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Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law and Parabolic Law Nonlinearities

By Khalil S. Al-Ghafri

Submitted: June 29th 2021Reviewed: September 15th 2021Published: October 16th 2021

DOI: 10.5772/intechopen.100469

Downloaded: 44

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, (G/G)-expansion method, trial solution approach, generalized extended tanh method, modified simple equation method, and improved extended tanh-equation method. For more details, readers are referred to references [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

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

iQt+η2Q+σFsQ2Q+2QQQ=0,i=1,2=2x2+2y2+2z2,E1

where the dependent variable Qxyztis a complex-valued wave profile and the independent variables x,yand zstand for spatial coordinates while tindicates temporal coordinate. The non-zero constants η,σ, and δaccount for the coefficients of the group velocity, non-Kerr nonlinearity, and resonant nonlinearity, respectively. The parameter splays an important role on manipulating the physical properties of distinct media and consequently affecting the behaviors of constructed solitons, see [26].

Here, we will consider two specific cases for the function Fsthat represent the effect of nonlinearity in the media. These two nonlinear influences include the Kerr law and parabolic law nonlinearities. Hence, Eq. (1) with the two laws of nonlinearity has the following forms

iQt+η2Q+σQ2Q+δ2QQQ=0,E2

and

iQt+η2Q+σQ2+ρQ4Q+δ2QQQ=0.E3

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 expΦξ-expansion method. Furthermore, Sedeeg et al. [28] studied the two models of 3D-RNLS equation given in (2) and (3) by applying the modified extended tanh method. Optical soliton solutions including dark, singular and combo solitons are extracted in addition to periodic solutions. Moreover, the exact solutions of the 3D-RNLS equation with Kerr law nonlinearity given in (2) has been examined by Hosseini et al. [29] by exploiting the new expansion methods based on the Jacobi elliptic equation. Recently, Hosseini et al. [30] studied the optical solitons and modulation instability of the models given in (2) and (3). Various forms of optical solitons are derived with the aid of the expaand hyperbolic function techniques.

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.

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2. Elucidation of solution method

Consider a nonlinear partial differential equation (NLPDE) for Qxyztin the form

PQQtQxQyQzQxxQyyQzz=0,E4

where Pis a polynomial in its arguments.

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

Qxt=qξ,ξ=xcosα+ycosβ+zcosγ+ct.E5

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

Hqqqq=0,E6

where prime denotes the derivative with respect to ξ. Then, integrate Eq. (6), if possible, to reduce the order of differentiation.

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

Uξ=a0+l=1malflξ+blglξ,E7

where a0,al,bl,l=12mare constants to be identified. The parameter m, which is a positive integer, can be determined by balancing the highest order derivative term with the highest order nonlinear term in Eq. (6).

The variables fξ,gξsatisfy the equations

fξ=εAg2ξ,gξ=AfξgξBAgξRBfξ,g2ξ=ε1A2RBfξ2f2ξ,E8

where Aand Bare arbitrary constants and ε=±1. The third equation in the system (8) represents the first integral which gives the relation between the functions fξand gξ.

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

f1ξ=RtanhA+Btanh,g1ξ=RsechA+Btanh,E9

demands ε=1.

f2ξ=RcothA+Bcoth,g2ξ=RcschA+Bcoth,E10

implies ε=1.

f3ξ=AAC+A2B2ξ,g3ξ=εA2B2AC+A2B2ξ,E11

provided R=0, where Cis an arbitrary constant.

The substitution of (7) along with (8) into Eq. (6) generates a polynomial in fiξgjξ. Equating each coefficient of fiξgjξin this polynomial to zero, yields a set of algebraic equations for ai,bj. Solving this system of equations, we can obtain many exact solutions of Eq. (4) according to (9)(11).

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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

Qxt=qξe,E12

where

ξ=xcosα+ycosβ+zcosγ+νt,φ=κxcosα+ycosβ+zcosγ+ωt.E13

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

cos2α+cos2β+cos2γη+δqω+ηκ2cos2α+cos2β+cos2γq+σq3=0,E14

and

ν+2ηκcos2α+cos2β+cos2γq=0.E15

From Eq. (15), we obtain

ν=2ληκ,E16

where λ=cos2α+cos2β+cos2γ. Hence, Eq. (14) reduces to the form

λη+δqω+ληκ2q+σq3=0.E17

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

λη+δqω+ληκ2q+σq3+ρq5=0,E18

and

ν+2ληκq=0.E19

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 qand integrate with respect to ξto arrive at

λη+δq2ω+ληκ2q2+σ2q4+ρ3q6+2μ=0,E20

where μis the integration constant. For convenience, we make use of the variable transformation given as

q2=V,E21

which leads to q2=V2/4V. Thus, Eq. (20), after manipulating, becomes

λη+δV2+8μV4ω+ληκ2V2+2σV3+43ρV4=0.E22
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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 qand q3, the solution of Eq. (17) reads

qξ=a0+a1fξ+b1gξ.E23

Substituting (23) together with Eqs. (8) into Eq. (17) gives rise to an equation having different powers of figj. Collecting all the terms with the same power of figjtogether and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations simultaneously leads to the following results.

Set I.If ε=1, then the following cases of solutions are retrieved.

Case I1.a0=a1=0, b1=±2λη+δA2B2σ, ω=λη+δR2ηκ2.

Qxyzt=±R2λη+δA2B2σsechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκte,E24

where λση+δA2B2>0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case I2.a0=±BRA2λη+δσ, a1=±A2B2A2λη+δσ, b1=0, ω=λ2η+δR2+ηκ2.

Qxyzt=±R2λη+δσB+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκte,E25

where λση+δ<0and φ=κxcosα+ycosβ+zcosγλ2η+δR2+ηκ2t.

Case I3.a0=±BRAλη+δ2σ, a1=±A2B2Aλη+δ2σ, b1=±λη+δA2B22σ,

ω=λη+δR22+ηκ2.
Qxyzt=±Rλη+δ2σB+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt±B2A2sechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκte,E26

where λση+δ<0, A2<B2and φ=κxcosα+ycosβ+zcosγλη+δR22+ηκ2t.

Set II.If ε=1, then the following cases of solutions are generated.

Case II1.a0=a1=0, b1=±2λη+δA2B2σ, ω=λη+δR2ηκ2.

Qxyzt=±R2λη+δA2B2σcschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκte,E27

where λση+δA2B2<0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case II2.a0=±BRA2λη+δσ, a1=±A2B2A2λη+δσ, b1=0, ω=λ2η+δR2+ηκ2.

Qxyzt=±R2λη+δσB+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκte,E28

where λση+δ<0and φ=κxcosα+ycosβ+zcosγλ2η+δR2+ηκ2t.

Case II3.a0=±BRAλη+δ2σ, a1=±A2B2Aλη+δ2σ, b1=±λη+δA2B22σ,

ω=λη+δR22+ηκ2.
Qxyzt=±Rλη+δ2σB+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκte,E29

where λση+δ<0, A2>B2and φ=κxcosα+ycosβ+zcosγλη+δR22+ηκ2t.

Set III.If R=0, then the following cases of solutions are created.

Case III1.a0=a1=0, b1=±2λη+δA2B2εσ, ω=ληκ2.

Case III2.a0=0, a1=±A2B2A2λη+δσ, b1=0, ω=ληκ2.

Case III3.a0=0, a1=±A2B2Aλη+δ2σ, b1=±λη+δA2B22εσ, ω=ληκ2.

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

Qxyzt=±2λη+δσA2B2AC+A2B2xcosα+ycosβ+zcosγ2ληκte,E30

where λση+δ<0and φ=κxcosα+ycosβ+zcosγληκ2t.

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

qξ=a0+l=12alflξ+blglξ.E31

Inserting (31) together with Eqs. (8) into Eq. (22) gives rise to an equation having different powers of figj. Collecting all the terms with the same power of figjtogether and equating each coefficient to zero, yields a set of algebraic equations. Solving these equations simultaneously leads to the following results.

Set I.If ε=1, then the following cases of solutions are obtained.

Case I1.b1=a2=b2=0, a0=Ra1A+B, a1=±A2B22A3λη+δρ,

ω=λη+δA+B2a02ηκ2a12a12, σ=2λη+δABA+B2a0Aa12, μ=0.

Qxyzt=R23λη+δρ1B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E32

where λρη+δ<0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case I2.b1=a2=b2=0, a0=Ra1AB, a1=±A2B22A3λη+δρ,

ω=λη+δAB2a02ηκ2a12a12, σ=2λη+δA+BAB2a0Aa12, μ=0.

Qxyzt=±R23λη+δρ1+B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E33

where λρη+δ<0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case I3.a1=a2=b2=0, a0=±Rb1A2B2, b1=±123λη+δA2B2ρ,

ω=λ5η+δA2B2a02+4ηκ2b124b12, σ=2λη+δA2B2a0b12, μ=λη+δA2B2a034b12.

Qxyzt=±R23λη+δρ1±A2B2sechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E34

where λρη+δ>0, A2>B2and φ=κxcosα+ycosβ+zcosγλη+δ5R24+ηκ2t.

Case I4.a2=b2=0, a0=Rb1AA+BA2B2, a1=±b1AA2B2, b1=±143λη+δA2B2ρ, ω=λη+δA+BA2a02+4ηκ2ABb124ABb12, σ=λη+δA+BAa02b12, μ=0.

Qxyzt=±R4A3λη+δA2B2ρA2B2tanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt±AsechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκtA2B2A+B12e,E35

where λρη+δ<0, A2<B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case I5.a2=b2=0, a0=±Rb1AABA2B2, a1=±b1AA2B2, b1=143λη+δA2B2ρ, ω=λη+δABA2a02+4ηκ2A+Bb124A+Bb12, σ=λη+δABAa02b12, μ=0.

Qxyzt=±R4A3λη+δA2B2ρA2B2tanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt±AsechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt+A2B2AB12e,E36

where λρη+δ<0, A2<B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case I6.a0=2λη+δA2+σb2R2σA2, a1=2BRb2A2, b1=±A2a0+R2b2RA2A2B2, a2=A2B2b2A2, ω=λη+δ5R24+ηκ2, ρ=3σ216λη+δR2, μ=λ2η+δ2R42σ.

Qxyzt=2λη+δR2σ1±A2B2sechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E37

where A2>B2and φ=κxcosα+ycosβ+zcosγλη+δ5R24+ηκ2t.

Case I7.b1=0, a0=A2a1A+BRb2RABA2, a1=2λη+δA2B2A+σBb2RσA2, a2=A2B2b2A2,

ω=λη+δR2ηκ2, ρ=3λη+δA2B22A24A2a12BRb22, μ=0.

Qxyzt=2λη+δR2σ1+B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E38

where φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case I8.b1=0, a0=A2a1+ABRb2RA+BA2, a1=2λη+δA2B2AσBb2RσA2, a2=A2B2b2A2,

ω=λη+δR2ηκ2, ρ=3λη+δA2B22A24A2a12BRb22, μ=0.

Qxyzt=2λη+δR2σ1B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E39

where φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case I9.a0=λη+δA+BA2σb2R22σA2, a1=ABA2a0+A+BR2b2A2R, b1=±λη+δR2σA2B2, a2=A2B2b2A2, ω=λη+δR24ηκ2, ρ=3σ24λη+δR2, μ=0.

Qxyzt=λη+δR22σ1+B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt±A2B2sechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E40

where A2<B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case I10.a0=λη+δABA2σb2R22σA2, a1=A+BA2a0+ABR2b2A2R, b1=±λη+δR2σA2B2, a2=A2B2b2A2, ω=λη+δR24ηκ2, ρ=3σ24λη+δR2, μ=0.

Qxyzt=λη+δR22σ1B+AtanhRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt±A2B2sechRxcosα+ycosβ+zcosγ2ληκtA+BtanhRxcosα+ycosβ+zcosγ2ληκt12e,E41

where A2<B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Set II.If ε=1, then the following cases of solutions are acquired.

Case II1.b1=a2=b2=0, a0=Ra1A+B, a1=±A2B22A3λη+δρ,

ω=λη+δA+B2a02ηκ2a12a12, σ=2λη+δABA+B2a0Aa12, μ=0.

Qxyzt=R23λη+δρ1B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E42

where λρη+δ<0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case II2.b1=a2=b2=0, a0=Ra1AB, a1=±A2B22A3λη+δρ,

ω=λη+δAB2a02ηκ2a12a12, σ=2λη+δA+BAB2a0Aa12, μ=0.

Qxyzt=±R23λη+δρ1+B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E43

where λρη+δ<0and φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case II3.a1=a2=b2=0, a0=±Rb1A2B2, b1=±123λη+δA2B2ρ,

ω=λ5η+δA2B2a02+4ηκ2b124b12, σ=2λη+δA2B2a0b12, μ=λη+δA2B2a034b12.

Qxyzt=±R23λη+δρ1±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E44

where λρη+δ>0, A2<B2and φ=κxcosα+ycosβ+zcosγλη+δ5R24+ηκ2t.

Case II4.a2=b2=0, a0=Rb1A+B, a1=±A2B24A3λη+δρ, b1=±Aa1A2B2A2B2, ω=λη+δA+B2a024ηκ2a124a12, σ=λη+δA+BA2B2a02Aa12, μ=0.

Qxyzt=R43λη+δρ1B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E45

where λρη+δ<0, A2>B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case II5.a2=b2=0, a0=Rb1AB, a1=±A2B24A3λη+δρ, b1=±Aa1A2B2A2B2, ω=λη+δAB2a024ηκ2a124a12, σ=λη+δABA2B2a02Aa12, μ=0.

Qxyzt=±R43λη+δρ1+B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E46

where λρη+δ<0, A2>B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case II6.a0=2λη+δA2σb2R2σA2, a1=2BRb2A2, b1=±2λη+δRσA2B2, a2=A2B2b2A2, ω=λη+δ5R24+ηκ2, ρ=3σ216λη+δR2, μ=λ2η+δ2R42σ.

Qxyzt=2λη+δR2σ1±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E47

where A2<B2and φ=κxcosα+ycosβ+zcosγλη+δ5R24+ηκ2t.

Case II7.b1=0, a0=A2a1+A+BRb2RABA2, a1=2λη+δA2B2AσBb2RσA2, a2=A2B2b2A2,

ω=λη+δR2ηκ2, ρ=3λη+δA2B22A24A2a1+2BRb22, μ=0.

Qxyzt=2λη+δR2σ1+B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E48

where φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case II8.b1=0, a0=A2a1ABRb2RA+BA2, a1=2λη+δA2B2A+σBb2RσA2,

a2=A2B2b2A2, ω=λη+δR2ηκ2, ρ=3λη+δA2B22A24A2a1+2BRb22, μ=0.

Qxyzt=2λη+δR2σ1B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E49

where φ=κxcosα+ycosβ+zcosγ+λη+δR2ηκ2t.

Case II9.a0=λη+δA+BA+2σb2R22σA2, a1=ABA2a0A+BR2b2A2R, b1=±λη+δR2σA2B2, a2=A2B2b2A2, ω=λη+δR24ηκ2, ρ=3σ24λη+δR2, μ=0.

Qxyzt=λη+δR22σ1+B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E50

where A2>B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

Case II10.a0=λη+δABA+2σb2R22σA2, a1=A+BA2a0ABR2b2A2R, b1=±λη+δR2σA2B2, a2=A2B2b2A2, ω=λη+δR24ηκ2, ρ=3σ24λη+δR2, μ=0.

Qxyzt=λη+δR22σ1B+AcothRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt±A2B2cschRxcosα+ycosβ+zcosγ2ληκtA+BcothRxcosα+ycosβ+zcosγ2ληκt12e,E51

where A2>B2and φ=κxcosα+ycosβ+zcosγ+λη+δR24ηκ2t.

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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 xand αwhile Figure 1(d) illustrates the relation between xand γ. Figure 2(a)(b) exhibit the 3D and 2D plots of resonant dark soliton given in the solution (29) of 3D-RNLS equation with Kerr-law nonlinearity. The obliqueness influence on the solution (29) is shown in Figure 2(c)(d). Additionally, Figure 3(a)(b) demonstrate the 3D and 2D plots of resonant soliton given in the solution (40) of 3D-RNLS equation with parabolic-law nonlinearity, where the wave profile describes a kink-shape soliton. It can be seen that Figure 3(c)(d) present the obliquity impact on the solution (40). Figure 4(a)(b) depict the 3D and 2D plots of resonant singular soliton given in the solution (48) of 3D-RNLS equation with parabolic-law nonlinearity, where the effect of obliqueness on this wave is illustrated in Figure 4(c)(d).

Figure 1.

(a)-(b) Resonant soliton and (c)-(d) obliqueness effect on resonant soliton corresponding to solution(24)withκ=0.5,η=δ=σ=1,R=A=2,B=1,α=β=γ=π/3,y=z=0,t=1.

Figure 2.

(a)-(b) Resonant soliton and (c)-(d) obliqueness effect on resonant soliton corresponding to solution(29)with the same values of parameters inFigure 1exceptσ=1.

Figure 3.

(a)-(b) Resonant soliton and (c)-(d) obliqueness effect on resonant soliton corresponding to solution(40)with the same values of parameters inFigure 1exceptA=1,B=2.

Figure 4.

(a)-(b) Resonant soliton and (c)-(d) obliqueness effect on resonant soliton corresponding to solution(48)with the same values of parameters inFigure 1.

One can obviously see from Figures 14 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.

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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.

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Conflict of interest

The author declares no conflict of interest.

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Khalil S. Al-Ghafri (October 16th 2021). Resonant Optical Solitons in (3 + 1)-Dimensions Dominated by Kerr Law and Parabolic Law Nonlinearities [Online First], IntechOpen, DOI: 10.5772/intechopen.100469. Available from:

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