A (3 + 1)-dimensional resonant nonlinear Schrödinger equation and its Jacobi elliptic and exponential function solutions

Journal Pre-proof A (3 + 1)-dimensional resonant nonlinear Schr¨odinger equation and its Jacobi elliptic and exponential function solutions K. Hosseini, M. Matinfar, M. Mirzazadeh

PII:

S0030-4026(20)30292-8

DOI:

https://doi.org/10.1016/j.ijleo.2020.164458

Reference:

IJLEO 164458

To appear in:

Optik

Received Date:

13 February 2020

Accepted Date:

21 February 2020

Please cite this article as: Hosseini K, Matinfar M, Mirzazadeh M, A (3 + 1)-dimensional resonant nonlinear Schr¨odinger equation and its Jacobi elliptic and exponential function solutions, Optik (2020), doi: https://doi.org/10.1016/j.ijleo.2020.164458

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier.

A (3+1)-dimensional resonant nonlinear Schrödinger equation and its Jacobi elliptic and exponential function solutions

K. Hosseini1*, M. Matinfar1, M. Mirzazadeh2*

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran 2 Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, P.C. 44891-63157 Rudsar-Vajargah, Iran

ro of

1

-p

*Corresponding Authors: [email protected], [email protected]

Abstract

re

The current study is concerned with a (3+1)-dimensional resonant nonlinear Schrödinger (3D-RNLS) equation with diverse applications in nonlinear optics and its exact solutions. In this regard, by

lP

adopting the new expansion methods based on the Jacobi elliptic equation, a number of exact solutions in terms of Jacobi elliptic and exponential functions to the 3D-RNLS equation with Kerr law nonlinearity are formally constructed. The outcomes of this paper undoubtedly demonstrate the

na

impressive capability of the new expansion methods to deal with nonlinear Schrödinger equations. Keywords: (3+1)-dimensional resonant nonlinear Schrödinger equation; Kerr law nonlinearity; New

ur

expansion methods; Jacobi elliptic and exponential function solutions

1. Introduction

It is known that nonlinear Schrödinger equations are of a key role in the vast areas of scientific

Jo

disciplines, particularly nonlinear optics. Although, the physical features of this class of nonlinear PDEs can be investigated by extracting and studying its exact solutions, searching such solutions is not easy. Today with the availability of symbolic computation packages, many effective methods [138] have been applied to overcome the difficulties in looking for exact solutions of nonlinear PDEs. In this article, we mainly focus on a (3+1)-dimensional resonant nonlinear Schrödinger equation with Kerr law nonlinearity as [1, 2] 𝑖𝑢𝑡 + 𝜂𝛻 2 𝑢 + 𝜎|𝑢|2 𝑢 + 𝛿 {

𝛻2 |𝑢| } |𝑢|

𝑢 = 0,

𝑢 = 𝑢(𝑥, 𝑦, 𝑧, 𝑡),

1

𝛻2 =

𝜕2 𝜕𝑥 2

+

𝜕2 𝜕𝑦 2

+

𝜕2 𝜕𝑧 2

,

(1)

and extract its exact solutions by adopting the new expansion methods based on the Jacobi elliptic equation [33-38]. In the above equation, the non-zero constants 𝜂, 𝜎, and 𝛿 are the coefficients of the group velocity, cubic nonlinearity, and resonant nonlinearity. Several scholars have dedicated their investigations for seeking exact solutions of the (3+1)-dimensional resonant nonlinear Schrödinger equation. For example, Sedeeg et al. [1] exerted the tanh method to retrieve generalized optical solitons of the 3D-RNLS equation and Ferdous et al. [2] obtained oblique resonant optical solitons of the 3D-RNLS equation by employing the generalized exp(−𝛷(𝜉))-expansion method. It is worth noting that the resonant nonlinear Schrödinger equations at lower dimensions have been investigated by a series of systematic methods; the interested reader is referred to see [3-15]. The organization of this paper is as follows: In Section 2, the reduced form of the 3D-RNLS equation with Kerr law

ro of

nonlinearity is derived. In Section 3, the exact solutions of the 3D-RNLS equation are constructed by considering the new expansion methods based upon the Jacobi elliptic equation. The concluding remarks are presented in the last section.

-p

2. The reduced form of the 3D-RNLS equation

In order to arrive at the reduced form of the 3D-RNLS equation with Kerr law nonlinearity, we apply

re

a complex transformation as follows

𝜖 = cos(𝛼)𝑥 + cos(𝛽)𝑦 + cos(𝛾)𝑧 + 𝑐𝑡.

lP

𝑢(𝑥, 𝑦, 𝑧, 𝑡) = 𝑈(𝜖)𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+𝜔𝑡) ,

(2)

By considering the above complex transformation (2) and Eq. (1), the reduced form of the 3D-RNLS equation with Kerr law nonlinearity can be written as

which

𝜗 = cos(𝛼)2 + cos(𝛽)2 + cos(𝛾)2 .

ur

𝑐 = −2𝜗𝜂𝑘,

(3)

na

𝜗(𝛿 + 𝜂)𝑈 ′′ (𝜖) − (𝜗𝜂𝑘 2 + 𝜔)𝑈(𝜖) + 𝜎𝑈 3 (𝜖) = 0,

3. Exact solutions of the 3D-RNLS equation with Kerr law nonlinearity

Jo

Exact solutions of the 3D-RNLS equation with Kerr law nonlinearity are established in this section by utilizing the new expansion methods based on the Jacobi elliptic equation. 3.1. Jacobi method

According to the new expansion method based upon the Jacobi elliptic equation and the terms 𝑈 ′′ (𝜖) and 𝑈(𝜖)3 , the solution of the 3D-RNLS equation with Kerr law nonlinearity can be written as 𝑈(𝜖) = 𝑎0 + 𝑎1

𝑓(𝜖) 1+𝑓(𝜖)2

+ 𝑎2

1−𝑓(𝜖)2 1+𝑓(𝜖)2

,

(4)

2

which 𝑎0 , 𝑎1 , and 𝑎2 are parameters to be determined and 𝑓(𝜖) is the Jacobi elliptic function satisfying the following ODE 𝑓 ′ (𝜖)2 = 𝐷 + 𝐸𝑓(𝜖)2 + 𝐹𝑓(𝜖)4 . By replacing the nontrivial solution (4) into Eq. (3) and exerting a series of calculations, we find −4𝜗𝐷𝛿𝑎2 − 4𝜗𝐷𝜂𝑎2 − 𝜗𝜂𝑘 2 𝑎0 − 𝜗𝜂𝑘 2 𝑎2 + 𝜎𝑎0 3 + 3𝜎𝑎0 2 𝑎2 + 3𝜎𝑎0 𝑎2 2 − 𝜔𝑎0 + 𝜎𝑎2 3 − 𝜔 𝑎2 = 0, −6𝜗𝐷𝛿𝑎1 − 6𝜗𝐷𝜂𝑎1 + 𝜗𝐸𝛿𝑎1 + 𝜗𝐸𝜂𝑎1 − 𝜗𝜂𝑘 2 𝑎1 + 3𝜎𝑎0 2 𝑎1 + 6𝜎𝑎0 𝑎1 𝑎2 + 3𝜎𝑎1 𝑎2 2 − 𝜔 𝑎1 = 0,

ro of

12𝜗𝐷𝛿𝑎2 + 12𝜗𝐷𝜂𝑎2 − 8𝜗𝐸𝛿𝑎2 − 8𝜗𝐸𝜂𝑎2 − 3𝜗𝜂𝑘 2 𝑎0 − 𝜗𝜂𝑘 2 𝑎2 + 3𝜎𝑎0 3 + 3𝜎𝑎0 2 𝑎2 + 3𝜎 𝑎0 𝑎1 2 − 3𝜎𝑎0 𝑎2 2 − 3𝜔𝑎0 + 3𝜎𝑎1 2 𝑎2 − 3𝜎𝑎2 3 − 𝜔𝑎2 = 0, 2𝜗𝐷𝛿𝑎1 + 2𝜗𝐷𝜂𝑎1 − 6𝜗𝐸𝛿𝑎1 − 6𝜗𝐸𝜂𝑎1 + 2𝜗𝐹𝛿𝑎1 + 2𝜗𝐹𝜂𝑎1 − 2𝜗𝜂𝑘 2 𝑎1 + 6𝜎𝑎0 2 𝑎1 + 𝜎 𝑎1 3 − 6𝜎𝑎1 𝑎2 2 − 2𝜔𝑎1 = 0,

8𝜗𝐸𝛿𝑎2 + 8𝜗𝐸𝜂𝑎2 − 12𝜗𝐹𝛿𝑎2 − 12𝜗𝐹𝜂𝑎2 − 3𝜗𝜂𝑘 2 𝑎0 + 𝜗𝜂𝑘 2 𝑎2 + 3𝜎𝑎0 3 − 3𝜎𝑎0 2 𝑎2 + 3𝜎 𝑎0 𝑎1 2 − 3𝜎𝑎0 𝑎2 2 − 3𝜔𝑎0 − 3𝜎𝑎1 2 𝑎2 + 3𝜎𝑎2 3 + 𝜔𝑎2 = 0,

-p

𝜗𝐸𝛿𝑎1 + 𝜗𝐸𝜂𝑎1 − 6𝜗𝐹𝛿𝑎1 − 6𝜗𝐹𝜂𝑎1 − 𝜗𝜂𝑘 2 𝑎1 + 3𝜎𝑎0 2 𝑎1 − 6𝜎𝑎0 𝑎1 𝑎2 + 3𝜎𝑎1 𝑎2 2 − 𝜔𝑎1 = 0,

re

4𝜗𝐹𝛿𝑎2 + 4𝜗𝐹𝜂𝑎2 − 𝜗𝜂𝑘 2 𝑎0 + 𝜗𝜂𝑘 2 𝑎2 + 𝜎𝑎0 3 − 3𝜎𝑎0 2 𝑎2 + 3𝜎𝑎0 𝑎2 2 − 𝜔𝑎0 − 𝜎𝑎2 3 + 𝜔 𝑎2 = 0.

lP

Now, the above nonlinear system is solved and the following results are obtained: Case 1. 𝑎0 = 0,

𝑎1 = ±4√−



𝑎0 = 0,

𝑎1 = 0,



𝑎0 = 0,

𝑎1 = 2√−

2𝜗(𝛿+𝜂) 𝜎

,

𝑎2 = 0,

na



𝑎2 = ±2√ 2𝜗(𝛿+𝜂)

ur

𝜎

,

2𝜗(𝛿+𝜂) 𝜎

𝜔 = −(8𝜗𝛿 + (𝜗𝑘 2 + 8𝜗)𝜂),

𝜔 = 4𝜗𝛿 + (4𝜗 − 𝜗𝑘 2 )𝜂,

,

2𝜗(𝛿+𝜂)

𝑎2 = ± √

𝜎

,

𝑚 = 1,

𝑚 = 1,

𝜔 = −(2𝜗𝛿 + (𝜗𝑘 2 + 2𝜗)𝜂),

𝑚 = 1,

when 𝐷 = 1, 𝐸 = −(𝑚2 + 1), and 𝐹 = 𝑚2 . Now, due to 𝑓(𝜖) = 𝑠𝑛(𝜖) and 𝑠𝑛(𝜖, 1) → tanh(𝜖), the

Jo

following exact solutions to the 3D-RNLS equation are constructed 𝑢1,2 (𝑥, 𝑦, 𝑧, 𝑡) = ±4 √−

2𝜗(𝛿+𝜂)

tanh(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝜎

1+tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)−(8𝜗𝛿+(𝜗𝑘

2 +8𝜗)𝜂)𝑡)

𝑢3,4 (𝑥, 𝑦, 𝑧, 𝑡) = ±2 √

2𝜗(𝛿+𝜂) 1−tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡) 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+(4𝜗𝛿+(4𝜗−𝜗𝑘 2 )𝜂)𝑡) 𝑒 , 𝜎 1+tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

3

,

𝑢5,6 (𝑥, 𝑦, 𝑧, 𝑡) = (2√− √

2𝜗(𝛿+𝜂)

tanh(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝜎

1+tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

±

2 2𝜗(𝛿+𝜂) 1−tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡) ) 𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)−(2𝜗𝛿+(𝜗𝑘 +2𝜗)𝜂)𝑡) . 𝜎 1+tanh2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

Case 2. 𝜗(𝛿+𝜂)

𝑎0 = ± √

2𝜎

,

𝑎1 = 0,

𝑎2 = ±

𝜗(𝛿+𝜂) 𝜗(𝛿+𝜂) 𝜎√ 2𝜎

,

1

𝜔 = 𝜗(−2𝜂𝑘 2 + 5𝛿 + 5𝜂), 2

1

𝑚= , 2

when 𝐷 = 1 − 𝑚2 , 𝐸 = 2𝑚2 − 1, and 𝐹 = −𝑚2 . Now, owing to 𝑓(𝜖) = 𝑐𝑛(𝜖), the following exact

𝜗(𝛿+𝜂)

𝑢7,8 (𝑥, 𝑦, 𝑧, 𝑡) = ±√

2𝜎

ro of

solutions to the 3D-RNLS equation are established ± 1

1 2 𝜗(𝛿+𝜂) 1−𝑐𝑛 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡,2) 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+(2𝜗(−2𝜂𝑘 2 +5𝛿+5𝜂))𝑡) 𝑒 . 1 𝜗(𝛿+𝜂) 1+𝑐𝑛2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡, ) 𝜎√ 2

2𝜎

𝑎0 = 0,

𝑎1 = ±4√−



𝑎0 = 0,

𝑎1 = 0,



𝑎0 = 0,

𝑎1 = 2√−

2𝜗(𝛿+𝜂) 𝜎

,

𝑎2 = ±2√ 𝜎

,

2𝜗(𝛿+𝜂) 𝜎

𝜔 = −(8𝜗𝛿 + (𝜗𝑘 2 + 8𝜗)𝜂), 𝜔 = 4𝜗𝛿 + (4𝜗 − 𝜗𝑘 2 )𝜂,

,

2𝜗(𝛿+𝜂)

𝑎2 = ± √

lP

2𝜗(𝛿+𝜂)

𝑎2 = 0,

re



-p

Case 3.

𝜎

,

𝑚 = 1,

𝑚 = 1,

𝜔 = −(2𝜗𝛿 + (𝜗𝑘 2 + 2𝜗)𝜂),

𝑚 = 1,

when 𝐷 = 𝑚2 , 𝐸 = −(𝑚2 + 1), and 𝐹 = 1. Now, due to 𝑓(𝜖) = 𝑛𝑠(𝜖) and 𝑛𝑠(𝜖, 1) → coth(𝜖), the

na

following exact solutions to the 3D-RNLS equation are constructed 𝑢9,10 (𝑥, 𝑦, 𝑧, 𝑡) = ±4 2𝜗(𝛿+𝜂)

coth(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝜎

1+coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

ur

√−

𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)−(8𝜗𝛿+(𝜗𝑘

2 +8𝜗)𝜂)𝑡)

Jo

𝑢11,12 (𝑥, 𝑦, 𝑧, 𝑡) = ±2 √

2𝜗(𝛿+𝜂) 1−coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡) 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+(4𝜗𝛿+(4𝜗−𝜗𝑘 2 )𝜂)𝑡) 𝑒 , 𝜎 1+coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝑢13,14 (𝑥, 𝑦, 𝑧, 𝑡) = (2√− √

2𝜗(𝛿+𝜂)

coth(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

𝜎

1+coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

±

2 2𝜗(𝛿+𝜂) 1−coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡) ) 𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)−(2𝜗𝛿+(𝜗𝑘 +2𝜗)𝜂)𝑡) . 𝜎 1+coth2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

Case 4. 4

,

𝜗(𝛿+𝜂)

𝑎0 = ± √

2𝜎

,

𝑎1 = 0,

𝑎2 = ∓

𝜗(𝛿+𝜂) 𝜗(𝛿+𝜂) 𝜎√ 2𝜎

,

1

𝜔 = 𝜗(−2𝜂𝑘 2 + 5𝛿 + 5𝜂), 2

1

𝑚= , 2

when 𝐷 = −𝑚2 , 𝐸 = 2𝑚2 − 1, and 𝐹 = 1 − 𝑚2 . Now, due to 𝑓(𝜖) = 𝑛𝑐(𝜖), the following exact solutions to the 3D-RNLS equation are established 𝜗(𝛿+𝜂)

𝑢15,16 (𝑥, 𝑦, 𝑧, 𝑡) = ±√

∓

2𝜎

1

1 2 𝜗(𝛿+𝜂) 1−𝑛𝑐 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡,2) 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+(2𝜗(−2𝜂𝑘 2 +5𝛿+5𝜂))𝑡) 𝑒 . 1 𝜗(𝛿+𝜂) 1+𝑛𝑐 2 (cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡, ) 𝜎√ 2

2𝜎

The three- and two-dimensional profiles of |𝑢2 (𝑥, 𝑦, 𝑧, 𝑡)| are plotted in Figure 1 by taking the

re

-p

ro of

parameters 𝜎 = 𝛿 = 1, cos(𝛽) = cos(𝛾) = 0.6, 𝑘 = 0.5, 𝜂 = −0.1, and 𝑥 = 𝑦 = 𝑡 = 0.

lP

Figure 1: The profiles of |𝑢2 (𝑥, 𝑦, 𝑧, 𝑡)| for 𝜎 = 𝛿 = 1, cos(𝛽) = cos(𝛾) = 0.6, 𝑘 = 0.5, 𝜂 = −0.1, and 𝑥 = 𝑦 = 𝑡 = 0.

na

3.2. Kudryashov method (New form)

Based upon the Kudryashov method and the balance number 𝑁 = 1, the solution of the 3D-RNLS equation with Kerr law nonlinearity can be written as 𝑎1 ≠ 0,

ur

𝑈(𝜖) = 𝑎0 + 𝑎1 𝐾(𝜖),

(5)

Jo

which 𝑎0 and 𝑎1 are parameters to be computed and 𝐾(𝜖) ia the solution of the following ODE 𝐾 ′ (𝜖)2 = 𝐾(𝜖)2 (1 − 𝜏𝐾(𝜖)2 ),

𝐾(𝜖) =

4𝐴 4𝐴2 𝑒 𝜖 +𝜏𝑒 −𝜖

,

𝜏 = 4𝐴𝐵.

Replacing the nontrivial solution (5) into Eq. (3) and utilizing a number of calculations yields −𝜗𝜂𝑘 2 𝑎0 + 𝜎𝑎0 3 − 𝜔𝑎0 = 0, 𝜗𝛿𝑎1 − 𝜗𝜂𝑘 2 𝑎1 + 𝜗𝜂𝑎1 + (3𝜎𝑎0 2 − 𝜔)𝑎1 = 0, 3𝜎𝑎0 𝑎1 2 = 0, 5

−2𝜗𝛿𝜏𝑎1 − 2𝜗𝜂𝜏𝑎1 + 𝜎𝑎1 3 = 0. By solving the above nonlinear system, the following results are derived 𝑎0 = 0,

𝑎1 = ±√

2𝜗𝜏(𝛿+𝜂) 𝜎

,

𝜔 = 𝜗(𝛿 − 𝜂𝑘 2 + 𝜂).

Now, the following exact solutions to the 3D-RNLS equation are constructed 𝑢1,2 (𝑥, 𝑦, 𝑧, 𝑡) = ±√

2𝜗𝜏(𝛿+𝜂)

4𝐴

𝜎

4𝐴2 𝑒 cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡 +4𝐴𝐵𝑒 −(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧−2𝜗𝜂𝑘𝑡)

× 𝑒 𝑖(𝑘(cos(𝛼)𝑥+cos(𝛽)𝑦+cos(𝛾)𝑧)+𝜗(𝛿−𝜂𝑘

2 +𝜂)𝑡)

.

ro of

The three- and two-dimensional profiles of |𝑢1 (𝑥, 𝑦, 𝑧, 𝑡)| are depicted in Figure 2 by considering the

lP

re

-p

parameters 𝐴 = 𝐵 = 𝜎 = 𝛿 = 1, cos(𝛽) = cos(𝛾) = 0.6, 𝑘 = 0.5, 𝜂 = −0.1, and 𝑥 = 𝑦 = 𝑡 = 0.

na

Figure 2: The profiles of |𝑢1 (𝑥, 𝑦, 𝑧, 𝑡)| for 𝐴 = 𝐵 = 𝜎 = 𝛿 = 1, cos(𝛽) = cos(𝛾) = 0.6, 𝑘 = 0.5,

ur

𝜂 = −0.1, and 𝑥 = 𝑦 = 𝑡 = 0.

Jo

4. Conclusion

Our concern in the present work was to retrieve exact solutions of a nonlinear model with diverse applications in nonlinear optics namely ‘’the (3+1)-dimensional resonant nonlinear Schrödinger equation’’. In this regard, a wide variety of exact solutions in terms of Jacobi elliptic and exponential functions to the 3D-RNLS equation with Kerr law nonlinearity were successfully extracted by exerting the new expansion techniques based upon the Jacobi elliptic equation. The findings presented herein undeniably reveal the impressive capability of the new expansion methods to handle nonlinear Schrödinger equations.

6

No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication.

Acknowledgment The first author (K. Hosseini) would like to acknowledge the financial support of the University of Mazandaran for this research work. References

ro of

1. A.K.H. Sedeeg, R.I. Nuruddeen, J.F. Gomez-Aguilar, Generalized optical soliton solutions to the (3+1)‑ dimensional resonant nonlinear Schrödinger equation with Kerr and parabolic law nonlinearities, Optical and Quantum Electronics 51 (2019) 173. 2. F. Ferdous, M.G. Hafez, A. Biswas, M. Ekici, Q. Zhou, M. Alfiras, S.P. Moshokoa, M. Belic, Oblique resonant optical solitons with Kerr and parabolic law nonlinearities and fractional temporal evolution by generalized exp(−𝜙(𝜉))-expansion, Optik 178 (2019) 439–448.

re

-p

3. M.G. Hafez, S.A. Iqbal, S. Akther, M.F. Uddin, Oblique plane waves with bifurcation behaviors and chaotic motion for resonant nonlinear Schrodinger equations having fractional temporal evolution, Results in Physics 15 (2019) 102778.

lP

4. A. Ali, A.R. Seadawy, D. Lu, Soliton solutions of the nonlinear Schrödinger equation with the dual power law nonlinearity and resonant nonlinear Schrödinger equation and their modulation instability analysis, Optik 145 (2017) 79–88. 5. A. Biswas, Y. Yildirim, E. Yasar, Q. Zhou, S.P. Moshokoa, M. Belic, Optical soliton perturbation with resonant nonlinear Schrödinger’s equation having full nonlinearity by modified simple equation method, Optik 160 (2018) 33–43.

na

6. H. Bulut, T.A. Sulaiman, H.M. Baskonus, Optical solitons to the resonant nonlinear Schrödinger equation with both spatio-temporal and inter-modal dispersions under Kerr law nonlinearity, Optik 163 (2018) 49–55.

ur

7. Y.X. Chen, Combined optical soliton solutions of a (1+1)-dimensional time fractional resonant cubic-quintic nonlinear Schrödinger equation in weakly nonlocal nonlinear media, Optik 203 (2020) 163898.

Jo

8. M. Ekici, Q. Zhou, A. Sonmezoglu, J. Manafian, M. Mirzazadeh, The analytical study of solitons to the nonlinear Schrödinger equation with resonant nonlinearity, Optik 130 (2017) 378–382. 9. M. Eslami, M. Mirzazadeh, B. Fathi Vajargah, A. Biswas, Optical solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients by the first integral method, Optik 125 (2014) 3107–3116.

10.

M. Ilie, J. Biazar, Z. Ayati, Resonant solitons to the nonlinear Schrödinger equation with different forms of nonlinearities, Optik 164 (2018) 201–209.

7

M. Inc, A.I. Aliyu, A. Yusuf, D. Baleanu, Optical solitons to the resonance nonlinear Schrödinger equation by Sine-Gordon equation method, Superlattices and Microstructures 113 (2018) 541– 549.

12.

M. Mirzazadeh, M. Eslami, B. FathiVajargah, A. Biswas, Optical solitons and optical rogons of generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity, Optik 125 (2014) 4246–4256.

13.

M. Mirzazadeh, M. Ekici, Q. Zhou, A. Biswas, Exact solitons to generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity, Optik 130 (2017) 178–183.

14.

H. Triki, T. Hayat, O.M. Aldossary, A. Biswas, Bright and dark solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients, Optics & Laser Technology 44 (2012) 2223–2231.

15.

E.M.E. Zayed, R.M.A. Shohib, Solitons and other solutions to the resonant nonlinear Schrodinger equation with both spatio-temporal and inter-modal dispersions using different techniques, Optik 158 (2018) 970–984.

16.

Q. Zhou, Q. Zhu, Y. Liu, H. Yu, P. Yao, A. Biswas, Thirring optical solitons in birefringent fibers with spatio-temporal dispersion and Kerr law nonlinearity, Laser Physics 25 (2015) 015402.

17.

A. Biswas, M.Z. Ullah, Q. Zhou, S.P. Moshokoa, H. Triki, M. Belic, Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle, Optik 145 (2017) 18–21.

18.

A. Biswas, A.H. Kara, M.Z. Ullah, Q. Zhou, H. Triki, M. Belic, Conservation laws for cubicquartic optical solitons in Kerr and power law media, Optik 145 (2017) 650–654.

19.

M. Saha, A.K. Sarma, A. Biswas, Dark optical solitons in power law media with time-dependent coefficients, Physics Letters A 373 (2009) 4438–4441.

20.

A. Biswas, Temporal 1-soliton solution of the complex Ginzburg–Landau equation with power law nonlinearity, Progress in Electromagnetics Research 96 (2009) 1–7.

21.

A.H. Bhrawy, A.A. Alshaery, E.M. Hilal, W.N. Manrakhan, M. Savescu, A. Biswas, Dispersive optical solitons with Schrödinger–Hirota equation, Journal of Nonlinear Optical Physics and Materials 23 (2014) 1450014.

22.

A.H. Bhrawy, A.A. Alshaery, E.M. Hilal, M. Savescu, D. Milovic, K.R. Khan, M.F. Mahmood, Z. Jovanoski, A. Biswas, Optical solitons in birefringent fibers with spatio-temporal dispersion, Optik 125 (2014) 4935–4944.

23.

A. Biswas, Dispersion-managed solitons in optical fibres, Journal of Optics A 4 (2002) 84–97.

Jo

ur

na

lP

re

-p

ro of

11.

24.

H. Triki, T. Hayat, O.M. Aldossary, A. Biswas, Bright and dark solitons for the resonant nonlinear Schrödinger’s equation with time-dependent coefficients, Optics and Laser Technology 44 (2012) 2223–2231.

25.

Q. Zhou, L. Liu, H. Zhang, M. Mirzazadeh, A.H. Bhrawy, E. Zerrad, S. Moshokoa, A. Biswas, Dark and singular optical solitons with competing nonlocal nonlinearities, Optica Applicata 46 (2016) 79–86.

26.

K. Hosseini, M.S. Osman, M. Mirzazadeh, F. Rabiei, Investigation of different wave structures to the generalized third-order nonlinear Scrödinger equation, Optik 206 (2020) 164259.

8

K. Hosseini, M. Mirzazadeh, Q. Zhou, Y. Liu, M. Moradi, Analytic study on chirped optical solitons in nonlinear metamaterials with higher order effects, Laser Physics 29 (2019) 095402.

28.

K. Hosseini, E. Yazdani Bejarbaneh, A. Bekir, M. Kaplan, New exact solutions of some nonlinear evolution equations of pseudoparabolic type, Optical and Quantum Electronics 49 (2017) 241.

29.

D. Kumar, K. Hosseini, F. Samadani, The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics, Optik 149 (2017) 439–446.

30.

A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods, Optical and Quantum Electronics 49 (2017) 278.

31.

K. Hosseini, A. Korkmaz, A. Bekir, F. Samadani, A. Zabihi, M. Topsakal, New wave form solutions of nonlinear conformable time-fractional Zoomeron equation in (2+1)-dimensions, Waves in Random and Complex Media (2019), doi: 10.1080/17455030.2019.1579393.

32.

K. Hosseini, F. Samadani, D. Kumar, M. Faridi, New optical solitons of cubic-quartic nonlinear Schrӧdinger equation, Optik 157 (2018) 1101–1105.

33.

H.C. Ma, Z.P. Zhang, A.P. Deng, A new periodic solution to Jacobi elliptic functions of MKdV equation and BBM equation, Acta Mathematicae Applicatae Sinica 28 (2012) 409–415.

34.

E.M.E. Zayed, R.M.A. Shohib, A. Biswas, Y. Yıldırım, F. Mallawi, M.R. Belic, Chirped and chirp-free solitons in optical fiber Bragg gratings with dispersive reflectivity having parabolic law nonlinearity by Jacobi’s elliptic function, Results in Physics 15 (2019) 102784.

35.

E.M.E.Zayed, M.E.M.Alngar, Optical solitons in birefringent fibers with Biswas–Arshed model by generalized Jacobi elliptic function expansion method, Optik 203 (2020) 163922.

36.

M.M.A. El-Sheikh, A.R. Seadawy, H.M. Ahmed, A.H. Arnous, W.B. Rabie, Dispersive and propagation of shallow water waves as a higher order nonlinear Boussinesq-like dynamical wave equations, Physica A 537 (2020) 122662.

37.

N.A. Kudryashov, Method for finding highly dispersive optical solitons of nonlinear differential equation, Optik (2019) 163550.

38.

K. Hosseini, M. Mirzazadeh, M. Ilie, S. Radmehr, Dynamics of optical solitons in the perturbed Gerdjikov–Ivanov equation, Optik (2020) 164350.

Jo

ur

na

lP

re

-p

ro of

27.

9