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

Open Access Published by De Gruyter Open Access November 21, 2020

New optical solitons of conformable resonant nonlinear Schrödinger’s equation

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

From the journal Open Physics

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

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.

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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.

New optical solitons of conformable resonant nonlinear Schrödinger’s equation

Abstract

1 Introduction

2 Analysis of the method

3 Exact solutions to the conformable resonant NLSE

4 Graphical representations

5 Conclusions and outlook

Acknowledgements

References

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