Analytical and approximate solutions of nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime

Lnre Akinyemi , Mehmet enol , M.S.Osmn

a Department of Mathematics, Lafayette College, Easton, PA, USA

b Nev ¸s ehir HacıBekta ¸s Veli University, Department of Mathematics, Nev ¸s ehir, Turkey

c Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

ABSTRACT The generalized Riccati equation mapping method (GREMM) is used in this paper to obtain different types of soliton solutions for nonlinear Schrödinger equation with higher dimension that existed in the regimes of anomalous dispersion.Later, we use the q-homotopy analysis method combined with the Laplace transform (q-HATM) to obtain approximate solutions of the bright and dark optical solitons.The q-HATM illustrates the solutions as a rapid convergent series.In addition, to show the physical behavior of the solutions obtained by the proposed techniques, the graphical representation has been provided with some parameter values.The findings demonstrate that the proposed techniques are useful, efficient and reliable mathematical method for the extraction of soliton solutions.

Keywords: q-homotopy analysis method Generalized Riccati equation mapping method Laplace transform

1.Introduction

The nonlinear Schrödinger equation (NLSE) is a prototype dis- persive NPDE that being evolved and analytically researched in several domains of physics for over 40 years.Among the most pop- ular governing equations for the dynamics of optical solitons, the NLSE is extremely important in the dynamics of nonlinear effects in optical communications [1–26] .There are various variants of the NLSE, which includes the nonlinear cubic-quintic terms, nonlinear derivative terms, nonlinear cubic-quintic-septimal terms, nonlin- ear cubic terms, and so forth [1] .The picosecond pulses in optical fibers have been represented using this equation.The applications further extend to different topics in scientific fields such as model- ing of deep water, plasma, nonlinear optical fibers, biophysics, con- densed matter physics and magneto-static spin waves [1,27–34] .To examine the dynamics of optical soliton propagation, there are sev- eral useful methods that have been applied, see

[35–58] for more detailed.

In this study, we intend to examine the nonlinear Schrödinger equation with higher dimension in the anomalous dispersion regime which is defined by [1]

whereu=u(x,y,t)is the complex envelope function associated with the optical-pulse electric field in a combing frame.t,xandyare the retarded time, normalized distance along the longitudi- nal axis of the fiber and normalized distance along the transverse axis of the fiber, respectively.β1,β2andβ4describethe effectsof the second-order dispersion.Finally,β3represents the Kerr nonlin- earity effect.Forβ2=β4= 0,andβ3= 1,(1) reduces to

which is the NLSE in the anomalous and normal dispersion regimes withandrespectively.In this present in- vestigation, the GREMM is first applied to get many soliton so- lutions with different structures.Then q-HATM which consist of q-HAM [59–62] and Laplace transform is further utilized to com- pute the approximate solutions to Eq.(1) .The parametersandn, which are convergence, are used by the q-HATM, offering greater flexibility in establishing and managing the convergence zone as well as the series solution of the convergence rate.In order to ad- dress the limitations such as enormous computation, performance, computer memory and others found in the classical method, q- HATM is suggested.The q-HATM has an immense ability to min- imize computational size and maintain a high degree of numer- ical solution accuracy.It is to be noted that for the special case when= -1 andn= 1,exact solution of the proposed model by q-HATM is also achieved with some valid conditions.It should be pointed out that Wazwaz in [1] have applied variational iteration method to obtain the bright and dark optical solitons for (1) .

The following is the order in which the paper will be com- pleted.Section 2 described the fundamental solution technique and its applicability to the model considered.The fundamental idea of q-HATM and it implementation to the proposed model are stated in Section 3 .The related findings are illustrated numerically in Section 4 and the conclusion in Section 5 .

2.The GREMM

This section presents the main algorithm for the GREMM [69] .Given a(2 + 1)-dimensional PDE as

We suggest the solitary wave transformation structure as

With the above transformation, we reduce (3) to the ODE

Assume that (5) solution is described by

wheregj(j= 0,1,···,Ω),ω1,ω2 andϑare constants to be eval- uated thereafter.The numberΩ＞0 is obtained upon balancing the nonlinear and highest-order derivative terms in (5) .In (6) , the functionΦ(ζ)satisfies the generalized Riccati equation provided as follows:

whereδ0,δ1andδ2are constants.The families of solution for (7) are set out as:

Here,ζ0is constant,P0 andQ0 are real numbers satisfyingP2-Q2＞0.Inserting Eqs.(6) and (7) into (5) with the coefficients ofΦj(ζ)to zero, we achieve a system of algebraic equations.By calculating the solutions of these algebraic equations and replace (6) with the above-mentioned solution sets, we swiftly get the ex- plicit solutions of (3) .

2.1. Application of the recommended techniques

Through the use of wave transformation in (4) , we reduce (1) to the real and imaginary parts couple nonlinear ODEs respectively as:

and

Upon setting the coefficient ofu′(ζ)in (9) to zero, we get

Balancingu′′(ζ)withu(ζ)3in (8) , we obtainΩ= 1,and from (6) , the solution is presumed as

Now we can easily focus on (8) solution procedure.

2.2. Application of the GREMM

If we plugging Eqs.(7) and (11) into (8) , and setting the coeffi- cient ofΦi(ξ)i= 0,1,···,3 to zero, we get

After solving the above system, in addition to (10) , we get the fol- lowing cases:

Case 1.

According to case (1) along with (11) and the sets of solutions defined in Section 2 , we reach the following solutions:

Forϱ=-4δ0δ2＞0 andδ1δ20 (orδ0δ20) :

Forδ0= 0 andδ1δ2:

whereζ=ω1x+ω2y+η1t,ϑ=θ1x+θ2y+η2t,ζ0is arbitrary constant,PandQare non-zero real constants satisfyingP2-Q2＞0.

Case 2.

Based on case (2) with the aid of (11) and the solutions set speci- fied in Section 2 , we state the following solutions:

Forϱ=-4δ0δ2＞0 andδ1δ20 (orδ0δ20 ):

Fig.1.The 2D, 3D and contour plots of (16) with β1 =, β2 = , β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , δ1 = δ2 = -1 , and δ0 = -1 .

whereζ=ω1x+ω2y+η1t,ϑ=θ1x+θ2y+η2t,ζ0is arbitrary constant,PandQare non-zero real constants satisfyingP2-Q2＞0.

2.3. The bright soliton solutions

The bright soliton solution can be obtain by proposing a solu- tion in the form

wheregis arbitrary real constant.Inserting (68) into (1) .Fur- thermore, setting coefficient of sechr(ω1x+ω2y+η1t),r= 1,3 t o zero, we get

With the solution of the imaginary part equals

The solutions of (69) are

Fig.2.The 2D, 3D and contour plots of (35) with β1 = , β2 = , β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , δ1 =δ2 = -1 , and δ0 = -1 .

With the use of Eqs.(68) and (71) , the bright soliton solution is defined as

3.Basic concept of the q-HATM

Here, the main idea of q-HATM [63–68] to NLSE with higher dimension in the anomalous dispersion regimes is presented.Con- sider (1) as

having initial condition

Firstly, start with applying Laplace transform to both sides of (73) to get

whereΨ=Ψ(x,y,t;q)andn≥1.We defineN(Ψ)as

Fig.3.The plots of the mixed bright-dark soliton solution in (a)-(c) q-HATM ( U (2) ), (d)-(f) exact and (g)-(i) comparison at t = 0 .1 in Section 3.1.1 with β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 .1 , y = 1 , θ1 = 0 .1 , θ2 = 0 .5 , n = 1 , and h = -1 .

Fig.4.The mixed bright-dark soliton profiles of q-HATM ( U (2) -solution) with different tin Section 3.1.1 with β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , n = 1 , and = -1 .

Fig.5.The 2D absolute error plots of q-HATM ( U (2) -solution) with the exact solution for the mixed bright-dark solition for β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , n = 1 , ħ = -1 , and t = 0 . 1 .

and

Accordingly, Eqs.(84) and (85) reduces to

The iterative terms ofu(x,y,t)are generated with the simplifica- tion of (87) and the q-HATM solution series is

Moreover, for the prescribed value ofnandrespectively, one get

Theorem 1.[68]Supposewecangetarealnumbersξsuchthat0＜ξ＜1satisfying

Inaddition,ifthetruncatedseriesU(Ω)(x,y,t;n;)establishedin(88)isusedasanapproximatesolutionfor(73),thenthemaximumabsolutetruncatederrorsiscalculatedas

Proof.Foraspecifiedvalueofandn,weget

Thiscompletestheproof.□

3.1. q-HATM solutions for (2 + 1) -dimensional NLSE

Here, we consider the initial conditions adopted from Eqs.(14) and (71) respectively.

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3.1.1.Usingthedarksolitonsolution

From Eqs.(13) and (14) withδ0=δ2= -1,δ1=andg1=g,the exact solution is

Using Eqs.(87) and (94) , one get

Upon solving the above equation with (94) , we obtain the follow- ing:

Other terms can be achieve for r = 2 , 3 , ···.

3.1.2.Using the bright soliton solution

From (72) , we have initial condition as

Fig.7.The bright soliton solution profiles of q-HATM ( U (2) ) with different tin Section 3.1.2 for β3 = β4 = 0 . 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , n = 1 , and = -1 .

Fig.8.The 2D absolute error plots of q-HATM ( U(2) ) with the exact solution for the bright solition at β3 = β4 = 0 . 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , n = 1 and t = 0 . 1 .

Fig.9.The -curve plots of q-HATM ( U(2) ) for the dark solition with β1 = - β2 = β3 = -1 , β4 = 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , and n = 1 .

Fig.10.The -curve plots of q-HATM ( U(2) ) for the bright solition solutions for β1 = - β2 = β3 = β4 = 0 . 1 , ω 1 = ω 2 = 0 . 1 , y = 1 , θ1 = 0 . 1 , θ2 = 0 . 5 , and n = 1 .

Upon solving the above equation with (96) , we have

Other terms can be acquire forr= 2,3,···.

4.Numerical results and discussions

This section examine the numerical simulations and their dis- cussions with the aid of proposed techniques for the given NLSE with higher dimension in the anomalous dispersion regime.The q-HATM three term solution series is given as

The GREMM present various kinds of soliton solutions which include the dark solition, singular, periodic, and other soliton-type solution.In Figs.1 and 2 , we present the graphical presentation of Eqs.(16) and (35) in 2D, 3D and contour plots which display a soliton-like solutions.Figs.3 , 4 , 6 , 7 present the three-term ap- proximation of the real part, imaginary part and absolute plots of the q-HATM and the exact solution.These plots include the 2D and 3D with some parameter values.In Figs.4 and 7 , the response of the q-HATM at different time “t′′ instance are displace in 2D.These plots present the bright and dark soliton solutions at different time levels.The 2D absolute error plots of q-HATM (U2) with the exact solution are illustrated in Figs.5 and 8 .These cited figures reveal the accuracy of q-HATM.

Choosing the auxiliary parameteris very critical in order to ensure quick convergence of the series solutions obtained by q- HATM.In Figs.9 and 10 , the-curves that facilitate the optimal selection of thevalues for concern model is outlined.In the-curves, the horizontal line guide shows theselection which confirms the option of selecting= -1 in this current study.In Figs.3 and 6 , we can identify that the q-HATM and exact solution presented is in good agreement which confirm that the proposed method is an effecient and reliable mathematical tool for handling nonlinear problems.

5.Concluding remarks

The present study involves the analytical and approximate so- lutions of NLSE with higher dimension which explain the anoma- lous dispersion regimes.We take advantage of the GREMM to sev- eral types of soliton solutions which includes the dark, singular, periodic, and rational solutions.The q-HATM, a powerful tool, is used to unfold the approximate solutions of the bright and dark soliton solutions.This procedure uses two convergence parame- ters,andn, whichprovidegreaterflexibilitytoinfluenceand manipulate the convergence region and also the convergence rate of the series solution.It is worth to be noting that for the spe- cial case when= -1 andn= 1,exact solution of the proposed model is achieved using q-HATM under some valid conditions.The plots captured in this study demonstrate simulation behavior and can assist researchers with some of the interesting and important implications of the model considered.This study ensures that the proposed methods are sufficiently effective and can be used to find solutions to other nonlinear models in science and engineering.

Declaration of Competing Interest

None.