a0 = a1

2(δ+

δ²−4β), b2 = 3

2a1b1δ−3a0b1, (25)

whereδ,k,b1, α2,b1anda1are arbitrary constants.

From eqs (25) and (17) we get the family of complex wave solutions according to eqs (6)–(10) as follows:

Ifβ =0 andδ²−4β >0, the complex dark-singular optical soliton solutions for eq. (11) can be obtained as follows:

(x,t)=

tanh √

δ²−4β

2 (x−ct+C)

+1 δ²−4βδ+δ²−4β

a1eⁱ⁽⁻^kx^+ω^t^+λ) 2

tanh

√

δ²−4β

2 (x−ct+C) δ²−4β+δ

. (26)

Ifβ =0 andδ²−4β <0, the complex periodic-singular solutions for eq. (11) can be obtained as follows:

(x,t)=

tan √

−δ²+4β

2 (x−ct+C)

−

δ²−4βδ−δ²+4β

a1eⁱ⁽⁻^kx^+ω^t^+λ) 2

tan

1/2

−δ²+4β (x−ct+C) −δ²+4β−δ , (27)

where=

−δ²+4β(δ+

δ²−4β).

Ifβ =0, δ =0 andδ²−4β >0, the complex bright singular soliton solutions for eq. (11) can be obtained as follows:

(x,t)=

δ+

δ²−4β

cosh(δ(x−ct+C))+sinh(δ(x −ct+C))+δ−

δ²−4β

a1eⁱ⁽⁻^kx^+ω^t^+λ) 2(cosh(δ(x−ct+C))+sinh(δ(x −ct+C))−1) .

(28) Ifβ =0, δ=0 andδ²−4β =0, the complex plane-wave solutions for eq. (11) can be obtained in the following form:

(x,t)=

(2−(x−ct +C) δ)

δ²−4β−2δ

a1eⁱ⁽⁻^kx^+ω^t^+λ)

2((x −ct +C) δ+2) . (29)

Ifβ=0, δ =0 andδ²−4β =0, the complex plane-wave solutions for eq. (11) can be obtained in the following form:

(x,t)= a₁

(x−ct+C)

δ²−4β+2+(x−ct+C) δ

eⁱ⁽⁻^kx^+ω^t^+λ)

2(x −ct +C) . (30)

Figure 3. Bright singular soliton solution when β = 0, δ=0 andδ²−4β >0.

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Figure 4. Plane-wave solution when β = 0, δ = 0 and δ²−4β=0.

Figure6shows the absolute dark-singular optical soliton solution (26) for (11) with suitable choice of parameters, figure7shows the absolute periodic-singular solutions (27) for (11) with suitable choice of parameters and fig- ure8shows the absolute bright-singular solutions (28) and figures9and10show the absolute plane-wave optical solutions (29) and (30) for (11) with suitable choice of parameters.

3.3 Parabolic law nonlinearity

According to parabolic law nonlinearity F(s)=b3s+ b4s²which appears in [8], eq. (13) converts to

AU_ζζ −BU+α2U³(b4U²+b3)=0. (31) Substitute the transformationU =V¹^/²in eq. (31). As before, we obtain the systems of parameters A, B, β, δ, α2, b₃, b₄,a₀anda₁. Solving it we obtain

A= −4

3a₁²α2b₄, B = −1

3α2b4(δa1−2a0)², β = a₀(δa₁−a₀)

a12 , b3= 4

3a1b4δ−8

3a0b4, (32) whereδ, k, b4, α2, a0anda1are arbitrary constants.

Figure 5. Plane-wave solution when β = 0, δ = 0 and δ²−4β =0.

Figure 6. Dark-singular solution when β = 0 and δ² − 4β >0.

From eqs (32) and (17) the family of complex wave solutions according to eqs (6)–(10) can be obtained as follows:

Ifβ =0 andδ²−4β >0, the complex dark-singular optical soliton solutions for eq. (11) can be obtained as follows:

(x,t)=

a₀

δ²−4βtanh √

δ²−4β

2 (x−ct+C)

−2a₁β+a₀δ δ²−4βtanh

√

δ²−4β

2 (x−ct +C)

+δ

e⁻ⁱ⁽^kx^−ω^t^−λ). (33)

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Figure 7. Periodic-singular solution when β = 0 and δ²−4β <0.

Figure 8. Bright-singular soliton solution when β = 0, δ=0 andδ²−4β >0.

Ifβ =0, δ = 0 andδ² −4β >0, the complex bright singular soliton solutions for eq. (11) can be obtained as follows:

(x,t)=

a0(cosh(δ(x−ct +C))+sinh(δ(x−ct+C)))+δa1−a0

cosh(δ(x−ct+C))+sinh(δ(x −ct+C))−1 e⁻ⁱ⁽^kx^−ω^t^−λ). (34) Figure 9. Plane-wave solution when β = 0, δ = 0 and δ²−4β =0.

Figure 10. Plane-wave solution when β = 0, δ = 0 and δ²−4β =0.

If β = 0, δ = 0 and δ² − 4β = 0, the complex plane-wave solutions for eq. (11) can be obtained in the following form:

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(x,t)= 1 2

2(4a₀−(2a₀−a₁)(x−ct+C))

(x−ct+C) δ+2 e⁻ⁱ⁽^kx^−ω^t^−λ). (35)

If β = 0, δ = 0 and δ² −4β = 0, the complex plane-wave solutions for eq. (11) can be obtained in the following form:

(x,t)=

a0(x−ct+C)+a1

x −ct +C eⁱ⁽⁻^kx^+ω^t^+λ). (36) Figure 11 shows the absolute dark-singular optical soliton solution (33) for (11) with suitable choice of parameters, figure12shows the absolute bright-singular solution (34) and figures 13 and14 show the absolute plane-wave optical solutions (35) and (36) for (11) with suitable choice of parameters.

4. Physical interpretations

It is important to point out that the solitons are formed from the delicate balance between the nonlinearity effect ofU³ and the dissipative effect ofU_ζζ. Solitons retain their identity after interacting with other solitons.

The method used in this paper provided us the new optical singular-soliton solutions (19)–(23), (26)–(30) and (33)–(36) via suitable choice of parameters. These solutions may be helpful for the numerical solvers. Also algebraic manipulation of this method is much easier than the other methods which use the symbolic com- putation “Maple”. The obtained solutions in this paper

Figure 11. Dark-singular solution when β = 0 and δ²− 4β >0.

have been checked using Maple by putting them back into the original equation (11).

Remark. The CGL model was solved by the modi- fied auxiliary equation method [6] and the sine-Gordon equation method [9] by comparing these results with our results in this paper and we show that the solutions obtained in this paper are new and cannot be found in the previous works.

Figure 12. Bright-singular soliton solution when β = 0, δ=0 andδ²−4β >0.

Figure 13. Plane-wave solution whenβ = 0, δ = 0 and δ²−4β =0.

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Figure 14. Plane-wave solution when β = 0, δ = 0 and δ²−4β=0.

5. Conclusions

In this work, we successfully applied the exp(−(ζ )) method to derive complex dark-singular and plane-wave solutions to the complex Ginzburg–Landau model with Kerr law, quadratic–cubic law and parabolic law nonlinearities and complex periodic-singular solution for this model with Kerr law and quadratic–cubic law nonlin-

earities. Some figures are included to understand some of the properties of the CGL model. The disadvantage of this method is that it fails to retrieve bright optical solitons for the model. The solutions obtained have a wide range of applications in optical fibres, nanoelectronics and optoelectronics. This method is useful to construct optical soliton solutions for most of the nonlinear physical phenomena.

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