# Impact of Fractional Derivative and Brownian Motion on the Solutions of the Radhakrishnan-Kundu-Lakshmanan Equation

Authors: Farah M. Al-Askar
Journal: Journal of Function Spaces
 (2023)

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## Abstract

The fractional-stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) is considered here. To attain new hyperbolic, elliptic, rational, and trigonometric stochastic-fractional solutions, we use two various methods such as the sine-cosine and the Jacobi elliptic function methods. The solutions acquired are important in understanding some interesting physical phenomena due to the significance of the Radhakrishnan-Kundu-Lakshmanan equation in designing the propagation of solitons through an optical fiber. Furthermore, we graph some of the obtained solutions in 3D to display the influence of fractional derivative and multiplicative noise on these solutions. Finally, we show that when the order of fractional derivative decreases, the surface shrinks, while the multiplicative noise stabilizes the solutions of FSRKLE a round zero.


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## Body

## 1. Introduction

Partial differential equations (PDEs) are found in several areas of applied science, including quantum mechanics, plasma physics, nonlinear optics, surface of water waves, hydrodynamics, molecular biology, fluid dynamics, elastic media, and biology. Obtaining solutions of PDEs is crucial for understanding physical phenomena. Therefore, many effective methods, including exp-function method [1], auxiliary equation [2], Darboux transformation [3], sine-cosine [4], Jacobi elliptic function [5], exp−ϕς-expansion [6], sine-Gordon expansion [7], G′/G-expansion [8–10], generalized Kudryashov [11], perturbation [12–14], extended trial equation [15, 16], Jacobi elliptic function [17, 18], Riccati equation [19], tanh-coth [20], homotopy perturbation [21], modified decomposition [22], and F-expansion [23], have been constructed to attain exact solutions of PDEs.Researchers and scientists have focused their attention over the last two decades on fractional differential equations (FDEs) that have been found to be more precise than classical differential equations in explaining complex physical phenomena in the real life. The idea of fractional derivative has been used to define various phenomena including fluid dynamics porous medium, signal processing, viscoelastic materials, ocean wave, electromagnetism, photonic, chaotic systems, wave propagation, optical fiber communication, plasma physics, and nuclear physics. Recently, Atangana and Goufo [24] have suggested the new conformable fractional derivative called beta-derivative. From this point, let us define the Atangana conformable derivative (ACD) for the function ψ:0,∞⟶ℝ of order β∈0,1 as follows:
(1)Dxβψx=limε⟶0ψx+εx+1/Γβ1−β−ψxε.The ACD satisfies the following properties for any constanta and b: (1) Dxβaφx+bψx=aDxβφx+bDxβψx, (2) Dxβa=0, (3) Dxβψθ=x+1/Γβ1−βdψ/dx, (4) If θ=a/βx+1/Γββ, then Dxβψθ=adψ/dθ.Stochastic partial differential equations (SPDEs), on the other hand, have been widely addressed as theoretical equations for spatial-temporal physical, chemical, and biological systems related to random perturbations. The significance of involving stochastic impacts in complex system modeling has been emphasized. For example, there is gaining awareness in using SPDEs to mathematically model complex phenomena in information systems, condensed matter physics, biology, climate systems, electrical and mechanical engineering, materials sciences, and finance.It is worth noting that two forms widely utilized for stochastic integral are Itô and Stratonovich [25]. Modeling problems primarily determine which form is acceptable; however, once that form is selected, an equivalent equation of the other form can be produced using the same solutions. As a result, the following relationship can be utilized to switch between Itô (written as ∫0tϕdW) and Stratonovich (written as ∫0tϕ∘dW):
(2)∫0tσϕsdWs=∫0tσϕs∘dWs−σ22∫0tϕsds,where Wt is a Brownian motion (BM).To satisfy a higher degree of quality agreement, the following stochastic Radhakrishnan-Kundu-Lakshmanan equation (FSRKLE) [26–28] is considered:
(3)idφ+γ1Dxxβφ−iγ2Dxβφ+γ3φ2φ−iγ4φDxβφ2−−iγ5Dxβφ2φ+iγ6Dxxxβφdt+iσφ∘dW=0,where φ∈ℂ, γk for k=1,2,3,4,5,6 are constants and σ is the noise strength and φ∘dW is multiplicative Brownian motion in the Stratonovich sense. Recently, many investigators have created exact solutions of FSRKLE (3), with β=0 and σ=0, using different methods such as extended simple equation method [29], first integral method [30], sine-cosine method [31], Lie group analysis [32], and trial equation method [33].The motivation of this article is to attain the exact solutions for FSRKLE (3). We use two separate approaches, the sine-cosine and the Jacobi elliptic function methods, to provide a wide range of solutions, including hyperbolic, trigonometric, rational, and elliptic functions. The acquired solutions are helpful for understanding several fascinating scientific events because of the significance of the RKL in describing the propagation of solitons through an optical fiber. Also, by creating 3D representations of the obtained FSRKLE (3) solutions, we examine the effect of BM on these solutions.The article is in the following format: in Section2, we determine the wave equation of the FSRKLE (3) by applying a suitable wave transformation. To develop the analytical solutions for the FSRKLE in Section 3, we use two different approaches (3). In Section 4, the impact of the BM on the solutions obtained is examined. The final section of the document is the conclusion.


## 2. Wave Equation for FSRKLE

To obtain the wave equation of the FSRKLE (3), the following transformation is utilized:
(4)φx,t=ψζeiqx,t−σWt−σ2t,ζ=1βx+1Γββ−νt,qx,t=−kβx+1Γββ+ωt,where the function ψ is deterministic and ν,k, and ω are unknown constants. Putting Equation (4) into Equation (3) and utilizing
(5)dφ=−νψ′+iωψ+12σ2ψ−σ2ψdt−σψdWeiqx,t−σWt−σ2t,=−νψ′+iωψdt−σψ∘dWeiqx,t−σWt−σ2t,where 1/2σ2ψ is the Itô correction term, and
(6)Dxβφ=ψ′−ikψeiqx,t−σWt−σ2t,Dxxβφ=ψ′′−2ikψ′−k2ψeiqx,t−σWt−σ2t,Dxxxβφ=ψ′′′−3ikψ′′−3k2ψ′+ik3ψeiqx,t−σWt−σ2tφDxβφ2=2ψ2ψ′eiqx,t−3σWt−3σ2t,Dxβφ2φ=3ψ2ψ′−ikψ3eiqx,t−3σWt−3σ2t,we get for imaginary part
(7)γ6k3ψ′′′−3γ6k2+γ2+2kγ1+νψ′−3γ5+2γ4ψ2ψ′e−2σWt−2σ2t=0,and for real part
(8)γ1+3kγ6ψ′′−k2γ1+kγ2−k3γ6ψ+γ3−kγ5ψ3e−2σWt−2σ2t=0.Taking expectationE· on both sides for Equations (7) and (8) and using
(9)EeσWt=eσ2/2t,we have
(10)γ6k3ψ′′′−3γ6k2+γ2+2kγ1+νψ′−3γ5+2γ4ψ2ψ′=0,(11)γ1+3kγ6ψ′′−ω+k2γ1+kγ2+k3γ6ψ−kγ5−γ3ψ3=0.Integrating Equation (10), we get
(12)γ6k3ψ′′−3γ6k2+γ2+2kγ1+νψ−γ5+23γ4ψ3=0.We obtain the next constraint conditions where the same functionψ achieves both Equations (11) and (12):
(13)γ1+3kγ6γ6=ω+k2γ1+kγ2+k3γ63γ6k2+γ2+2kγ1+ν=3kγ5−γ33γ5+2γ4,whenever
(14)γ3=−3γ5γ1+γ1γ4+6kγ6γ5+3kγ6γ43γ6,(15)ω=8k3γ62+8k2γ1γ6+2kγ12+2kγ2γ6+γ1γ2+ν3kγ6+γ1γ6.Plugging Equation (14) into Equation (11), we have the wave equation as follows:
(16)ψ′′−ℏ1ψ3−ℏ2ψ=0,where
(17)ℏ1=3γ5γ1+γ1γ4+9kγ6γ5+3kγ6γ43γ6γ1+3kγ6,(18)ℏ2=9k3γ62+9k2γ1γ6+2kγ12+3kγ2γ6+γ1γ2+ν3kγ6+γ1γ6γ1+3kγ6.


## 3. The Exact Solutions of the FSRKLE

We employ two various methods such as the Jacobi elliptic function [18] and sine-cosine [4], to determine the exact solutions to Equation (16). As a consequence, we can obtain the solutions of the FSRKLE (3).


### 3.1. Jacobi Elliptic Function Method

We suppose the solutions of Equation (16) has the type
(19)ψζ=a+bsnθζ,where snθζ=snθζ,m, for 0<m<1, is Jacobi elliptic sine function and a,b, and θ are undefined constants. Differentiate Equation (19) twice, we get
(20)ψ′′ζ=−m2+1bθ2snθζ+2m2bθ2sn3θζ.Putting Equations (19) and (20) into Equation (16), we obtain
(21)2m2bθ2−ℏ1b3sn3θζ−3ℏ1ab2sn2θζ−m2+1bθ2+3ℏ1a2b+ℏ2bsnθζ−ℏ1a3+aℏ2=0.Equating each coefficient ofsnθζn to zero, we have for n=0,1,2,3,
(22)ℏ1a3+aℏ2=0,m2+1bθ2+3ℏ1a2b+ℏ2b=0,3ℏ1ab2sn2=0,2m2bθ2−ℏ1b3=0.The outcomes of solving the previous equations are(23)a=0,b=±−2m2ℏ2m2+1ℏ1,θ2=−ℏ2m2+1.As a result, using (19), the solution of Equation (16) is
(24)ψζ=±−2m2ℏ2m2+1ℏ1sn−ℏ2m2+1ζ.Hence, the exact solution of the FSRKLE (3) is
(25)φx,t=±−2m2ℏ2m2+1ℏ1sn−ℏ2m2+11βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2<0 and ℏ1>0. If m⟶1, then solution (25) tends to
(26)φx,t=±−ℏ2ℏ1tanh−ℏ221βx+1Γββ−νteiqx,t−σWt−σ2t.Analogously, we can replacesn in (19) with cnξ=cnξ,m and dnξ,m=dnξ,m to obtain the following solutions of Equation (16):
(27)ψζ=±−2m2ℏ22m2−1ℏ1cnℏ22m2−1ζ,ψζ=±−2m2ℏ22−m2ℏ1dnℏ22−m2ζ.Thus, the exact solutions of the FSRKLE (3) are as follows:
(28)φx,t=±−2m2ℏ22m2−1ℏ1cn−ℏ22m2−11βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2/2m2−1>0, ℏ1<0, and
(29)φx,t=±−2m2ℏ22−m2ℏ1dnℏ22−m21βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2>0, ℏ1<0, respectively. If m⟶1, then the Equations (28) and (29) tends to
(30)φx,t=±−2ℏ2ℏ1sechℏ21βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2>0, ℏ1<0.


### 3.2. Sine-Cosine Method

Suppose the solutionψ of Equation (16) takes the form
(31)ψζ=AYn,where
(32)Y=cosBζorY=sinBζ.Setting Equation (31) into Equation (16), we get
(33)−AB2−n2Yn+nn−1Yn−2−ℏ1A3Y3n−ℏ2AYn=0,rewriting the above equation
(34)ℏ2A−AB2n2Yn+nn−1AB2Yn−2+ℏ1A3Y3n=0.Balancing the term ofY in Equation (34), we obtain
(35)n−2=3n⇒n=−1.Plugging Equation (35) into Equation (34),
(36)ℏ2A−AB2Y−1+ℏ1A3+2AB2Y−3=0.We get by setting each coefficient ofY−3 and Y−1 equal to zero
(37)ℏ2A−AB2=0,(38)ℏ1A3+2AB2=0.By solving Equations (37) and (38), we get
(39)B=ℏ2andA=−2ℏ2ℏ1.Hence, the solution of Equation (16) is
(40)ψζ=AsecBζorψζ=AcscBζ.Depending on the sign ofℏ1 and ℏ2, there are numerous cases:Case 1.
Ifℏ2>0 and ℏ1<0, then FSRKLE (3) has the solutions
(41)φx,t=−2ℏ2ℏ1secℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(42)φx,t=−2ℏ2ℏ1cscℏ21βxβ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 2.
Ifℏ2<0 and ℏ1<0, then the analytical solutions of FSRKLE (3) have the form
(43)φx,t=i2ℏ2ℏ1sech−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(44)φx,t=2ℏ2ℏ1csch−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 3.
Ifℏ2<0 and ℏ1>0, then the solutions of FSRKLE (3) are
(45)φx,t=−2ℏ2ℏ1sech−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(46)φx,t=−i−2ℏ2ℏ1csch−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 4.
Ifℏ2>0 and ℏ1>0, then FSRKLE (3) has the solutions
(47)φx,t=i2ℏ2ℏ1secℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(48)φx,t=i2ℏ2ℏ1cscℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,where ℏ1,ℏ2 are defined in (17).Remark 1.
Settingβ=1 and σ=0 in Equations (41), (42), and (45), we get the identical solutions as asserted in [31].


## 3.1. Jacobi Elliptic Function Method

We suppose the solutions of Equation (16) has the type
(19)ψζ=a+bsnθζ,where snθζ=snθζ,m, for 0<m<1, is Jacobi elliptic sine function and a,b, and θ are undefined constants. Differentiate Equation (19) twice, we get
(20)ψ′′ζ=−m2+1bθ2snθζ+2m2bθ2sn3θζ.Putting Equations (19) and (20) into Equation (16), we obtain
(21)2m2bθ2−ℏ1b3sn3θζ−3ℏ1ab2sn2θζ−m2+1bθ2+3ℏ1a2b+ℏ2bsnθζ−ℏ1a3+aℏ2=0.Equating each coefficient ofsnθζn to zero, we have for n=0,1,2,3,
(22)ℏ1a3+aℏ2=0,m2+1bθ2+3ℏ1a2b+ℏ2b=0,3ℏ1ab2sn2=0,2m2bθ2−ℏ1b3=0.The outcomes of solving the previous equations are(23)a=0,b=±−2m2ℏ2m2+1ℏ1,θ2=−ℏ2m2+1.As a result, using (19), the solution of Equation (16) is
(24)ψζ=±−2m2ℏ2m2+1ℏ1sn−ℏ2m2+1ζ.Hence, the exact solution of the FSRKLE (3) is
(25)φx,t=±−2m2ℏ2m2+1ℏ1sn−ℏ2m2+11βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2<0 and ℏ1>0. If m⟶1, then solution (25) tends to
(26)φx,t=±−ℏ2ℏ1tanh−ℏ221βx+1Γββ−νteiqx,t−σWt−σ2t.Analogously, we can replacesn in (19) with cnξ=cnξ,m and dnξ,m=dnξ,m to obtain the following solutions of Equation (16):
(27)ψζ=±−2m2ℏ22m2−1ℏ1cnℏ22m2−1ζ,ψζ=±−2m2ℏ22−m2ℏ1dnℏ22−m2ζ.Thus, the exact solutions of the FSRKLE (3) are as follows:
(28)φx,t=±−2m2ℏ22m2−1ℏ1cn−ℏ22m2−11βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2/2m2−1>0, ℏ1<0, and
(29)φx,t=±−2m2ℏ22−m2ℏ1dnℏ22−m21βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2>0, ℏ1<0, respectively. If m⟶1, then the Equations (28) and (29) tends to
(30)φx,t=±−2ℏ2ℏ1sechℏ21βx+1Γββ−νteiqx,t−σWt−σ2t,for ℏ2>0, ℏ1<0.


## 3.2. Sine-Cosine Method

Suppose the solutionψ of Equation (16) takes the form
(31)ψζ=AYn,where
(32)Y=cosBζorY=sinBζ.Setting Equation (31) into Equation (16), we get
(33)−AB2−n2Yn+nn−1Yn−2−ℏ1A3Y3n−ℏ2AYn=0,rewriting the above equation
(34)ℏ2A−AB2n2Yn+nn−1AB2Yn−2+ℏ1A3Y3n=0.Balancing the term ofY in Equation (34), we obtain
(35)n−2=3n⇒n=−1.Plugging Equation (35) into Equation (34),
(36)ℏ2A−AB2Y−1+ℏ1A3+2AB2Y−3=0.We get by setting each coefficient ofY−3 and Y−1 equal to zero
(37)ℏ2A−AB2=0,(38)ℏ1A3+2AB2=0.By solving Equations (37) and (38), we get
(39)B=ℏ2andA=−2ℏ2ℏ1.Hence, the solution of Equation (16) is
(40)ψζ=AsecBζorψζ=AcscBζ.Depending on the sign ofℏ1 and ℏ2, there are numerous cases:Case 1.
Ifℏ2>0 and ℏ1<0, then FSRKLE (3) has the solutions
(41)φx,t=−2ℏ2ℏ1secℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(42)φx,t=−2ℏ2ℏ1cscℏ21βxβ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 2.
Ifℏ2<0 and ℏ1<0, then the analytical solutions of FSRKLE (3) have the form
(43)φx,t=i2ℏ2ℏ1sech−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(44)φx,t=2ℏ2ℏ1csch−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 3.
Ifℏ2<0 and ℏ1>0, then the solutions of FSRKLE (3) are
(45)φx,t=−2ℏ2ℏ1sech−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(46)φx,t=−i−2ℏ2ℏ1csch−ℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t.Case 4.
Ifℏ2>0 and ℏ1>0, then FSRKLE (3) has the solutions
(47)φx,t=i2ℏ2ℏ1secℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,or
(48)φx,t=i2ℏ2ℏ1cscℏ21βx+1Γββ−νtei−k/βx+1/Γββ+ωt−σWt−σ2t,where ℏ1,ℏ2 are defined in (17).Remark 1.
Settingβ=1 and σ=0 in Equations (41), (42), and (45), we get the identical solutions as asserted in [31].


## 4. Effect of BM and Fractional Derivative on the Solutions

Here, the impact of BM and the fractional derivative on the exact solutions of the FSRKLE (3) is described. Fix the constants γ1=γ2=γ4=γ5=γ6=1,k=−1,ν=3, and m=0.5. Hence, γ3=5/3,ℏ1=4/3, and ℏ2=−1. Now, we present some diagrams for various value of σ (intensity of noise) and for t∈0,5, x∈0,6. We apply the MATLAB to simulate the solutions of Equation (3).Firstly, the impact of noise: in Figure 1, when σ=0, we note that the surface fluctuates.Figure 1
3D diagram of Equations (25) and (45) with σ=0.In Figure2, if the noise appeared, then after small transit behaviors, the surface gets more planer when the intensity of noise increases as follows:Figure 2
3D diagram of Equations (25) and (45) with σ=1,2.Secondly, the impact of fractional derivative: in Figures 3 and 4, if σ=0, we can observe that as β increases, the surface extends:Figure 3
3D diagram of Equation (25) with σ=0 and various β.Figure 4
3D diagram of Equation (45) with σ=0 and various β.


## 5. Conclusions

In this paper, we obtained the exact solutions of the fractional-stochastic Radhakrishnan-Kundu-Lakshmanan Equation (3). To obtain rational, elliptic, trigonometric, and hyperbolic stochastic solutions, we used two different methods: the Jacobi elliptic function and the sine-cosine. Because of the priority of the FSRKLE in fluid dynamics and plasma physics, the results produced are useful for understanding some exciting physical phenomena. Finally, we plotted the obtained solutions using MATLAB tools to provide a number 3D diagram to demonstrate the impact of fractional derivative and multiplicative noise on these solutions. In future work, we can consider the FSRKLE (3) with additive noise.



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