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Revista mexicana de física

Print version ISSN 0035-001X

Rev. mex. fis. vol.65 n.6 México Nov./Dec. 2019  Epub May 30, 2020

http://dx.doi.org/10.31349/revmexfis.65.658 

Research

in Optics

Analytical treatment in optical metamaterials with anti-cubic law of nonlinearity by improved exp(-Ω(η))-expansion method and extended sinh-Gordon equation expansion method

Onur Alp Ilhana 

Jalil Manafianb 
http://orcid.org/0000-0001-7201-6667

aDepartment of Mathematics, Faculty of Education, Erciyes University, 38039-Melikgazi-Kayseri, Turkey. e-mail: oailhan@erciyes.edu.tr.

bDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran. e-mail: j-manafianheris@tabrizu.ac.ir, Tel:+989147738166 .

Abstract

Here, the improved exp(-Ω(η))-expansion method and extended sinh-Gordon equation expansion method are being applied on (1+2)-dimensional non-linear Schrödinger equation (NLSE), optical metamaterials, with anti-cubic nonlinearity. Materials like photovoltaic-photorefractive, polymer and organic consists of spatial solitons and optical nonlinearities, which can be identified by seeking help from NLSE with anti-cubic nonlinearity. Abundant exact traveling wave solutions consisting of free parameters are established in terms of bright, dark, singular, kink-singular, and combined dark-bright soliton solutions. Various arbitrary constants obtained in the solutions help us to discuss the graphical behavior of solutions and also grants flexibility to formulate solutions that can be linked with a large variety of physical phenomena. Moreover, graphical representation of solutions are shown vigorously in order to visualize the behavior of the solutions acquired for the equation.

Keywords: Solitons; metamaterials; anti-cubic nonlinearity; improved exp(-Ω(η))-expansion method; extended sinh-Gordon equation expansion method

PACS: 02.60.Lj; 02.70.Wz; 02.90.+p; 04.30.Nk

1. Introduction

The nonlinear dynamics that describes the propagation of pulses in optical metamaterials (MMs) is given by the nonlinear Schrödinger equation (NLSE). In the presence of parabolic law nonlinearity, with an additional anti-cubic nonlinear term and perturbation terms that include inter-modal dispersion (IMD), self-steepening (SS) as well as nonlinear dispersion (ND), the governing equation reads [1-6]

iqt+aqxx+b1q|-4+b2q|2+b3q|4 q=i [αqx+β(|q|2q)x+ v (|q|2)xq]+θ1(q|2q)xx+ θ2q|2qxx+θ3q2qxx*. (1)

In Eq. (1), the unknown or the dependent variable q(x, t) represents the wave profile, while x and t are the spatial and temporal variables respectively. The first and second terms are the linear temporal evolution term and group velocity dispersion (GVD), while the third term introduces the anti-cubic nonlinear term, fourth and fifth terms account for the parabolic law nonlinearity, and sixth, seventh and eighth terms represent the IMD, SS, and ND respectively. Finally, the last three terms with 𝜃 k for k = 1, 2, 3 appear in the context of metamaterials [7, 8].

Metamaterials are basically artificially structured materials which are made from assemblies of multiple elements fashioned from composite materials such as metals or plastics. In just a few years, the field of optical metamaterials has emerged as one of the most exciting topics in the science of light, with stunning and unexpected outcomes that have fascinated scientists and the general public alike. Its applications' include superlenses, super-resolution devices, and negative-indexed materials. Such applications necessitate the presence of unnatural materials with properties that can fit into these applications and others. The study of solitons in optical metamaterials is trending as a hotspot in the field of optical materials. There has been a significant amount of results that are reported in this field. However, there is still a long way to go. There are many unanswered questions than answers. This paper will quell the thirst partially. In the past, solitons in optical metamaterials have been studied with various forms of non-Kerr laws of nonlinearity where several integration schemes have been implemented [10-28]. The interested reader also read herein references [29-45]. This paper is going to revisit the study of solitons in optical metamaterials for a specific form of nonlinear medium. This is of anti-cubic (AC) type. There are three forms of integration algorithms that will be applied to extract soliton solutions to metamaterials with AC nonlinearity. These schemes will retrieve bright, dark and singular soliton solutions that will be very important in the study of optical materials. These solitons will appear with constraint conditions that are otherwise referred to as existence criteria of the soliton parameters. After a quick introduction to the model, the integration techniques will be applied and the details are enumerated in the subsequent sections.

In order to solve Eq. (1), the starting hypothesis is [7, 8]

qx,t=uηx,texp(iϕx,t), (2)

where

η(χ, t) = k (x - vt), (3)

and the phase component 𝜙 is given by

ϕ(x, t) = -κx + ωt + θ, (4)

In Eqs. (2) and (3), u(x,t) represents the amplitude portion of the soliton, and k and v are inverse width and velocity of soliton. From (4), κ is the frequency of the soliton, ω is the wave number of the soliton and finally θ is the phase constant. Inserting (2) into (1) and then decomposing into real and imaginary parts yield a pair of relations. Imaginary part gives

v=-α-2aκ, (5)

and

3β + 2ν - 2κ(3θ1 + θ2 - θ3) = 0, (6)

while real part leads to

ak2u´´-ω+ak2+ ακu+b1u-3

+b2-βκ+κ2θ1+κ2θ2+κ2θ3u3+b3u5

-3k2θ1+k2θ2+k2θ3u2u´´

-6k2θ1u(u´)2=0. (7)

To acquire an analytic solution, the transformations θ 1 = 0 and θ 2 = - θ 3 are applied in Eq. (7), and gives

ak2u" - (ω + ακ2 + ακ)u + b1u-3+ (b2 - βκ)u3 + b3u5 = 0, (8)

where

3β + 2ν + 4κθ3 = 0. (9)

In order to obtain closed-form solutions, we employ the transformation given by

u=v12, (10)

that will reduce Eq. (8) into the ordinary differential equation (ODE)

ak2 (2vv" - v'2) + 4b1 - 4(ω + ακ2 + ακ)υ2+ 4(b2 - βκ)ν3+4b3v4 = 0. (11)

The generalized (G'/G)-expansion approach [29-31] will now be applied, in the subsequent section, to Eq. (11) to retrieve bright, dark and singular soliton solutions to the NLSE with AC nonlinearity (1). Biswas and coworkers investigated the extended nonlinear Schrödinger equation [49], the nonlinear Schrödinger equation with parabolic law nonlinearity [50], the perturbed nonlinear Schrödinger equation with five different forms of nonlinearity [51], the Schrödinger-Hirota equation in birefringent fiber [52], the Gerdjikov-Ivanov equation [53], the complex Ginzburg-Landau equation [54] and obtained new exact solutions including different forms of optical solitons. Also, authors of [55-57] studied the nonlinear Schrödinger equation and obtained optical solitons with help of the trial and extended trial equation methods. Finally, bright optical soliton solutions from resonant nonlinear Schrödinger's equation has been gained by the aid of semi-inverse variational principle by Biswas et al. [58].

This paper is organized as follows: Sec. 2 presented a brief discussion about the improved exp(-Ω(η))-expansion method and its application for solving the aforementioned equation. Moreover, Sec. 3 and its sub-sections deal with the applications of the extended sinh-Gordon equation expansion method (EShGEEM) to look for new singular, kink-singular, and combined dark-bright soliton solutions. Physical significance by graphical presentation of some of the obtained solutions is given in Sec. 4. Also, a conclusion is given in Sec. 5.

2. The improved method exp(-Ω(η))-expansion

This section elucidates a systematic explanation of exp(-Ω(η))-expansion method [59, 60] so that it can be further applied to optical metamaterials with anti-cubic non-linearity in order to furnish its exact solutions:

Step 1. The following nonlinear partial differential equation (NLPDE)

N(u, ux, ut, uxx, utt,...) =0, (12)

can be transformed into an (ODE)

QU,BU´, -BuU´, B2U´´, B2v2U´ ,=0, (13)

by using the suitable transformation η = B (x - vt), where B and v are the free parameters which would be calculated subsequently.

Step 2. Assuming the solution of the ODE to be of the form:

Uη=j=0NAjFj(η)j=0MBjFj(η), (14)

where F (η) = exp (-Ω(η)) and A j (0 ≤ j ≤ N ),B j (0 ≤ j ≤ M), are constants to be determined, such that A N , BM 0, and, Ω = Ω(η) satisfying the ODE given below

Ω' = μF-1(η) + F (η)+ λ. (15)

The special cases formed from the solutions [61, 62] of the ODE given in Eq. (15) are mentioned below:

Solution-1: If μ ≠ 0 and λ 2 - 4μ > 0, then we have

Ωη=In (-λ2-4μ2μ×tanhλ2-4μ2η+E-λ2μ), (16)

where E is integral constant.

Solution-2: If μ ≠ 0 and λ2 - 4μ < 0, then we have

Ωη=In (-λ2-4μ2μ×tan-λ2-4μ2η+E-λ2μ). (17)

Solution-3: If μ = 0, λ 0, and λ2 - 4μ > 0, then we get

Ωη=-In λexpλη+E-1 . (18)

Solution-4: If μ ≠ 0, λ 0, and λ2 - 4μ = 0, then we get

Ωη=In-2λη+E+4λ2(η+E). (19)

Solution-5: If μ = 0, λ = 0, and λ2 - 4μ = 0, then we get

Ω(η) = 1η (η + E), (20)

where A j (0 ≤ j ≤ N), B j (0 ≤ j ≤ M), λ and μ are also the constants to be explored later. The values N and M are determined by equalizing the maximum order nonlinear term and the maximum order partial derivative term appearing in (13). If N and M are the rational, then the appropriate transformations can be applied to conquer these hurdles.

Step 3. Putting (14) into Eq. (13) as well as the values of N and M determined in previous step into (14). Gathering coefficients of all the powers of F (η), then equating every coefficient with zero, we derive a set of over-determined nonlinear algebraic equations for A 0 , B 0 , A 1 , B 1,... , A N , B M , λ, and μ.

Here, it is important to note that E is the integration constant. We have the following relations as

vηδFN-M, (21)

v´ηδFN-M-1 F´=-δμFN-M-1+ λFN-M+FN-M+1-δFN-M+1, (22)

(v´η)2δ2F2N-2M+2, (23)

(vη)4δ2F4N-4M. (24)

where 𝛿 = (An/Bm). Balancing v' 2 with v 4 in Eq. (11) yields

F2N-2M+2(v´ξ)2=(vη)4F4N-4N (25)

We can determine values of N and M as follows:

2N - 2M + 2 = 4N - 4M  N = M + 1. (26)

Case I: N=2, M=1

The improved exp(-𝜙)-expansion method (IEFM) allows us to recruit the substitutions

vη=A0+A1Fη+A2F2(η)B0+B1F(η)=A2+A1eΩ+A0e2ΩB0e2Ω+B1eΩ . (27)

Plugging (27) along with (15) into Eq. (11) and equating all the coefficients of powers of F (η) to be zero, one gains a system of algebraic equations. Solving this system by the help of Maple yields

Set I:

A0=3b1B14+A12b3(A1-A2λ)2A2b3(A1-A2λ)2,A1=A1, A2=A2, B0=B0, B1=B1,   k=2A2B1-b33a

κ=3b2B12+8A1B1b3B0-4b3A2λB13B12β, ω=-19(A1-A2λ)2B14β2 Σ1+Σ2+Σ3+Σ4+Σ5,

Σ1=-2λA1A264ab32B02A22+ 9b2B14ab2+αβ+6b3B125A1B1-4A2B0αβ+2ab2+24b3A12B124ab3+β2-4A2b3A1B0B140ab3+9β2,

Σ2=b3A24B12λ416ab3+3β2,

Σ3= -4λ3 A23 B1b3(3B12αβ+2ab2-2A2B08ab3+3β2+6A1B14ab3+β2)

Σ4=λ2A2264ab32B02A22-48A2ab3b2B0B12-72 A2b3β2 A1B0B1-256A2ab32A1B0B1-24A2αb3βB0B12+9αβb2B14+48αb3βA1B-13+9ab22B14+96ab3b2A1B13+208ab32A12B12+54b3β2A12B12,

Σ5= A1264ab32B02A22+24b3B12A1B1-A2B0αβ+2ab2+b3A12B1264ab3+15β2-8A2b3A1B0B116ab3+3β2+9b2B14ab2+αβ,

μ=3B15b1+b3(A2λ-A1)2(4B0A22λ-4A1B0A2+B1A12)4A22b3(A1-A2λ)2B1,

λ=λ, (28)

where A1 , A2 , B0 and B1 are arbitrary constants. Imposing the solution set (28) into (27), the solution formula of Eq. (11) can be concluded in the following cases:

Subcase IA:

By the help of (16), the exact solutions to the model are deducted as

qη=A2-A1λ2-4μ2μtanhλ2-4μ2η~+λ2μ+3b1B14+A12b3A1-A2λ2A2b3(A1-A2λ)2λ2-4μ2μtanhλ2-4μ2η~+λ2μ2B0λ2-4μ2μtanhλ2-4μ2η~+λ2μ2-B1λ2-4μ2μtanhλ2-4μ2η~+λ2μ12

×ei-3b2B12+8 A1B1b3-8A2b3B0-4b3A2λB13B12β x- 19(A1-A2λ)2B14β2Σ1+Σ2+Σ3+Σ4+Σ5t+θ (29)

where

η~=2A2B1-b33ax+α+2a3b2B12+8A1B1b3-8A2b3B0-4b3A2λB13B12βt+E.

It should be noted that these solitons exist for

λ2-4μ=λ2-3B15b1+b3A2λ-A124B0A22λ-4A1B0A2+B1A12A22b3A1-A2λ2B1>0.

Subcase IB:

By the help of (17), the exact solutions to the model are deducted as

qη=A2+A1λ2-4μ2μtan4μ-λ22η~-λ2μ+3b1B14+A12b3(A1-A2λ)2A2b3A1-A2λ2λ2-4μ2μtan4μ-λ22η~-λ2μ2B04μ-λ22μtan4μ-λ22η~-λ2μ2+B14μ-λ22μtan4μ-λ22η~-λ2μ12

×ei-3b2B12+8 A1B1b3-8 A2b3B0-4b3A2λB13B12βx-19(A1-A2λ)2B14β2Σ1+Σ2+Σ3+Σ4+Σ5t+θ (30)

where

η~=2A2B1-b33ax+α+2a3b2B12+8A1B1b3-8A2b3B0-4b3A2λB13B12βt+E.

It should be noted that these solitons exist for

λ2-4μ=λ2-3B15b1+b3A2λ-A124B0A22λ-4A1B0A2+B1A12A22b3A1-A2λ2B1<0.

Subcase IC:

By the help of (18), the exact solutions to the model are obtained as

qη=λ2A2+λA1(exp(η+E))-1)+3b1B14+A12b3(A1-A2λ)2A2b3(A1-A2λ)2(exp (λ(η+E))-1)2 B0(expλη+E-1)2+λB1(expλη+E-1)12

×ei-3b2B12+8 A1B1b3-8 A2b3B0-4b3A2λB13B12βx-19A1-A2λ2B14β2Σ1+Σ2+Σ3+Σ4+Σ5t+θ (31)

where

η=2A2B1-b33ax+α+2a3b2B12+8A1B1b3-8A2b3B0-4b3A2λB13B12β.

It should be emphasized that these solitons exist for λ ≠ 0.

Subcase ID:

By the help of (19), the exact solutions to the model are gained as

qη=A2-A12λη+E+4λ2(η+E)+3b1B14+A12b3(A1-A2λ)2A2b3(A1-A2λ)22λη+E+4λ2(η+E)2B0-2λη+E+4λ2(η+E)2-B12λη+E+4λ2(η+E)12

×ei-3b2B12+8 A1B1b3-8 A2b3B0-4b3A2λB13B12βx-19(A1-A2λ)2B14β2Σ1+Σ2+Σ3+Σ4+Σ5t+θ (32)

where

η=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-8 A2b3B0-4b3A2λB13B12βt.

It should be noted that these solitons exist for

3B15b1+b3A2λ-A124B0A22λ-4A1B0A2+B1A12-λ2A22B1=0.

Subcase IE:

By the help of (20), the exact solutions are obtained as

qη=A2+A1η+E+3b1B14+A14b3A2b3A12(η+E)2B0(η+E)2+B1(η+E)12ei-3b2B12+8 A1B1b3-8 A2b3B03B12βx-Σ59 A12B14β2t+θ,

where

η=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-8 A2b3B03B12βt.

It should be pointed out that these solitons exist for 3B15b1+b3A13B1A1-4B0A2=0.

Set II:

A0=B0(A1B1-A2B0)B12, A1=A1, A2=A2, B0=B0, B1=B1,  k=2A2B1-b33A

κ=3b2B12+8 A1B1b3-16 A2b3B03B12β,

ω=19 B14β2ab2+6β2Σ2,

Σ1=9 B14b2ab2+βα+2A1B1b3A1B1-4A2B032ab3+9β2+2 A22B02b3128ab3+33β2+24B12b3A1B1-2A2B02ab2+βα,

Σ2=B14-3b3b1±B12A12b3-4B1A2B0b3A1+3A22B02b3,

μ=1B12A22b3B14-3b3b1±B12A12b3-4B1A2B0b3A1+3A22B02b3,

λ=λ, (34)

where A 1, A 2, B 0 and B 1 are arbitrary constants. Plugging the solution set (34) into (27), the solution formula of Eq. (11) can be concluded in the following:

Subcase IIA:

By the help of (16), the exact solutions are in the following:

qη=A2-A1λ2-4μ2μtanhλ2-4μ2η~+λ2μ+B0A1B1-A2B0B12λ2-4μ2μtanhλ2-4μ2η~+λ2μ2B0λ2-4μ2μtanhλ2-4μ2η~+λ2μ2-B1λ2-4μ2μtanhλ2-4μ2η~+λ2μ12

×ei-3b2B12+8 A1B1b3-16 A2b3B03B12βx-19B14β2Σ1+6β2Σ2t+θ (35)

where

η~=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-16 A2b3B03B12βt+E.

It should be emphasized that these solitons exist for

λ2-4μ=λ2-4B12A22b3B14-3b3b1±B12A12b3-4B1A2B0b3A1+3A22B02b3>0.

Subcase IIB:

By the help of (17), the exact solutions are concluded in the form:

qη=A2+A1λ2-4μ2μtan4μ-λ22η~-λ2η+B0A1B1-A2B0B12λ2-4μ2μtan4μ-λ22η~-λ2μ2B04μ-λ22μtan4μ-λ22η~-λ2μ2+B14μ-λ22μtan4μ-λ22η~-λ2μ12

×ei-3b2B12+8 A1B1b3-16 A2b3B03B12βx-19B14β2Σ1+6β2Σ2t+θ (36)

where

η~=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-16 A2b3B03B12βt+E.

It should be noted that these solitons exist for

λ2-4μ=λ2-4B12A22b3B14-3b3b1±B12A12b3-4B1A2B0b3A1+3A22B02b3<0.

Subcase IIC:

By the help of (18), the exact solutions are given as

qη=λ2A2+λA1expλη+E-1+B0A1B1-A2B0B12expλη+E-12B0expλη+E-12+λB1expλη+E-112

×ei-3b2B12+8 A1B1b3-16 A2b3B03B12βx-19B14β2Σ1+6β2Σ2t+θ (37)

where

η~=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-16 A2b3B03B12βt.

It should be pointed out that these solitons exist for λ = 0.

Subcase IID:

By the help of (19), the exact solutions are gained as

qη=A2-A12λη+E+4λ2η+E+B0A1B1-A2B0B122λη+E+4λ2η+E2B0-2λη+E+4λ2η+E2-B12λη+E+4λ2η+E12 ei-3b2B12+8 A1B1b3-16 A2b3B03B12βx-19B14β2Σ1+6β2Σ2t+θ (38)

where

η=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-16 A2b3B03B12βt.

It should be noted that these solitons exist for

B12A22b3λ2=4B14-3b3b1±B12A12b3-4B1A2B0b3A1+3A22B02b3.

Subcase IIE:

By the help of (20), the exact solutions are concluded as

qη=A2-A1η+E+B0A1B1-A2B0B12η+E2B0η+E2+B1η+E12 ei-3b2B12+8 A1B1b3-16 A2b3B03B12βx-19B14β2Σ1+6β2Σ2t+θ, (39)

where

η=2A2B1-b33ax+α+2a3b2B12+8 A1B1b3-16 A2b3B03B12βt.

It should be noted that these solitons exist for

B14-3b3b1=b3 B12A12-4B1A2B0A1+3A22B02

Set III:

A0=A0, A1=A1,A2=A2,  B0=12λB1,   B1=-27b3b13A14-8A12A0A2+16A02A2243b1,

κ=24b3A0A1A2+3A0A2b2B1-8b3A13-3b2B1A123B1βA0A2-A12,

ω=-19B12β2A0A2-A122Σ1+Σ2,

Σ1=9B12b2A0A2-A122ab2+βα+2b3A12-A12+3A0A2232ab3+9β2+24B1b3A1-A12+3A0A2A0A2-A122ab2+βα,   Σ2=-24b3A03β2A23,

μ=-A02A0A2-A12,  λ=-2A0A1A0A2-A12,  κ=2A2B14A0A2-A12b1a,   Δ=λ2-4μ=4A03A2A0A2-A122, (40)

where A 0 , A 1 and A 2 are arbitrary constants. Substituting the solution set (40) into (27), the solution formula of Eq. (11) can be written in the following cases:

Subcase IIIA:

By the help of (16), the exact solutions can be stated as

qη=A2+A1A2A0tanhA0A0A2A0A2-A12η~-A1A0+A0A1A0tanhA0A0A2A0A2-A12η~-A1A0212λB1A2A0tanhA0A0A2A0A2-A12η~-A1A02+B1A2A0tanhA0A0A2A0A2-A12η~-A1A012

×ei-24b3A0A1A2+3A0A2b2B1-8b3A13-3b2B1A123B1βA0A2-A12x-19B12β2A0A2-A122Σ1+Σ2t+θ (41)

where

η~=2A2B14A0A2-A12b1ax+α+2a24b3A0A1A2+3A0A2b2B1-8b3A13- 3b2B1A123B1βA0A2-A12t+E,

and A 0 A 2 > 0.

Subcase IIIB:

By the help of (17), the exact solutions can be obtained as

qη=A2+A1A2A0tanA0-A0A2A0A2-A12η~+A1A0+A0A2A0tanA0-A0A2A0A2-A12η~-A1A0212λB1A2A0tanA0-A0A2A0A2-A12η~-A1A02+B1A2A0tanA0-A0A2A0A2-A12η~-A1A012

×ei-24b3A0A1A2+3A0A2b2B1-8b3A13-3b2B1A123B1βA0A2-A12x-19B12β2A0A2-A122Σ1+Σ2t+θ (42)

where

η~=2A2B14A0A2-A12b1ax+α+2a24b3A0A1A2+3A0A2b2B1-8b3A13-3b2B1A123B1βA0A2-A12t+E,

with A0A<0.

Subcase IIIC:

By the help of (20), the exact solutions to the model as follows:

qη=A2+A1η+E12λB1η+E2+B1η+E12ei-8b3A13+3b2B1A123B1βA12x+19B12β2A14Σ1+Σ2t+θ (43)

where

η=-2A2B1A12b1ax+α+2a8b3A13+3b2B1A123B1βA12t.

It should be pointed out that these solitons for A 0 = 0.

3. Algorithm of the extended sinh-Gordon equation expansion method

Take the sinh-Gordon equation as:

uxt=αsinhu, (44)

where u=u(x,t) and α is a constant. By utilizing the relations u(x,t) =U(ξ) and ξ = kx - ωt, then Eq. (44) transforms to the following NODE:

U´´=-αkωsinhU. (45)

By using of the integrating process of Eq. (45) gets

U2´2=-αkω sinh2U2+p, (46)

where p is an integration constant.

Putting U2 = 𝜔(ξ), and - α/kω = q in Eq. (46), we gain

ω´=p+qsinh2(ω), (47)

by choosing the parameters p and q in Eq. (47) can be found the further results of solutions [63-67]:

Case-I: Take p = 0 and q = 1, then Eq. (47) gets

ω´=sinhω (48)

Simplifying Eq. (48), the following results are gained [63]:

sinh ω= ±isech (ξ), cosh (ω) = - tanh (ξ) (49)

and

sinh (ω) = ±csch (ξ), cosh (ω) = - coth (ξ). (50)

where, i = -1 represent an imaginary number.

Case-II: Take p = 1 and q = 1 , then Eq. (47) gets

ω' = cosh (ω). (51)

Again, simplifying Eq. (51), the following results are achieved [63]:

sinh (ω) = tan (ξ), cosh (ω) = ±sec (ξ) (52)

and

sinh (ω) = - cot (ξ), cosh (ω) = ±csc (ξ). (53)

We assume that the given nonlinear PDE

F (n, ut, ux, utt, uxx, uxt,.) =0, t > 0. (54)

In Eq. (54), u = u(x, t) is an unknown function, F is a polynomial in u(x, t) and its various partial derivatives are involved. Consider the following transformation:

u (x, t) = U (ξ) ξ = kx - ωt. (55)

Implementing the transformation of (55) into Eq. (54), then Eq. (54) converted to the following NODE

GU,-ωU´, kU´, ω2U´´, k2U´´,.=0, (56)

where G is a polynomial of U = U (ξ) and its derivatives with respect to ξ.

Suppose the exact solution be as

Uω=j=1Ncoshj-1ω ×Bjsinhω+Ajcoshω+A0. (57)

Base on the relations (48)-(50), then (57) can be written in the following forms

Uξ=j=1N-tanhξj-1 × ±iBjsechξ-Ajtanhξ+A0,  (58)

and

Uξ=j=1N-cothξj-1 × ±Bjcschξ-Ajcothξ+A0.  (59)

Similarly, base on the relations (51)-(53), then (57) can be written in the following forms

Uξ=j=1N±secξj-1 × Bjtanξ±Ajsecξ+A0,  (60)

and

Uξ=j=1N±cscξj-1 × -Bjcotξ±Ajcscξ+A0,  (61)

the value of N can be determined by using the homogeneous balance principle.

3.1. The EShGEEM

3.1.1. For Case-I: Eq. (48)

Base on the Eqs. (57)-(59), the solution of Eq. (11) can be presented as

U ξ= ±iB1 sech ξ- A1 tanh (ξ) + A0, (62)

and

U ξ= ±B1  csch ξ- A1 coth (ξ) + A0, (63)

and so

U (ω) = B1 sinh (ω) + A1 cosh (ω) + A0, (64)

where A 1 0 or B 1 0.

Inserting (64) and its derivatives into Eq. (11), we can obtain the following results by solving the nonlinear algebra system with the help of symbolic computation package as:

Set I-1:

A0=0,  A1=0,  B1=B1,    k=2B1 -b3a,    κ=b2β,    ω=-b2(ab2+αβ)β2.

The bright and singular soliton solutions for the Eq. (1) are achieved as

q1,1x,t=-iB1sech2B1-b3ax+α+2ab2βt12  ei-b2βx-b2(ab2+αβ)β2t+θ, (65)

and

q1,2x,t=-iB1csch2B1-b3ax+α+2ab2βt12  ei-b2βx-b2(ab2+αβ)β2t+θ. (66)

Set I-2:

A0=1-2B1,  A1=0,   B1=B1,    κ=2B1-b3a,      κ=b2β+-2b3B1β,

ω=-4ab3B1b2+ab22+2αβb3B1+αb2β-2B12ab32-2β2.

The bright and singular soliton solutions for the Eq. (1) are presented as

q2,1x,t=1-2B1-iB1sech2B1-b3ax+α+2ab2β+2a-2b3B1βt12

× ei-b2β+-2b3B1βx-4ab3B1b2+ab22+2αβb3B1+αb2β-2B12ab32-2β2t+θ, (67)

and

q2,2x,t=1-2B1-iB1csch2B1-b3ax+α+2ab2β+2a-2b3B1βt12

× ei-b2β+-2b3B1βx-frac4ab3 B1b2+ab22+2αβb3B1+αb2β-2B12ab32-2β2t+θ (68)

Set I-3:

A0=0,  A1=A1,  B1=0,    k=2A1 -b3a,    κ=b2β,    ω=-b2(ab2+αβ)β2.

Therefore, we conclude the following dark and singular soliton solutions for the Eq. (1) respectively:

q3,1x,t=-A1tanh2A1-b3ax+α+2ab2βt12ei-b2β x- b2ab2+αββ2t+θ, (69)

and

q3,2x,t=-A1coth2A1-b3ax+α+2ab2βt12ei-b2β x- b2ab2+αββ2t+θ, (70)

Set I-4:

A0=±A1,    A1=A1,   B1=0,    κ=2A1-b3a,   κ=b2±2b3A1β,

ω=- 4ab32A12±4ab3A1b2+ab22±2αβb3A1+αb2ββ2.

The dark and singular soliton solutions for the Eq. (1) are deduced as

q4,1x,t=±A1-A1tanh2A1-b3ax+α+2ab2±2b3A1βt12×ei-b2±2b3A1β x- 4ab32A12±4ab3A1b2+ab22±2αβb3A1+αb2ββ2t+θ, (71)

and

q4,2x,t=±A1-A1coth2A1-b3ax+α+2ab2±2b3A1βt12×ei-b2±2b3A1β x- 4ab32A12±4ab3A1b2+ab22±2αβb3A1+αb2ββ2t+θ. (72)

Set I-5:

A0=0,   A1=±B1,   B1=B1,     κ=4B1-b3a,   κ=b2β,   ω=-b2(ab2+αβ)β2.

The combined dark-bright and singular soliton solutions for the Eq. (1) are concluded as

q5,1 x,t=B1tanh4B1-b3ax+α+2ab2βt-iB1sech4B1-b3ax+α+2ab2βt12 ×ei-b2β x- b2(ab2+αβ)β2t+θ, (73)

and

q5,2 x,t=B1coth4B1-b3ax+α+2ab2βt-iB1csch4B1-b3ax+α+2ab2βt12 ×ei-b2β x- b2(ab2+αβ)β2t+θ. (74)

Set I-6:

A0±B1,   A1=±B1,   B1=B1,    κ=4B1-b3a,    κ=b2±2b3B1β,

ω=-ab2β±2αβb3B1+ ab22 ±4ab2b3B1+4B12ab32β2.

Thus, we deduce the following combined dark-bright and singular soliton solutions for the Eq. (1) respectively:

q6,1 x,t=± B1B1tanh4B1-b3ax+α+2ab2±2b3B1βt-iB1sech4B1-b3ax+α+2ab2±2b3B1βt12 ×ei-b2±2b3B1β x- αb2β±2αβb3B1+ab22±4ab2b3B1+4B12ab32β2t+θ, (75)

and

q6,2 x,t=± B1B1coth4B1-b3ax+α+2ab2±2b3B1βt-iB1csch4B1-b3ax+α+2ab2±2b3B1βt12 ×ei-b2±2b3B1β x- αb2β±2αβb3B1+ab22±4ab2b3B1+4B12ab32β2t+θ. (76)

3.1.2. For Case-II: Eq. (48)

Base on the Eqs. (60)-(61), the solution of Eq. (11) can be presented as

V ξ= B1tanξ ± A1secξ+ A0, (77)

And

V ξ= -B1cotξ ± A1cscξ+ A0, (78)

and so

V ω= B1sinh ω+ A1coshω+ A0, (79)

where A 1 0 or Bi ≠ 0.

Plugging (94) and its derivatives into Eq. (11), we can gain the following results by solving the nonlinear algebra system with the help of symbolic computation package as:

Set I-1:

A0=0,   A1=0,    B1=B1,     κ=2B1-b3a,      κ=b2β,     ω=- b2(ab2+αβ)β2.

The periodic and singular periodic solutions for the Eq. (1) are shown as

q7,1 x,t= B1tan2B1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ. (80)

and

q7,2 x,t= B1cot2B1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ. (81)

Set I-2:

A0=  iB1,   A1=0,   B1=B1,    κ=2B1  -b3a ,   κ= b2+2ib3B1β,

ω= - 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2.

Therfore, we conclude the following periodic and singular periodic solutions for the Eq. (1) respectively:

q8,1 x,t= iB1+B1tan2B1-b3ax+α+2ab2+2ib3B1βt12 ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ, (82)

and

q8,2 x,t= iB1+B1cot2B1-b3ax+α+2ab2+2ib3B1βt12 ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ. (83)

Set I-3:

A0= 0,   A1=A1,   B1=0,    κ=2A1  -b3a ,   κ= b2β,   ω=-b2(ab2+αβ)β2.

The periodic and singular periodic solutions for the Eq. (1) are presented as

q9,1 x,t=-A1sec2A1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ, (84)

And

q9,2 x,t=-A1csc2A1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ. (85)

Set I-4:

A0=12A1,    A1=A1,    B1=0,    κ=2A1-b3 a,    κ=b2+2b3A1β,

ω=- 22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32β2.

The periodic and singular periodic solutions for the Eq. (1) are shown as

q10,1 x,t=12 A1-A1sec2A1-b3ax+α+2ab2+2b3A1βt12 ei-b2+2b3A1β x- 22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32β2t+θ, (86)

and

q10,2 x,t=12 A1-A1csc2A1-b3ax+α+2ab2+2b3A1βt12 ei-b2+2b3A1β x- 22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32β2t+θ, (87)

Set I-5:

A0= 0,   A1=± B1,   B1=B1,    κ=4B1  -b3a ,   κ= b2β,   ω=-b2ab2+αββ2.

Therefore, we conclude the following combined periodic-singular and singular periodic solutions for the Eq. (1) respectively:

q11,1 x,t= B1tan2A1-b3ax+α+2ab2βtsec2A1-b3ax+α+2ab2βt12× ei-b2β x- b2(ab2+αβ)β2t+θ, (88)

and

q11,2 x,t= B1cot2A1-b3ax+α+2ab2βtcsc2A1-b3ax+α+2ab2βt12× ei-b2β x- b2(ab2+αβ)β2t+θ. (89)

Set I-6:

A0=iB1,   A1=±B1,  B1=B1,   κ=4B1-b3a,   κ=b2+2ib3B1β,

ω=-4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2.

Thus, we deduce the following combined periodic-singular and singular periodic solutions for the Eq. (1) respectively:

q11,1 x,t= iA1+B1tan2A1-b3ax+α+2ab2+2ib3B1βtB1sec2A1-b3ax+α+2ab2+2ib3B1βt12× ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ, (90)

and

q11,2 x,t= iA1+B1cot2A1-b3ax+α+2ab2+2ib3B1βtB1csc2A1-b3ax+α+2ab2+2ib3B1βt12× ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ, (91)

3.1.3. For Case-II: Eq. (48)

Base on the Eqs. (60)-(61), the solution of Eq. (11) can be presented as

V ξ= B1 tanξ ± A1secξ + Α0, (92)

and

V ξ= B1 cotξ ± A1cscξ + Α0, (93)

and so

V ω= B1 sinhω+ A1coshω + Α0, (94)

where A 1 0 or B 1 0.

Plugging (94) and its derivatives into Eq. (11), we can gain the following results by solving the nonlinear algebra system with the help of symbolic computation package as:

Set I-1:

A0=0,    A1=0,   B1=B1,    κ=2B1-b3a,    κ=b2β,    ω=-b2(ab2+αβ)β2.

The periodic and singular periodic solutions for the Eq. (1) are shown as

q7,1 x,t=B1tan2B1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ, (95)

and

q7,2 x,t=B1cot2B1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ, (96)

Set I-2:

A0=iB1,   A1=0,   B1=B1,   κ=2B1-b3a,    κ=b2+2ib3B1β,

ω=-4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2.

Therfore, we conclude the following periodic and singular periodic solutions for the Eq. (1) respectively:

q8,1 x,t=iB1+B1tan2B1-b3ax+α+2ab2+2ib3B1βt12× ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ, (97)

and

q8,2 x,t=iB1+B1cot2B1-b3ax+α+2ab2+2ib3B1βt12× ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ. (98)

Set I-3:

A0=0,   A1=A1,   B1=0,   κ=2A1-b3a,    κ=b2β,   ω=- b2(ab2+αβ)β2.

The periodic and singular periodic solutions for the Eq. (1) are presented as

q9,1 x,t=-A1sec2A1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ, (99)

And

q9,2 x,t=-A1csc2A1-b3ax+α+2ab2βt12 ei-b2β x- b2(ab2+αβ)β2t+θ. (100)

Set I-4:

A0=12A1,   A1=A1,   B1=0,   κ=2A1-b3a,    κ=b2+2b3A1β

ω=- 22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32β2.

The periodic and singular periodic solutions for the Eq. (1) are shown as

q10,1x,t=12A1-A1sec2A1-b3ax+α+2ab2+2b3A1βt12×ei-b2+2b3A1βx-22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32βt+θ, (101)

and

q10,2x,t=12A1-A1csc2A1-b3ax+α+2ab2+2b3A1βt12×ei-b2+2b3A1βx-22ab3A1b2+ab22+2αβb3A1+αb2β+2A12ab32βt+θ, (102)

Set I-5:

A0=0,    A1=±B1,    B1=B1,    κ=4B1-b3a,   κ=b2β,    ω=-b2(ab2+αβ)β2.

Therefore, we conclude the following combined periodic-singular and singular periodic solutions for the Eq. (1) respectively:

q11,1 x,t=B1tan2A1-b3ax+α+2ab2βt sec2A1-b3ax+α+2ab2βt12 ×ei-b2β x- b2(ab2+αβ)β2t+θ, (103)

and

q11,2 x,t=B1cot2A1-b3ax+α+2ab2βt csc2A1-b3ax+α+2ab2βt12 ×ei-b2β x- b2(ab2+αβ)β2t+θ. (104)

Set I-6:

A0=iB1,    A1=±B1,   B1=B1,    κ=4B1-b3a,    κ=b2+2ib3B1β

ω=- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2.

Thus, we deduce the following combined periodic-singular and singular periodic solutions for the Eq. (1) respectively:

q11,1 x,t=iA1+B1tan2A1-b3ax+α+2ab2+2ib3B1βt  B1sec2A1-b3ax+α+2ab2+2ib3B1βt12 ×ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ, (105)

and

q11,2 x,t=iA1+B1cot2A1-b3ax+α+2ab2+2ib3B1βt  B1csc2A1-b3ax+α+2ab2+2ib3B1βt12 ×ei-b2+2ib3B1β x- 4iab3B1b2+ab22+2iαβb3B1+αb2β-4B12ab32β2t+θ

4. Physical significance by graphical presentation of some of the obtained solutions

In this portion of the paper, we seek to draw attention on the physical conduct of some of the obtained solutions. We have examined the nature of the solutions by assuming appropriate values of involved unknown parameters and plotted these solutions graphically. Resultant solutions for the Eq. (1) consist of kink, singular, dark, combined dark-bright, kink-singular, and bell soliton solutions as well as periodic solutions. Solitons are a unique kind of solitary wave having a very important property of not losing its identity upon the interaction with various other solitons. Particle-like structures and extended structures such as magnetic monopoles, domain walls as well as cosmic strings, whose repercussions lie in the cosmology of the early universe, are exhibited by soliton solutions. The obtained solutions including the bright soliton (65), singular soliton (66), dark soliton (69), combined dark-bright soliton (73), periodic wave solutions (95) and (99), and periodic-singular wave solution (103). Graphical representations of the accomplished solutions are as follows Figs. (1-7):

Figure 1 Graph of (65) by taking B 1 = 1, a = 1, b 2 = 0.5, b 3 = -0.6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0.1), blue (t = 0.5) and green (t = 1)). 

Figure 2 Graph of (66) by taking B 1 = 1, a = 1, b 2 = 0:5, b 3 = -0:6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0:1), blue (t = 0:5) and green (t = 1)). 

Figure 3 Graph of (69) by taking A 1 = 1, a = 1, b 2 = 0.5, b 3 = -0.6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0.1), blue (t = 0.5) and green (t = 1)) 

Figure 4 Graph of (73) by taking A 1 = 1, a = 1, b 2 = 0.5, b 3 = -0.6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0.1), blue (t = 1.5) and green (t = 5)). 

Figure 5 Graph of (95) by taking A 1 = 1, a = 1, b 2 = 0.5, b 3 = -0.6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0.1), blue (t = 0.5) and green (t = 1)). 

Figure 6 Graph of (99) by taking A 1 = 1, a = 1, b 2 = 0.5, b 3 = -0.6, α = 2, β = 3, θ = 1 (a) 3D plot and (b)) 2D plot (red (t = 0.1), blue (t = 0.5) and green (t = 1)). 

Figure 7 Graph of (103) by taking B 1 = 1, a = 1, b 2 = 0:5, b 3 = -0:6, α = 2, β = 3, θ = 2 (a) 3D plot and (b)) 2D plot (red (t = 0:1), blue (t = 0:5) and green (t = 1)). 

Remark 4.1

Foroutan et al. [4] employed the generalized G'/G-expansion method to obtain the solitary wave solutions of the NLSE with dual power law nonlinearity. On the other hand, authors of [5] have used improved tan (𝜙(𝜉)/2)-expansion method with the aim of exploring new solutions of Eq. (1). The two methods mentioned above assume different forms of solutions which contain functions of ξ. These functions of η in exp (-Ω(η)) satisfies an ODE with hyperbolic, trigonometric, and rational functions. On the other hand, the ODE used in improved tan(𝜙(𝜉)/2) -expansion method and the generalized G ' /G-expansion method contain polynomial functions and all the cases of solutions of the ODE have not been explored in [4, 5]. Some of the obtained complex exponential solutions are in harmony with the already existing solutions [5]. The number of arbitrary constants in the solutions mentioned in [5] is less than those obtained in this paper. The extra arbitrary parameters might have potential applications in various fields of science which would make our solutions better than the already published ones. Thus, it can be stated that solutions obtained in this manuscript bestow valuable supplements to the existing literature.

5. Conclusion

In this work, we have investigated the nonlinear Schrödinger equation with anti-cubic nonlinearity by employing the improved exp(-Ω(η))-expansion method and extended sinh-Gordon equation expansion method. These methods have been lately augmented in order to explore more and more exact solutions for various NLPDEs. The richness of these obtained solutions lies in the existence of abundant arbitrary constants, having local structures, whose numerous applications prevail in various fields namely; signal propagation in optical fibers, wave propagation, IMD, SS as well as ND etc. It is worthy to note that several already published solutions of optical metamaterials, with anti-cubic nonlinearity, can be recovered by taking appropriate values of the involved arbitrary constants. Various explicit physical structures such as solitary wave, kink, and bell-shaped wave solutions are graphically represented distinctly which reveal the competence and high applicability of the obtained solutions. These solutions may be helpful in validating several numerical schemes and their accuracy.

Acknowledgments

The authors would like to express their sincere thanks to the Reviewer for his valuable comments and suggestions of the original manuscript.

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Received: March 25, 2019; Accepted: June 11, 2019

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