Solitons and other solutions for coupled nonlinear Schrödinger equations using three different techniques

https://doi.org/10.1007/s12043-019-1762-y

Solitons and other solutions for coupled nonlinear Schrödinger equations using three different techniques

ELSAYED M E ZAYED¹, ABDUL-GHANI AL-NOWEHY^{2 ,∗}and MONA E M ELSHATER¹

1Mathematics Department, Faculty of Sciences, Zagazig University, Zagazig, Egypt

2Mathematics Department, Faculty of Education and Science, Taiz University, Taiz, Yemen

∗Corresponding author. E-mail: alnowehy2010@yahoo.com

MS received 25 July 2018; revised 3 December 2018; accepted 13 December 2018; published online 11 April 2019 Abstract. In this paper, we apply three different techniques, namely, the sine–cosine method, the new extended auxiliary equation method and the modified simple equation method for constructing many new exact solutions with parameters as well as bright–dark, singular and other soliton solutions of the coupled nonlinear Schrödinger equations. The solutions of these coupled nonlinear equations are compared with the well-known results.

Keywords. Sine–cosine method; new extended auxiliary equation method; modified simple equation method;

exact solutions; solitons and other solutions; coupled nonlinear Schrödinger equations.

PACS Nos 02.30.Jr; 02.30.Hq; 04.20.Jb; 05.45.Yv

1. Introduction

It is of fundamental importance to investigate the exact travelling wave solutions, solitary wave solu- tions and periodic wave solutions which play prominent roles in practical applications in the study of non- linear physical phenomena. Nonlinear wave phenom- ena, such as fluid mechanics, plasma physics, optical fibres, biology, solid-state physics, chemical physics etc. appear in various scientific and engineering fields.

In recent decades, many effective methods, such as the exp-function method [1–3], the simplest equation method [4,5], the Jacobi elliptic function expansion method [6–10], the Weierstrass elliptic function method [11–13], the tanh-function method [14–18], the gener- alised new auxiliary equation method [19], the (G/G,1/G)-expansion method [20–22], the new map- ping method [23,24], the extended trial equation method [25,26], the new extended auxiliary equation method [27–29], the sine–cosine method [30–34], the modified simple equation method [35–40], the Riccati–Bernoulli sub-ODE method [41], the exp(–ϕ(ζ))-expansion tech- nique [42], the new factorisation technique [43], the theory of local fractional and local fractional Riccati dif- ferential equation method [44–48], the weighted energy method [49], the complete discrimination system for polynomial method, trial equation method [50], the Hirota bilinear method, theta function identities [51],

the singular manifold method [52] etc. have been established to obtain the exact travelling wave solutions of the nonlinear evolution equations in mathematical physics.

The objective of this paper is to apply three effec- tive methods, namely, the sine–cosine method, the new extended auxiliary equation method and the modi- fied simple equation method for solving the following coupled system of nonlinear Schrödinger equations [53]:

i1y+1

21tt +1

21x x+(|1|²+2|2|²)1 =0, (1.1) i2y+1

22x x +(|2|²+2|1|²)2 =0, (1.2) where y is the longitudinal coordinate, and t and x are the temporal and the transverse spatial coordinates, respectively. Equations (1.1) and (1.2) have been dis- cussed in [53] using the Weierstrass elliptic function expansion method, where new doubly periodic solu- tions in terms of Weierstrass elliptic functions have been obtained. To our knowledge, eqs (1.1) and (1.2) are not investigated elsewhere using the three methods pro- posed in this paper.

This paper is organised as follows. In §2, the mathematical analysis of (1.1) and (1.2) is given. In

§3–5, we solve eqs (1.1) and (1.2) using the sine–cosine method, the new extended auxiliary equation method and the modified simple equation method. In §6, some graphical representations of our results are presented. In

§7, some conclusions are drawn.

2. Mathematical analysis

In order to solve eqs (1.1) and (1.2) we make the trans- forms:

1(x,y,t)=1(ξ)exp[iϕ1(x,y,t)], (2.1) 2(x,y,t)=2(ξ)exp[iϕ2(x,y,t)], (2.2) where i(ξ) and ϕi(x,y,t) are real functions, such that ξ = k(x+l y+λt), ϕi(x,y,t) = αix +βiy + γit(i =1,2),k=0,l, λ, αi, βi andγi are real param- eters to be determined later. Substituting (2.1) and (2.2) into eqs (1.1) and (1.2), respectively, and separating the imaginary and real parts, we have

Im:

α1= −(l+λγ1), α2= −l, Re:

k²

2(λ²+1)1+1³

−

β1+1 2γ₁²+1

2α₁²

1+2₂²1=0 (2.3) and

k²

2₂+₂³−

β2+ 1 2α₂²

2+2₁²2=0, (2.4) where

=d/dξ

.If we make the ansatz

2=c1, (2.5)

wherecis a constant, then system (2.4) holds if 1+λ²

c = β1+(1/2)γ₁²+(1/2)α²₁

c(β2+(1/2)α₂²) = 1+2c² c³+2c,

(2.6) which leads to the relations

c²= 2λ²+1 1−λ² , β1=(1+λ²)

β2+1

2α₂²

−1 2γ₁²−1

2α²₁, λ² =1. (2.7) Therefore, under ansatz (2.5) and condition (2.7), we have the nonlinear ODE:

k² 21−

β2+l²

2

1+ 3

1−λ²

1³=0. (2.8)

3. Solving eq. (2.8) using the sine–cosine method According to the sine method [30–34], eq. (2.8) has the formal solution

1(ξ)=

γ1sin^β1(γ2ξ), |ξ| ≤ π γ2,

0, otherwise. . (3.1)

Substituting (3.1) into (2.8) we have

−k²

2 γ₂²β₁²γ1sin^β1(γ2ξ) +k²

2γ2²β1²γ1(β1−1)sin^β1−2(γ2ξ)

−

β2+l² 2

γ1sin^β1(γ2ξ) + 3γ₁³

1−λ² sin^3β1(γ2ξ)=0. (3.2) By balancing the exponent of sin functions in (3.2), we get 3β1 = β1 −2 and hence β1 = −1. Substituting β1 = −1 into (3.2), we get

γ1 = ±

2 3

β2+l²

2

(1−λ²), γ2 = ±

−2 k²

β2+l²

2

, (3.3)

provided(β2+(l²/2)) < 0 and (1−λ²) < 0.Now, eq. (2.8) has the periodic solution

1(ξ)= ± 2

3

β2+l² 2

(1−λ²) 1/2

×csc

⎛

⎝ξ −2

k²

β2+l² 2

⎞⎠. (3.4)

Similarly, if we use the cosine method 1(ξ)=

γ1cos^β1(γ2ξ),|ξ| ≤(π/2γ2)

0, otherwise . (3.5)

Then we get the periodic solution 1(ξ)= ±

2 3

β2+l²

2

(1−λ²) 1/2

×sec

⎛

⎝ξ −2

k²

β2+l² 2

⎞⎠. (3.6)

If csc(ix) = −icsch(x), then (3.4) is reduced to the singular soliton solution

1(ξ)= ± −2

3

β2+l² 2

(1−λ²) 1/2

×csch

⎛

⎝ξ

2 k²

β2+l²

2

⎞⎠, (3.7)

provided(β2+(l²/2)) >0 and(1−λ²) <0.

Also, if sec(ix) = sech(x),then (3.6) is reduced to the bright soliton solution

1(ζ )= ± 2

3

β2+l² 2

(1−λ²) 1/2

×sech

⎛

⎝ξ

2 k²

β2+l²

2

⎞⎠, (3.8)

provided(β2+(l²/2)) >0 and(1−λ²) >0.

4. Solving eq. (2.8) using the new extended auxiliary equation method

According to the new extended auxiliary equation method [27–29], balancing ₁ with ₁³ in eq. (2.8) yields the balance number N = 1. Consequently, eq. (2.8) has the following formal solution:

1(ξ)=a0+a1F(ξ)+a2F²(ξ), (4.1) wherea0,a1anda2are constants to be determined, such thata₂ = 0, while the function F(ξ) satisfies the fol- lowing first-order equation:

F²(ξ)=c0+c2F²(ξ)+c4F⁴(ξ)+c6F⁶(ξ), (4.2) where cj (j =0,2,4,6) are constants to be deter- mined. Substituting (4.1) along with (4.2) into eq. (2.8), collecting the coefficients of each power Fⁱ(ξ)

F(ξ)j

(i =0,1,2, . . . ,6, j=0,1), and set- ting these coefficients to zero, we have the following algebraic equations:

F⁶(ξ): 1

2(λ²−1)[3k²a1c6λ²−18a1a₂²−3k²a1c6]=0, F⁵(ξ):

1

2(λ²−1)[8k²a2c6λ²−6a₂³−8k²a2c6] =0, F⁴(ξ):

1

2(λ²−1)[−18a²₁a2−6k²a2c4−18a0a₂² +16k²a2c4λ²]=0,

F³(ξ):

1

2(λ²−1)[−2k²a1c4+2k²a1c4λ²−6a₁³

−36a0a1a2] =0, F²(ξ):

1

2(λ²−1)[2β2a1(1−λ²)−l²a1λ²

−k²a1c2(1−λ²)+ l²a1−18a²₀a1] =0, F(ξ):

1

2(λ²−1)[ −l²a₀λ²+2β2a₀−2k²a₂c₀−2β2a₀λ²

−6a₀³+2k²a2c0λ²+l²a0] =0, F⁰(ξ):

1

2(λ²−1)[ −2β2a2λ²+2β2a2−l²a2λ²−18a0a₁²

−4k²a2c2+4k²a2c2λ²+l²a2−18a²₀a2] =0. (4.3) Solving the system of algebraic equation (4.3) with the aid of Maple, we obtain the following results:

a0 = k²c4(λ²−1)

3a₂ , a1 =0, a2=a2, β2 = −1

2a2[2k⁴c²₄(λ²−1)+l²a₂²−4k²c2a₂²], c0 = −k²c4

9a₂⁴ [k²c4(λ²−1)²−3a₂²c2(λ²−1)], c₆ = 3a₂²

4k²(λ²−1). (4.4)

It is well known [27–29] that eq. (4.2) has the solution F(ξ)= 1

2 −c4

c6 (1± f(ξ)) 1/2

, (4.5)

where f(ξ)can be expressed through the Jacobi elliptic functions sn(ξ,m),cn(ξ,m), dn(ξ,m)and so on. Here, 0 < m <1 is the modulus of the Jacobi elliptic func- tions. Substituting (4.4) along with (4.5) into (4.1), we get the following Jacobi elliptic function solutions of eq. (2.8):

1(ξ)= ±

k²c4(λ²−1) 3a₂

f(ξ). (4.6)

Then we have the following results:

(1) If

c0 = c₄³(m²−1)

32c²₆m² , c2= c²₄(5m²−1)

16c6m² , c6 >0,

then 1(ξ)= ±

k²c4(λ²−1) 3a2

sn

ξ

k²c²₄(λ²−1) 3a₂²m²

(4.7) or

1(ξ)= ±

k²c4(λ²−1) 3a2m

ns

ξ

k²c²₄(λ²−1) 3a₂²m²

. (4.8) Ifm →1,then we have the dark soliton solution:

1(ξ)= ±

k²c4(λ²−1) 3a2

tanh

ξ

k²c₄²(λ²−1) 3a²₂

(4.9) and the singular soliton solution:

1(ξ)=±

k²c₄(λ²−1) 3a2

coth

ξ

k²c²₄(λ²−1) 3a²₂

, (4.10) provided(λ²−1) >0.

(2) If

c0= c³₄(1−m²)

32c²₆ , c2= c²₄(5−m²)

16c6 , c6 >0, then 1(ξ)= ±

k²c4(λ²−1)m 3a₂

sn

ξ

k²c²₄(λ²−1) 3a²₂

(4.11) or

1(ξ)= ±

k²c₄(λ²−1) 3a2

ns

ξ

k²c²₄(λ²−1) 3a₂²

, (4.12) provided (λ²−1) > 0 . Ifm → 1,then we have the same dark soliton solution (4.9).

(3) If c0= c³₄

32m²c²₆, c2= c²₄(4m²+1)

16c6m² , c6 <0, then

1(ξ)= ±

k²c4(λ²−1) 3a2

cn

ξ

−k²c²₄(λ²−1) 3a₂²m²

(4.13) or

1(ξ)= ±

k²c4(λ²−1)√ 1−m² 3a₂

×sd

ξ

−k²c²₄(λ²−1) 3a₂²m²

. (4.14)

Ifm→1,then we have the bright soliton solution 1(ξ)=±

k²c4(λ²−1) 3a2

sech

ξ

−k²c²₄(λ²−1) 3a²₂

, (4.15) provided(λ²−1) <0.

(4) If

c₀ = c³₄m²

32c²₆(m²−1), c₂ = c₄²(5m²−4)

16c6(m²−1), c₆ <0, then

1(ξ)= ±

k²c₄(λ²−1) 3a2

√1−m²

dn

ξ

k²c₄²(λ²−1) 3a₂²(m²−1)

(4.16) or

1(ξ)= ±

k²c4(λ²−1) 3a2

nd

ξ

k²c₄²(λ²−1) 3a₂²(m²−1)

, (4.17) provided(m²−1)(λ²−1) >0.

(5) If

c0 = c³₄

32c²₆(1−m²), c2 = c₄²(4m²−5)

16c6(m²−1), c6 >0, then

1(ξ)= ±

k²c4(λ²−1) 3a2

nc

ξ

k²c₄²(λ²−1) 3a₂²(1−m²)

(4.18) or

1(ξ)= ±

k²c4(λ²−1) 3a2

√1−m²

ds

ξ

k²c₄²(λ²−1) 3a₂²(1−m²)

, (4.19) provided(1−m²)(λ²−1) > 0.Ifm → 0,then we have the periodic wave solution

1(ξ)= ±

k²c₄(λ²−1) 3a₂

sec

ξ

k²c²₄(λ²−1) 3a²₂

(4.20) and we have the periodic wave solution

1(ξ)= ±

k²c4(λ²−1) 3a2

csc

ξ

k²c²₄(λ²−1) 3a²₂

, (4.21) provided(λ²−1) >0.

(6) If c₀= m²c³₄

32c²₆, c₂ = c²₄(m²+4)

16c6 , c₆ <0, then

1(ξ)= ±

k²c₄(λ²−1) 3a2

dn

ξ

−k²c₄²(λ²−1) 3a₂²

(4.22) or

1(ξ)= ±

k²c4(λ²−1)√ 1−m² 3a2

×nd

ξ

−k²c²₄(λ²−1) 3a²₂

, (4.23)

provided(λ²−1) <0.

5. Solving eq. (2.8) using the modified simple equation method

To this aim, balancing ₁ with₁³ in eq. (2.8) yields the balance number N =1. According to the modified simple equation method [35–40], eq. (2.8) has the formal solution

1(ξ)= A₀+A₁ ϕ(ξ)

ϕ(ξ)

, (5.1)

where A0 and A1 are constants to be determined, such that A₁ = 0 and ϕ(ξ) = 0. Substituting (5.1) into (2.8), and collecting all the coefficients of powers of ϕ^−j(j =0,1,2,3) and setting them to zero, we have the following algebraic equations:

ϕ⁰(ξ):

−

β2+l² 2

A0+ 3

1−λ²A³₀ =0, (5.2) ϕ⁻¹(ξ):

A1

k²

2 ϕ−

β2+l² 2

ϕ+ 9A²₀ 1−λ²ϕ

=0, (5.3) ϕ⁻²(ξ):

−3

2k²ϕ+ 9A0A1

1−λ²ϕ =0, (5.4)

ϕ⁻³(ξ):

k²+ 3A²₁

1−λ² =0. (5.5)

On solving the two algebraic equations (5.2) and (5.5), we have the following results:

A0=0 or A0 = ±

1

3(1−λ²)

β2+l² 2

,

A1= ±

−k²(1−λ²)

3 , (5.6)

provided(β2+(l²/2)) <0, (1−λ²) <0. Case1: A0 =0, A1= ±

−k²(1−λ²)

3 .

In this case, eqs (5.3) and (5.4) reduce to k²

2 ϕ(ξ)−

β2+l² 2

ϕ(ξ)=0 (5.7)

and

ϕ(ξ)=0. (5.8)

From (5.7) and (5.8), we obtain ϕ(ξ)=0.

This is a contradiction becauseϕ(ξ)should be a non- zero value. Hence, it is rejected.

Case2: A0 = ±

1

3(1−λ²)(β2+^l₂²), A1= ±

−k²(1−λ²)

3 .

In this case, eqs (5.3) and (5.4) reduce to k²ϕ(ξ)+4

β2+l²

2

ϕ(ξ)=0 (5.9)

and ϕ(ξ)−2

k

−

β2+l² 2

ϕ(ξ)=0. (5.10) From (5.9) and (5.10), we obtain

ϕ(ξ)=c1exp

⎡

⎣2ξ k

−

β2+l² 2

⎤⎦. (5.11)

From (5.10) and (5.11), we obtain ϕ(ξ)= −c1

−k² 4

β2+(l²/2)

×exp

⎡

⎣2ξ k

−

β2+l² 2

⎤⎦. (5.12)

Consequently, ϕ(ξ)= −k²c1

4

β2+(l²/2)exp

⎡

⎣2ξ k

−

β2+l² 2

⎤⎦+c2,

(5.13) wherec1 andc2 are constants of integration. Then, we have the new exact solution

1(ζ )= ±

1

3(1−λ²)

β2+l² 2

⎡

⎢⎣1− 2 exp

(2ξ/k)

−

β2+(l²/2) exp

(2ξ/k)

−(β2+(l²/2))

−(4(β2+(l²/2))c2)/(k²c1)

⎤

⎥⎦.

(5.14) In particular, if we setc2/c1=k²/4(β2+(l²/2)),then

we have the singular soliton solution:

1(ξ)= ∓

1

3(1−λ²)

β2+l² 2

×coth

⎛

⎝ξ

−(β2+(l²/2)) k²

⎞

⎠ (5.15)

and if we setc2/c1 = (−k²/4(β2+(l²/2))),then we have the dark soliton solution

1(ξ)= ∓

1

3(1−λ²)

β2+l² 2

×tanh

⎛

⎝ξ

−(β2+(l²/2)) k²

⎞

⎠, (5.16)

where(β2+(l²/2)) <0, (1−λ²) <0.

6. Graphical representation of some solutions of eq. (2.8)

In this section, we present the graphical representa- tion of the original equations. Let us now examine

Figure 1. Plot of the singular soliton solution of (3.7) with β2=4,l=0, λ=2,k=1.

figures 1–6 as they illustrate some of our solutions obtained in this paper. To this aim, we select some spe- cial values of the obtained parameters, e.g. in some of the solutions (3.7), (3.8), (4.7), (4.13), (4.22) and (5.16) of the coupled nonlinear Schrödinger equations (1.1)

Figure 2. Plot of the bright soliton solution of (3.8) with β2=3,l =0, λ=1/2,k=2.

Figure 3. Plot of the Jacobi elliptic function solution of (4.7) withl=0,λ=1/2,k=c4=a2=1,m=1/3.

Figure 4. Plot of the Jacobi elliptic function solution of (4.13) withl=0, λ=1/2,k=c4=a2=1,m=1/3.

Figure 5. Plot of the Jacobi elliptic function solution of (4.22) withl=0, λ=1/2,c4=1,k=a2=2,m=1/4.

and (1.2), where −10 < x,t < 10. For more conve- nience, the graphical representations of these solutions are shown in figures1–6.

From these figures, one can see that the obtained solu- tions possess the Jacobi elliptic function solutions, the dark soliton solutions, the singular soliton solutions, the solitary wave solutions and the periodic wave solutions.

Also, these figures show the behaviour of these solu- tions, providing some perspective to the readers on how the behaviour solutions are produced.

Figure 6. Plot of the dark soliton solution of (5.25) with β2= −3,l=0, λ=k=2.

7. Conclusions

We have derived many Jacobi elliptic function solutions, solitary wave solutions and periodic wave solutions of the coupled nonlinear Schrödinger equations (1.1) and (1.2) using the sine–cosine method, the new extended auxiliary equation method and the modified simple equation method. In this paper, on comparing our results with the well-known results obtained in [53], we deduced that our results are new and not found else- where. The three methods employed in this paper are concise and effective powerful mathematical tools for obtaining exact solutions of other nonlinear evolution equations. Finally, the results of this study have been checked using Maple by putting them back into the original equations.

Acknowledgements

The authors wish to thank the editor and the referees for their comments on this paper.

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