## Abstract

This study investigates the optical solitons of of (3+1)-dimensional resonant nonlinear Schrödinger (3D-RNLS) equation with the two laws of nonlinearity. The two forms of nonlinearity are represented by Kerr law and parabolic law. Based on complex transformation, the traveling wave reduction of the governing model is derived. The projective Riccati equations technique is applied to obtain the exact solutions of 3D-RNLS equation. Various types of waves that represent different structures of optical solitons are extracted. These structures include bright, dark, singular, dark-singular and combined singular solitons. Additionally, the obliquity effect on resonant solitons is illustrated graphically and is found to cause dramatic variations in soliton behaviors.

### Keywords

- Optical solitons
- 3D-RNLS equation
- Kerr law and parabolic law nonlinearities
- Projective Riccati equations method
- Obliquity influence

## 1. Introduction

Soliton is one of the important nonlinear waves that has been under intensive investigation in the physical and natural sciences. It has been noticed that solitons play a significant role on describing the physical phenomena in various branches of science, such as optical fibers, plasma physics, nonlinear optics, and many other fields [1, 2, 3, 4, 5]. For example, solitons in the field of nonlinear optics are known as optical solitons and have the capacity to transport information through optical fibers over transcontinental and transoceanic distances in a matter of a few femtoseconds [6, 7]. Moreover, it is found that the efficient physical properties of solitons may support the improvement on photonic and optoelectronic devices [8, 9]. Further to this, optical solitons can be exploited widely in optical communication and optical signal processing systems [10, 11].

The formation of solitons is essentially caused due to a delicate balance between dispersion and nonlinearity in the medium. Understanding the dynamics of solitons can be performed through focusing deeply on one model of the nonlinear Schrödinger family of equations with higher order nonlinear terms [12, 13]. Thus, various studies in literatures scrutinized the resonant nonlinear Schrödinger equation which is mainly the governing model that describes soliton propagation and Madelung fluids in many nonlinear media. Several integration schemes have been implemented to examine the behavior of solitons such as ansatz method, semi-inverse variational principle, simplest equation approach, first integral method, functional variable method, sine-cosine function method, (

The present study concentrates on the investigation of resonant optical solitons in (3 + 1)-dimensions with two types of nonlinear influences, namely, Kerr law and parabolic law nonlinearities. In particular, we shed light on the model of (3 + 1)-dimensional resonant nonlinear Schrödinger (3D-RNLS) equation given in the form

where the dependent variable

Here, we will consider two specific cases for the function

and

The first model given in Eq. (2) is the 3D-RNLS equation dominated by the Kerr law nonlinearity and is found to have applications in the optical fiber and water waves when the refractive index of the light is proportional to the intensity. The second model presented in Eq. (3) is the 3D-RNLS equation with the parabolic law nonlinearity which arises in the context of nonlinear fiber optics.

In literatures, there are some studies that dealt with the 3D-RNLS equation to find exact solutions. For example, Ferdous et al. [27] investigated the conformable time fractional 3D-RNLS equation with Kerr and parabolic law nonlinearities. Different structures of oblique resonant optical solitons have been obtained by using the generalized

The aim of current work is to derive the optical solitons of 3D-RNLS equation presented in (2) and (3). The mathematical technique applied to solve the models is based on a finite series expressed in terms of the solution of projective Riccati equations. The paper is organized as follows. In Section 2, we analyze the idea of implementing the proposed method. In Section 3, the traveling wave reduction of (2) and (3) is extracted. Then, Section 4 displays the derivation of resonant optical solitons in (3 + 1)-dimensions. In Section 5, the main outlook of results and remarks are presented. Finally, the conclusion of work is given in Section 6.

## 2. Elucidation of solution method

Consider a nonlinear partial differential equation (NLPDE) for

where

Since we seek for exact traveling wave solutions, we introduce the wave variables

Inserting (5) into Eq. (4), one can find the following ordinary differential equation (ODE)

where prime denotes the derivative with respect to

Now, assume that the solution of Eq. (6) can be expressed in the finite series of the form

where

The variables

where

The set of Eqs. (8) is found to admit the following solutions

demands

implies

provided

The substitution of (7) along with (8) into Eq. (6) generates a polynomial in

## 3. Traveling wave reduction for Eqs. (2) and (3)

In order to tackle the complex models of 3D-RNLS equation with Kerr law and parabolic law nonlinearities given in (2) and (3), we embark on analyzing their structures by using the wave transformation of the form

where

### 3.1 Traveling wave reduction for Eq. (2)

Applying transformation (12), the 3D-RNLS equation with Kerr law nonlinearity given in (2) is broken down into real and imaginary parts as

and

From Eq. (15), we obtain

where

### 3.2 Traveling wave reduction for Eq. (3)

Similarly, we utilize the wave transformation (12) to the 3D-RNLS equation with parabolic law nonlinearity given in (3) which is decomposed to real and imaginary parts as

and

From Eq. (19), we come by the expression given in (16). To seek a closed form solution, the structure of Eq. (18) has to be rearranged. Thus, we multiply Eq. (18) by

where

which leads to

## 4. Optical soliton solutions of 3D-RNLS equation with Kerr law and parabolic law nonlinearities

Now, we aim to employ the projective Riccati equations method given in Section 2 to extract the exact resonant optical soliton solutions with Kerr law and parabolic law nonlinearities for 3D-RNLS equations given in (2) and (3). Basically, the proposed technique will be implemented to Eqs. (17) and (20) and then their obtained solutions will be inserted into (12) so as to derive the optical solitons of the models discussed in this work.

### 4.1 Oblique resonant solitons of 3D-RNLS equation with Kerr law nonlinearity

According to the expansion given in (7) and the balance between the terms

Substituting (23) together with Eqs. (8) into Eq. (17) gives rise to an equation having different powers of

** Set I.**If

where

where

where

** Set II.**If

where

where

where

** Set III.**If

Herein, these three cases in the Set III provide the solution of the form

where

### 4.2 Oblique resonant solitons of 3D-RNLS equation with parabolic law nonlinearity

Based on the expansion given in (7), we consider that the solution to Eq. (22) takes the form

Inserting (31) together with Eqs. (8) into Eq. (22) gives rise to an equation having different powers of

** Set I.**If

where

where

where

where

where

where

where

where

where

where

** Set II.**If

where

where

where

where

where

where

where

where

where

where

## 5. Results and remarks

To give a clear insight into the behavior of resonant optical solitons, the graphical representations for some of the extracted soliton solutions are presented. Besides, the obliqueness influence on the resonant solitons is examined. Thus, we display the 3D and 2D plots of the absolute of these solutions by selecting different values of the model parameters. For example, Figure 1(a)–(b) present the 3D and 2D plots of resonant soliton for the solution given in (24) of 3D-RNLS equation with Kerr-law nonlinearity. It is clear from the graph that the wave profile represents bright soliton. Figure 1(c)–(d) display the 3D plot for the effect of obliquity on the resonant soliton given in (24), where Figure 1(c) shows the relation between

One can obviously see from Figures 1–4 that the obliqueness influences the behavior of resonant solitons, where the structure of solitons is changed remarkably with the variation of obliqueness parameters. Further to this, it is noticed that the amplitude of the resonant solitons decreases and the width rises with the increase of obliqueness as shown in the 2D graphs.

## 6. Conclusions

This work scoped the behavior of optical solitons of 3D-RNLS equation. The dominant nonlinearity in the model is considered to have two forms which are Kerr law and parabolic law. The resonant solitons are derived with the aid of projective Riccati equations method. Various forms of wave structures are retrieved such as bright, dark, singular, kink, dark-singular and combined singular solitons. The influence of obliquity on resonant solitons is also examined. It is found that the change in the obliqueness parameters leads to a noticeable variation on the behavior of optical soliton waves. In addition to this, the amplitude of the resonant solitons undergoes a reduction, but their width is enhanced as the obliqueness is increased. The results obtained in this work are entirely new and it may be useful to understand the dynamics of resonant solitons affected by obliqueness in different nonlinear media such as optical fiber and Madelung fluids.