New optical solitons of conformable resonant nonlinear Schrödinger’s equation

Hadi Rezazadeh; Reza Abazari; Mostafa M. A. Khater; Mustafa Inc; Dumitru Baleanu

doi:10.1515/phys-2020-0137

Article Open Access

New optical solitons of conformable resonant nonlinear Schrödinger’s equation

Hadi Rezazadeh , Reza Abazari , Mostafa M. A. Khater , Mustafa Inc and Dumitru Baleanu

Published/Copyright: November 21, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Open Physics Volume 18 Issue 1

Abstract

Sardar subequation approach, which is one of the strong methods for solving nonlinear evolution equations, is applied to conformable resonant Schrödinger’s equation. In this technique, if we choose the special values of parameters, then we can acquire the travelling wave solutions. We conclude that these solutions are the solutions obtained by the first integral method, the trial equation method, and the functional variable method. Several new traveling wave solutions are obtained including generalized hyperbolic and trigonometric functions. The new derivation is of conformable derivation introduced by Atangana recently. Solutions are illustrated with some figures.

Keywords: Sardar subequation method; conformable derivative; resonant Schrödinger’s equation

1 Introduction

From mathematical point of view, nonlinear phenomena are one of the most important subjects of investigation that arise in various branches of sciences such as physics of solid-state and plasma, chemical kinematics, optical fibers, fluid mechanics, and biology. Usually, the mathematical modelling of nonlinear phenomena leads to nonlinear evolution equations (NLEEs). Obtaining the solutions of NLEEs can be a great help in studying the behavior of these phenomena. Among the possible solutions to NLEEs, specific form of solutions may be contingent only on a single combination of variables such as traveling wave solutions. Traveling wave solution is a special wave solution that translates in a particular direction with the addition of retaining a fixed shape. Exact traveling wave solutions of NLEEs have a very substantial contribution in physical models and have progressively been very important tools.

In the literature, there are several methods for acquiring traveling solutions to the NLEEs, such as first integral [1,2], exp(−φ(χ))-expansion [3,4], Jacobi elliptic functions [5,6], modified Khater [7,8], generalized Kudryashov [9,10], modified auxiliary equation [11,12], new extended direct algebraic method [13,14]], functional variable [15,16], sub-equation [17,18], (G′/G)-expansion [19,20] and others [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].

Nonlinear Schrödinger equations (NLSEs) are important physical models illustrating the dynamics of optical soliton promulgation in optical fibers. Lu et al. implemented two recent methods, namely, the generalized Kudryashov method and the generalized Riccati equation mapping method, for the higher-order NLSE [10]. Rezazadeh et al. applied a new extended direct algebraic method for solving the nonlinear conformable fractional Schrödinger–Hirota equation [14]. The space–time fractional perturbed NLSEs under the Kerr law nonlinearity are studied via the extended sinh-Gordon expansion method by Sulaiman et al. [25]. For a deeper discussion about NLSEs and employed methods for their solutions, we refer the reader to refs. [10,14,25] and references therein. In this study, we first describe briefly the mathematical concept of the Sardar subequation method, then we use it to investigate the traveling wave solutions of resonant NLSEs having conformable derivative of order υ Î ( 0 , 1 ) . In particular, we consider this equation in optical fibers with dual-power law nonlinearity given by [39,40]:

(1) i u t ( v ) + τ u x x + ( θ | u | + γ | u | 2 ) u + μ | u | x x | u | u = 0 , 0 < v ≤ 1 ,

where u t ( v ) is the conformable derivative operator of order v ∈ ( 0 , 1 ) , defined by the following:

(2) u t ( v ) ( x , t ) = lim h → 0 u x , t + h t + 1 Γ ( v ) 1 − v − u ( x , t ) h .

Atangana presented this derivative operator [41] and later showed that it satisfies multiplication and chain rules in refs. [42,43]. Yépez-Martínez et al. [44] introduced a new traveling wave for solving conformable equations. Later on, many effective methods were proposed, see, e.g., [45,46,47,48,49,50,51,52,53,54,55,56,57] for obtaining the exact traveling wave solutions of various nonlinear conformable evolution equations.

The rest of the article is organized as follows: Section 2 deals with description of the Sardar subequation method. In Section 3, several types of new traveling wave solutions of the conformable resonant NLSE using the conformable derivative are obtained. In Section 4, the graphical representations of obtained solutions are discussed. In Section 5, the conclusion part is given.

2 Analysis of the method

This section includes a brief description of the Sardar subequation method, which was first formulated by Inc et al. [58].

Let us assume that the NLEE for q ( x , t ) is written as:

(3) F ( q , q t , q x , q t t , … ) = 0 ,

where F is a polynomial. Using the new transformation, q ( x , t ) = q ( η ) , η = x − λ v t + 1 Γ ( v ) v , where λ is the wave speed, we can rewrite equation (3) as a nonlinear ordinary differential equation:

(4) G ( q , q η , q η η , q η η η , … ) = 0 .

We assume that equation (4) has the formal solution

(5) q ( η ) = ∑ j = 0 Ξ Ω i ℘ j ( η ) , Ω Ξ ≠ 0 ,

where Ξ is a positive integer, in most cases, which will be determined, and Ω j are arbitrary constants which are determined such that ℘ ( η ) be solution of the equation:

(6) ℘ ′ ( η ) 2 = ρ + a ℘ 2 ( η ) + b ℘ 4 ( η ) ,

where a , b and ρ are real constants. We know that (6) admits the solutions:

Case I: If a > 0 and ρ = 0 , then

℘ 1 ± ( η ) = ± − p q a b sech p q a η , ( b < 0 ) ,

℘ 2 ± ( η ) = ± p q a b csch p q a η , ( b > 0 ) ,

where

sech p q ( η ) = 2 p e η + q e − η , csch p q ( η ) = 2 p e η − q e − η .

Case II: If a < 0 , b > 0 and ρ = 0 , then

℘ 3 ± ( η ) = ± − p q a b sec p q − a η ,

℘ 4 ± ( η ) = ± − p q a b csc p q − a η ,

where

sec p q ( η ) = 2 p e i η + q e − i η , csc p q ( η ) = 2 i p e i η − q e − i η .

Case III: If a < 0 , b > 0 and ρ = a 2 4 b , then

℘ 5 ± ( η ) = ± − a 2 b tanh p q − a 2 η ,

℘ 6 ± ( η ) = ± − a 2 b coth p q − a 2 η ,

℘ 7 ± ( η ) = ± − a 2 b tanh p q − 2 a η ± i p q sech p q − 2 a η ,

℘ 8 ± ( η ) = ± − a 2 b coth p q − 2 a η ± p q csch p q − 2 a η ,

℘ 9 ± ( η ) = ± − a 8 b tanh p q − a 8 η + coth p q − a 8 η ,

where

tanh p q ( η ) = p e η − q e − η p e η + q e − η , coth p q ( η ) = p e η + q e − η p e η − q e − η .

Case IV: If a > 0 , b > 0 and ρ = a 2 4 b , then

℘ 10 ± ( η ) = ± a 2 b tan p q a 2 η ,

℘ 11 ± ( η ) = ± a 2 b cot p q a 2 η ,

℘ 12 ± ( η ) = ± a 2 b tan p q 2 a η ± p q sec p q 2 a η ,

℘ 13 ± ( η ) = ± a 2 b cot p q 2 a η ± p q csc p q 2 a η ,

℘ 14 ± ( η ) = ± a 8 b tan p q a 8 η + cot p q a 8 η ,

where

tan p q ( η ) = − i p e i η − q e − i η p e i η + q e − i η , cot p q ( η ) = i p e i η + q e − i η p e i η − q e − i η .

Substituting equation (5) into equation (4) and using equation (6) and collecting all terms with the same order of ℘ i ( η ) together and equating each coefficient of the resulting polynomial to zero yield a set of algebraic equations for a , b , λ , Ω i ( i = 0 , 1 , 2 , … , Ξ ) , which can be solved by Maple to determine values of constants a , b , λ , Ω i ( i = 0 , 1 , 2 , … , Ξ ) . Then, substituting these constants and the known solutions of equation (6) into equation (5), we obtain the exact solutions of NLEE (3).

3 Exact solutions to the conformable resonant NLSE

Since u = u ( x , t ) in equation (1) is a complex function, we suppose that

(7) u ( x , t ) = U ( η ) e i − κ x + ω v t + 1 Γ ( v ) v + η 0 ,

(8) η = x − λ v t + 1 Γ ( v ) v ,

where κ represents the soliton frequency and ω is the soliton wave number, while η 0 represents the phase constant. Therefore, substituting this hypothesis into (1) and decomposing into real and imaginary parts yield the following two equations:

(9) λ + 2 κ τ = 0 ,

(10) ( τ + μ ) U ″ − ( ω + κ 2 τ ) U + θ U 2 + γ U 3 = 0 ,

where the prime denotes the derivation with respect to η .

Balancing the terms of U 3 and U ″ in equation (10) gives Ξ = 1 . Hence, from equation (10), we obtain

(11) U ( η ) = Ω 0 + Ω 1 ℘ ( η ) , Ω 1 ≠ 0 .

Substituting equation (11) into equation (10) and collecting all terms with the same order of ℘ ( η ) together and setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for a , b , ω , Ω 0 and Ω 1 :

(12) − ω Ω 0 − κ 2 τ Ω 0 + δ Ω 0 3 + θ Ω 0 2 = 0 , − ω Ω 1 + 2 θ Ω 0 Ω 1 + Ω 1 μ a − κ 2 τ Ω 1 + Ω 1 τ a + 3 δ Ω 0 2 Ω 1 = 0 , 3 δ Ω 0 Ω 1 2 + θ Ω 1 2 = 0 , 2 Ω 1 μ b + 2 Ω 1 τ b + δ Ω 1 3 = 0 .

Solving the above set of algebraic equations, by Maple, we acquire the following result:

(13) Ω 0 = − 1 3 θ δ , Ω 1 = ± − 2 b ( τ + μ ) δ , a = 1 9 θ 2 δ ( τ + μ ) , ω = − 1 9 9 κ 2 τ δ + 2 θ 2 δ .

Therefore, the solutions of equation (1) are as follows.

Case I: If θ 2 δ ( τ + μ ) > 0 and ρ = 0 , then

u 1 ± ( x , t ) = − 1 3 θ δ 1 ± − 2 p q sech p q 1 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 2 ± ( x , t ) = − 1 3 θ δ 1 ± − 2 p q csch p q 1 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 .

Case II: If θ 2 δ ( τ + μ ) < 0 and ρ = 0 , then

u 3 ± ( x , t ) = − 1 3 θ δ 1 ± 2 p q sec p q − 1 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 4 ± ( x , t ) = − 1 3 θ δ 1 ± 2 p q csc p q − 1 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 .

Case III: If θ 2 δ ( τ + μ ) < 0 , b > 0 and ρ = 1 4 b ( θ 2 δ ( τ + μ ) ) 2 , then

u 5 ± ( x , t ) = − 1 3 θ δ 1 ± tanh p q − 1 18 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 6 ± ( x , t ) = − 1 3 θ δ 1 ± coth p q − θ 2 18 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 7 ± ( x , t ) = − 1 3 θ δ 1 ± tanh p q − 2 9 θ 2 δ x + 2 κ τ v t + 1 Γ ( v ) v ± i p q sech p q − 2 9 θ 2 δ ( τ + μ ) x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 ,

u 8 ± ( x , t ) = − 1 3 θ δ 1 ± coth p q − 2 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v ± p q csch p q − 2 9 θ 2 δ ( τ + μ ) x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 ,

u 9 ± ( x , t = − 1 3 θ δ 1 ± 1 2 tanh p q × − 1 72 θ 2 δ ( τ + μ ) x + 2 κ τ v t + 1 Γ ( v ) v + coth p q − 1 72 θ 2 δ ( τ + μ ) x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 .

Case IV: If θ 2 δ ( τ + μ ) > 0 , b > 0 and ρ = 1 4 b ( θ 2 δ ( τ + μ ) ) 2 , then

u 10 ± ( x , t ) = − 1 3 θ δ 1 ± i tan p q 1 18 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 11 ± ( x , t ) = − 1 3 θ δ 1 ± i cot p q 1 18 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v × t + 1 Γ ( v ) v + η 0 ,

u 12 ± ( x , t ) = − 1 3 θ δ 1 ± i tan p q 2 9 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v ± p q sec p q 2 9 θ 2 δ ( τ + μ ) x + 2 κ τ v t v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 ,

u 13 ± ( x , t ) = − 1 3 θ δ 1 ± i cot p q 2 9 θ 2 δ ( τ + μ ) x + 2 κ τ v t + 1 Γ ( v ) v ± p q csc p q 2 9 θ 2 δ ( τ + μ ) x + 2 κ τ v t v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 ,

u 14 ± ( x , t ) = − 1 3 θ δ 1 ± i 2 tan p q 1 72 θ 2 δ ( τ + μ ) × x + 2 κ τ v t + 1 Γ ( v ) v + cot p q 1 72 θ 2 δ ( τ + μ ) x + 2 κ τ v t v × exp i − κ x − 1 9 9 κ 2 τ δ + 2 θ 2 δ v t + 1 Γ ( v ) v + η 0 .

4 Graphical representations

The three-dimensional (3D) plots for the modulus, real and imaginary parts of the traveling wave solutions u 1 + ( x , t ) , u 4 − ( x , t ) and u 9 + ( x , t ) are displayed in Figures 1(a), 2(a) and 3(a), respectively. Figures 1(b), 2(b) and 3(b) also demonstrate the shape of contour plot indicated by the modulus, real and imaginary parts of the traveling wave solutions u 1 + ( x , t ) , u 4 − ( x , t ) and u 9 + ( x , t ) . Furthermore, the two-dimensional (2D) line plots of the modulus, real and imaginary parts of the traveling wave solutions u 1 + ( x , t ) , u 4 − ( x , t ) and u 9 + ( x , t ) are presented in Figures 1(c), 2(c) and 3(c) with t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 .

$Figure 1 (a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + {u}_{1}^{+} , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + {u}_{1}^{+} and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + {u}_{1}^{+} at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , t=0,t=0.2,t=0.4,t=0.6,t=0.8,t=1, respectively, when μ = 1.5 , τ = 1 , θ = 1.5 , κ = 0.75 , \mu =1.5,\tau =1,\theta =1.5,\kappa =0.75, δ = 1.2 , p = 1.2 , q = 1.1 , η 0 = 1 \delta =1.2,p=1.2,q=1.1,{\eta }_{0}=1 and v = 1 . v=1.$

Figure 1

(a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 1 + at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , respectively, when μ = 1.5 , τ = 1 , θ = 1.5 , κ = 0.75 , δ = 1.2 , p = 1.2 , q = 1.1 , η 0 = 1 and v = 1 .

$Figure 2 (a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 − {u}_{4}^{-} , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 + {u}_{4}^{+} and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 + {u}_{4}^{+} at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , t=0,t=0.2,t=0.4,t=0.6,t=0.8,t=1, respectively, when μ = − 1.25 , τ = 2 , θ = 1.5 , κ = 1.5 , \mu =-1.25,\tau =2,\theta =1.5,\kappa =1.5, δ = 2 , p = 0.96 , q = 0.95 , η 0 = 0 \delta =2,p=0.96,q=0.95,{\eta }_{0}=0 and v = 0.95 . v=0.95.$

Figure 2

(a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 − , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 + and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 4 + at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , respectively, when μ = − 1.25 , τ = 2 , θ = 1.5 , κ = 1.5 , δ = 2 , p = 0.96 , q = 0.95 , η 0 = 0 and v = 0.95 .

$Figure 3 (a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 9 + {u}_{9}^{+} and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 9 + {u}_{9}^{+} at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , t=0,t=0.2,t=0.4,t=0.6,t=0.8,t=1, respectively, when μ = 0.75 , τ = 1 , θ = 1 , κ = 1 , δ = − 1 , \mu =0.75,\tau =1,\theta =1,\kappa =1,\delta =-1, p = 1 , q = 1 , η 0 = 1.2 p=1,q=1,{\eta }_{0}=1.2 and v = 0.9 . v=0.9.$

Figure 3

(a) 3D-plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of , (b) the contour plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 9 + and (c) 2D-polar plot of the modulus (left), real (middle) and imaginary (right) parts of the travelling wave solution of u 9 + at t = 0 , t = 0.2 , t = 0.4 , t = 0.6 , t = 0.8 , t = 1 , respectively, when μ = 0.75 , τ = 1 , θ = 1 , κ = 1 , δ = − 1 , p = 1 , q = 1 , η 0 = 1.2 and v = 0.9 .

5 Conclusions and outlook

In this paper, Sardar subequation approach, which is one of the strong methods for solving NLEEs, is applied to conformable resonant Schrödinger’s equation. We showed that two types of new traveling wave solutions including the generalized hyperbolic and trigonometric functions for the conformable resonant NLEEs via the conformable derivatives were successfully found out by using the Sardar subequation method. In order to represent the resulting solutions some figures are plotted. Maple is used for mathematical computation results. Applicability and simplicity of the employed method in this paper show that many other conformable equations can be solved in a similar manner. We will report these results in future research studies.

Acknowledgements

This research work was supported by a research grant from the Amol University of Special Modern Technologies, Amol, Iran.

References

[1] Raza N, Arshed S, Sial S. Optical solitons for coupled Fokas–Lenells equation in birefringence fibers. Mod Phys Lett B. 2019;33(26):1950317.10.1142/S0217984919503172Search in Google Scholar

[2] Arshed S, Raza N. Optical solitons perturbation of Fokas–Lenells equation with full nonlinearity and dual dispersion. Chin J Phys. 2020;63:314–24.10.1016/j.cjph.2019.12.004Search in Google Scholar

[3] Raza N. New optical solitons in nonlinear negative-index materials with Bohm potential. Indian J Phys. 2019;93(5):657–63.10.1007/s12648-018-1234-0Search in Google Scholar

[4] Raza N, Afzal U, Butt AR, Rezazadeh H. Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities. Optical Quant Electron. 2019;51(4):107.10.1007/s11082-019-1813-0Search in Google Scholar

[5] Zubair A, Raza N. Bright and dark solitons in (n + 1)-dimensions with spatio-temporal dispersion. J Opt. 2019;48(4):594–605.10.1007/s12596-019-00572-8Search in Google Scholar

[6] Raza N, Zubair A. Bright, dark and dark-singular soliton solutions of nonlinear Schrödinger’s equation with spatio-temporal dispersion. J Mod Opt. 2018;65(17):1975–82.10.1080/09500340.2018.1480066Search in Google Scholar

[7] Khater MM, Park C, Lu D, Attia RA. Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation. Adv Differ Equ. 2020;2020(1):1–12.10.1186/s13662-019-2475-8Search in Google Scholar

[8] Khater MM, Attia RA, Lu D. Computational and numerical simulations for the nonlinear fractional Kolmogorov–Petrovskii–Piskunov (FKPP) equation. Phys Scr. 2020;95(5):055213.10.1088/1402-4896/ab76f8Search in Google Scholar

[9] Qian L, Attia RA, Qiu Y, Lu D, Khater MM. The shock peakon wave solutions of the general Degasperis–Procesi equation. Int J Mod Phys B. 2019;33(29):1950351.10.1142/S021797921950351XSearch in Google Scholar

[10] Lu D, Seadawy AR, Khater MM. Structures of exact and solitary optical solutions for the higher-order nonlinear Schrödinger equation and its applications in mono-mode optical fibers. Mod Phys Lett B. 2019;33(23):1950279.10.1142/S0217984919502798Search in Google Scholar

[11] Alderremy AA, Attia RA, Alzaidi JF, Lu D, Khater M. Analytical and semi-analytical wave solutions for longitudinal wave equation via modified auxiliary equation method and Adomian decomposition method. Therm Sci. 2019;23(6):1943–57.10.2298/TSCI190221355ASearch in Google Scholar

[12] Rezazadeh H, Korkmaz A, Eslami M, Mirhosseini-Alizamini SM. A large family of optical solutions to Kundu–Eckhaus model by a new auxiliary equation method. Optical Quant Electron. 2019;51(3):84.10.1007/s11082-019-1801-4Search in Google Scholar

[13] Tozar A, Kurt A, Tasbozan O. New wave solutions of time fractional integrable dispersive wave equation arising in ocean engineering models. Kuwait J Sci. 2020;47(2):22–33.Search in Google Scholar

[14] Rezazadeh H, Mirhosseini-Alizamini SM, Eslami M, Rezazadeh M, Mirzazadeh M, Abbagari S. New optical solitons of nonlinear conformable fractional Schrödinger–Hirota equation. Optik. 2018;172:545–53.10.1016/j.ijleo.2018.06.111Search in Google Scholar

[15] Çenesiz Y, Tasbozan O, Kurt A. Functional variable method for conformable fractional modified KdV–ZK equation and Maccari system. Tbilisi Math J. 2017;10(1):117–25.10.1515/tmj-2017-0010Search in Google Scholar

[16] Eslami M, Rezazadeh H, Rezazadeh M, Mosavi SS. Exact solutions to the space-time fractional Schrödinger–Hirota equation and the space-time modified KDV–Zakharov–Kuznetsov equation. Optical Quant Electron. 2017;49(8):279.10.1007/s11082-017-1112-6Search in Google Scholar

[17] Kurt A. New analytical and numerical results for fractional Bogoyavlensky–Konopelchenko equation arising in fluid dynamics. Appl Mathematics-A J Chin Universities. 2020;35(1):101–12.10.1007/s11766-020-3808-9Search in Google Scholar

[18] Atilgan E, Senol M, Kurt A, Tasbozan O. New wave solutions of time-fractional coupled Boussinesq–Whitham–Broer–Kaup equation as a model of water waves. China Ocean Eng. 2019;33(4):477–83.10.1007/s13344-019-0045-1Search in Google Scholar

[19] Jamshidzadeh S, Abazari R. Solitary wave solutions of three special types of Boussinesq equations. Nonlinear Dyn. 2017;88(4):2797–805.10.1007/s11071-017-3412-6Search in Google Scholar

[20] Abazari R, Jamshidzadeh S. Exact solitary wave solutions of the complex Klein–Gordon equation. Optik. 2015;126(19):1970–5.10.1016/j.ijleo.2015.05.056Search in Google Scholar

[21] Kurt A. New periodic wave solutions of a time fractional integrable shallow water equation. Appl Ocean Res. 2019;85:128–35.10.1016/j.apor.2019.01.029Search in Google Scholar

[22] Bulut H, Sulaiman TA, Baskonus HM, Rezazadeh H, Eslami M, Mirzazadeh M. Optical solitons and other solutions to the conformable space-time fractional Fokas–Lenells equation. Optik. 2018;172:20–7.10.1016/j.ijleo.2018.06.108Search in Google Scholar

[23] Bulut H, Sulaiman TA, Baskonus HM. Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattice Microst. 2018;123:12–9.10.1016/j.spmi.2017.12.009Search in Google Scholar

[24] Sulaiman TA, Baskonus HM, Bulut H. Optical solitons and other solutions to the conformable space-time fractional complex Ginzburg–Landau equation under Kerr law nonlinearity. Pramana. 2018;91(4):58.10.1007/s12043-018-1635-9Search in Google Scholar

[25] Sulaiman TA, Bulut H, Baskonus HM. Optical solitons to the fractional perturbed NLSE in nano-fibers. Discret & Contin Dynl Syst-S. 2019;763–9.Search in Google Scholar

[26] Gao W, Rezazadeh H, Pinar Z, Baskonus HM, Sarwar S, Yel G. Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique. Optical Quant Electron. 2020;52(1):1–13.10.1007/s11082-019-2162-8Search in Google Scholar

[27] Raza N, Aslam MR, Rezazadeh H. Analytical study of resonant optical solitons with variable coefficients in Kerr and non-Kerr law media. Optical Quant Electron. 2019;51(2):59.10.1007/s11082-019-1773-4Search in Google Scholar

[28] Korkmaz A, Hepson OE, Hosseini K, Rezazadeh H, Eslami M. Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class. J King Saud Univ-Sci. 2020;32(1):567–74.10.1016/j.jksus.2018.08.013Search in Google Scholar

[29] Biswas A, Rezazadeh H, Mirzazadeh M, Eslami M, Zhou Q, Moshokoa SP, et al. Optical solitons having weak non-local nonlinearity by two integration schemes. Optik. 2018;164:380–4.10.1016/j.ijleo.2018.03.026Search in Google Scholar

[30] Rezazadeh H, Manafian J, Khodadad FS, Nazari F. Traveling wave solutions for density-dependent conformable fractional diffusion–reaction equation by the first integral method and the improved tan(1/2φ(ξ))-expansion method. Optical Quant Electron. 2018;50(3):121.10.1007/s11082-018-1388-1Search in Google Scholar

[31] Abazari R, Jamshidzadeh S, Biswas A. Solitary wave solutions of coupled Boussinesq equation. Complexity. 2016;21(S2):151–5.10.1002/cplx.21791Search in Google Scholar

[32] Abazari R. Application of extended tanh function method on KdV–Burgers equation with forcing term. Rom J Phys. 2014;59(1–2):3–11.10.1155/2014/948072Search in Google Scholar

[33] Akgül A. A novel method for a fractional derivative with non-local and non-singular kernel. Chaos Soliton Fract. 2018;114:478–82.10.1016/j.chaos.2018.07.032Search in Google Scholar

[34] Akgül A, Modanli M. Crank–Nicholson difference method and reproducing kernel function for third order fractional differential equations in the sense of Atangana–Baleanu Caputo derivative. Chaos Soliton Fract. 2019;127:10–6.10.1016/j.chaos.2019.06.011Search in Google Scholar

[35] Akgül EK. Solutions of the linear and nonlinear differential equations within the generalized fractional derivatives. Chaos: an interdisciplinary. J Nonlinear Sci. 2019;29(2):023108.Search in Google Scholar

[36] Baleanu D, Fernandez A, Akgül A. On a fractional operator combining proportional and classical differintegrals. Mathematics. 2020;8(3):360.10.3390/math8030360Search in Google Scholar

[37] Akgül A, Cordero A, Torregrosa JR. Solutions of fractional gas dynamics equation by a new technique. Math Methods Appl Sci. 2020;43(3):1349–58.10.1002/mma.5950Search in Google Scholar

[38] Akgül A. Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell–Eyring non-Newtonian fluid. J Taibah Univ Sci. 2019;13(1):858–63.10.1080/16583655.2019.1651988Search in Google Scholar

[39] Gao W, Yel G, Baskonus HM, Cattani C. Complex solitons in the conformable (2 + 1)-dimensional Ablowitz–Kaup–Newell–Segur equation. Aims Math. 2020;5(1):507–21.10.3934/math.2020034Search in Google Scholar

[40] Panda SK, Abdeljawad T, Ravichandran C. A complex valued approach to the solutions of Riemann–Liouville integral, Atangana–Baleanu integral operator and non-linear telegraph equation via fixed point method. Chaos Soliton Fract. 2020;130:109439.10.1016/j.chaos.2019.109439Search in Google Scholar

[41] Kumar D, Singh J, Baleanu D. Analysis of regularized long-wave equation associated with a new fractional operator with Mittag–Leffler type kernel. Phys A: Stat Mech its Appl. 2018;492:155–67.10.1016/j.physa.2017.10.002Search in Google Scholar

[42] Cattani C. On the existence of wavelet symmetries in archaea DNA. Comput Math Methods Med. 2012;2012:673934.10.1155/2012/673934Search in Google Scholar PubMed PubMed Central

[43] Yang XJ, Gao F. A new technology for solving diffusion and heat equations. Therm Sci. 2017;21(1 Part A):133–40.10.2298/TSCI160411246YSearch in Google Scholar

[44] Yang XJ, Baleanu D, Lazaveric MP, Cajic MS. Fractal boundary value problems for integral and differential equations with local fractional operators. Therm Sci. 2015;19(3):959–66.10.2298/TSCI130717103YSearch in Google Scholar

[45] Cattani C, Rushchitskii YY. Cubically nonlinear elastic waves: wave equations and methods of analysis. Int Appl Mech. 2003;39(10):1115–45.10.1023/B:INAM.0000010366.48158.48Search in Google Scholar

[46] Cattani C. A review on harmonic wavelets and their fractional extension. J Adv Eng Computation. 2018;2(4):224–38.10.25073/jaec.201824.225Search in Google Scholar

[47] Yokuş A, Gülbahar S. Numerical solutions with linearization techniques of the fractional Harry Dym equation. Appl Math Nonlinear Sci. 2019;4(1):35–42.10.2478/AMNS.2019.1.00004Search in Google Scholar

[48] Al-Ghafri KS, Rezazadeh H. Solitons and other solutions of (3 + 1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov equation. Appl Math Nonlinear Sci. 2019;4(2):289–304.10.2478/AMNS.2019.2.00026Search in Google Scholar

[49] Biswas A. Soliton solutions of the perturbed resonant nonlinear Schrodinger’s equation with full nonlinearity by semi-inverse variational principle. Quant Phys Lett. 2012;1(2):79–89.Search in Google Scholar

[50] Eslami M, Mirzazadeh M, Biswas A. Soliton solutions of the resonant nonlinear Schrödinger’s equation in optical fibers with time-dependent coefficients by simplest equation approach. J Mod Opt. 2013;60(19):1627–36.10.1080/09500340.2013.850777Search in Google Scholar

[51] Atangana A, Goufo D, Franc E. Extension of matched asymptotic method to fractional boundary layers problems. Math Probl Eng. 2014;2014:107535.10.1155/2014/107535Search in Google Scholar

[52] Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math. 2015;13:1.10.1515/math-2015-0081Search in Google Scholar

[53] Atangana A, Alqahtani RT. Modelling the spread of river blindness disease via the caputo fractional derivative and the beta-derivative. Entropy. 2016;18(2):40.10.3390/e18020040Search in Google Scholar

[54] Yépez-Martínez H, Gómez-Aguilar JF, Baleanu D. Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Optik. 2018;155:357–65.10.1016/j.ijleo.2017.10.104Search in Google Scholar

[55] Yépez-Martínez H, Gómez-Aguilar JF. Fractional sub-equation method for Hirota-Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana’s conformable derivative. Waves Random Complex Media. 2019;29(4):678–93.10.1080/17455030.2018.1464233Search in Google Scholar

[56] Ghanbari B, Gomez JF. The generalized exponential rational function method for Radhakrishnan–Kundu–Lakshmanan equation with β-conformable time derivative. Rev Mexicana de Física. 2019;65(5):503–18.10.31349/RevMexFis.65.503Search in Google Scholar

[57] Uddin MF, Hafez MG, Hammouch Z, Baleanu D. Periodic and rogue waves for Heisenberg models of ferromagnetic spin chains with fractional beta derivative evolution and obliqueness. Waves Random Complex Media. 2020;1–15; 10.1080/17455030.2020.1722331.Search in Google Scholar

[58] Inc M, Rezazadeh H, Baleanu D. New solitary wave solutions for variants of the (3 + 1)-Dimensional Wazwaz–Benjamin–Bona–Mahony equations. Front Phys. 2020; 10.3389/fphy.2020.00332 Search in Google Scholar

Received: 2020-04-20

Revised: 2020-08-29

Accepted: 2020-09-01

Published Online: 2020-11-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0137

Keywords for this article

Sardar subequation method; conformable derivative; resonant Schrödinger’s equation

Creative Commons

BY 4.0