# Traveling Wave Solution of the Kaup–Boussinesq System with Beta Derivative Arising from Water Waves

Authors: Dan Chen; Zhao Li
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

The main purpose of this paper is to construct the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves. By using the complete discriminant system method of polynomial, the rational function solution, the trigonometric function solution, the exponential function solution, and the Jacobian function solution of the Kaup–Boussinesq system with beta derivative are obtained. In order to further explain the propagation of the Kaup–Boussinesq system with beta derivative in water waves, we draw its three-dimensional diagram, two-dimensional diagram, density plot, and contour plot by using Maple software.


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## Body

## 1. Introduction

The Kaup–Boussinesq system was first proposed by Kaup after adding a first-order nonlinear term to the Boussinesq equation when he studied it in 1975. For more than 40 years, the study of predecessors on the Kaup–Boussinesq system has covered a wide range of fields and made many achievements. Especially, the study of traveling wave solutions of the Kaup–Boussinesq system have become a very important research field [1–6]. Many important methods are also used to construct traveling wave solutions [7–12] and optical soliton solutions [13–17] of the Kaup–Boussinesq system. However, as far as we can tell from the literature, the solutions obtained mainly focus on hyperbolic function solutions, trigonometric function solutions, and rational function solutions. On the contrary, practical problems from the fields of physics, communication, control, and engineering technology are usually simulated by fractional partial differential equations [18–23]. Therefore, the study of traveling wave solution of the fractional Kaup–Boussinesq system is a very important and challenging problem for scientists.The Kaup–Boussinesq system with beta derivative can be described as follows [24, 25]:(1)DtβΩ−φxxx−2Ωφx=0,Dtβφ−Ωx−φ2x=0,where Ω=Ωx,t and φ=φx,t represent the height of the water surface above the horizontal bottom and the horizontal velocity field, respectively. Dtβ stands for the beta fractional derivative. In [24], Wang and his collaborators obtained the traveling wave of system (1) by using the auxiliary equation method. In [25], Kilic and Inc studied the time fractional Kaup–Boussinesq system in the sense of the modified Riemann–Liouville derivative by using first integral method. The abovementioned literature has not obtained the Jacobian function solution, so in this paper, we will use the complete discriminant system to study system (1).Next, we give the definition of beta fractional derivative.Definition 1.
(see [26]). Let f:0,∞⟶R. Then, the beta derivative of ft of order β is defined as(2)Tβft=dβftdtβ=limε⟶0ft+εt+1/Γβ1−β−ftε,β∈0,1.
The layout of this article is as follows. In Section2, we obtain the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves. In Section 3, we give a summary.


## 2. Traveling Wave Solution of (1)

In order to obtain the traveling wave solution of (1), we first make the following traveling wave transformation:(3)Ωx,t=ϑζ,φx,t=θζ,ζ=x−λβt+1Γββ.Substituting (3) into (1), we obtain(4)−λϑ′−θ‴−2ϑϑ′=0,−λθ′−ϑ′−2θϑ′=0.Integrating both sides of the second equation of (4) with respect to ζ at the same time and making the integral constant zero, we get(5)ϑ=−λθ−θ2.Substituting (5) into the first equation of (4), then it can be rewritten as(6)λ2+6λθ+6θ2θ′−θ‴=0.Integrating equation (6) once with respect to ζ, we get(7)θ″−λ2θ−3λθ2−2θ3+b1=0,where b1 is the integral constant.Multiplying both sides of equation (7) by θ′ and integrating it once, then equation (7) can be obtained:(8)12θ′2−12λ2θ2−λθ3−12θ4+b1θ=b2,where b2 is the integral constant.Then, equation (8) can be reduced as(9)θ′2=θ4+2λθ3+λ2θ2+a1θ+a0,where a1=−2b1,a0=2b2.For equation (9), we assume that(10)ω=θ+λ2,ζ1=ζ.Substituting (10) into (9), we obtain(11)ωζ12=Fω=ω4+Q2ω2+Q1ω+Q0,where Q2=λ2,Q1=a1,Q0=λ4/16−a1λ/2+a0.According to the complete discriminant system of polynomials [27], the integral representation of (11) can be expressed as follows:(12)±ζ1−ζ0=∫dωω4+Q2ω2+Q1ω+Q0,where ζ0 is an integral constant.Suppose thatFω=ω4+Q2ω2+Q1ω+Q0, then its complete discrimination system is as follows:(13)D1=4,D2=−Q2,D3=−2Q23+8Q2Q0−9Q12,D4=−Q23Q12+4Q24Q0+36Q2Q12Q0−32Q22Q02−274Q14+64Q03,E2=9Q22−32Q2Q0.Next, according to the complete discriminant system method of polynomials, we will give all single wave solutions of the Kaup–Boussinesq system with beta derivative.Case 1.
WhenD2=D3=D4=0, that is, ωζ12=ω4. The solutions of (1) can be presented as follows:(14)Ω1x,t=−1x−λ/βt+1/Γββ−ζ02+λ24,ϕ1x,t=−1x−λ/βt+1/Γββ−ζ0−λ2.Case 2.
WhenD2<0,D3=D4=0, that is, ωζ12=ω2+u22, where u is the positive real number. The solutions of (1) can be presented as follows:(15)Ω2x,t=−Q22tan2Q22x−λβt+1Γββ−ζ0+λ24,ϕ2x,t=Q22tanQ22x−λβt+1Γββ−ζ0−λ2.
By selecting appropriate parameters, we can easily draw three-dimensional, two-dimensional, density plot, and contour plot of the solutionsϕ2x,t and Ω2x,t of system (1) (see Figures 1 and 2).Figure 1
The graphics ofϕ2x,t in equation (15) at a0=3,a1=0,λ=2,b1=0,andb2=3/2 (a) 3D surface (b) 2D surface (c) Density plot (d) Contour plot.
(a)(b)(c)(d)Figure 2
The graphics ofΩ2x,t in equation (15) at a0=3,a1=0,λ=2,b1=0,andb2=3/2 (a) 3D surface (b) 2D surface (c) Density plot (d) Contour plot.
(a)(b)(c)(d)Case 3.
WhenD2D3<0,D4=0, Fω has a double root and two conjugate complex roots, that is,(16)ωζ12=ω−α2ω−κ2+γ2.
The solutions of (1) can be presented as follows:(17)Ω3x,t=−e±α−κ2+γ2x−λ/βt+1/Γββ−ζ0−τ+α−κ2+γ22−τ2e±α−κ2+γ2x−λ/βt+1/Γββ−ζ0−τ2−12+λ24,ϕ3x,t=e±α−κ2+γ2x−λ/βt+1/Γββ−ζ0−τ+α−κ2+γ22−τe±α−κ2+γ2x−λ/βt+1/Γββ−ζ0−τ2−1−λ2,where τ=α−2κ/α−κ2+γ2; the above solutions are solitary wave solutions.Case 4.
WhenD2>0,D3>0,D4=0, Fω has four real roots, that is,(18)ωζ12=ω−x1ω−x2ω−x3ω−x4,where x1,x2,x3,x4 are real numbers satisfying x1>x2>x3>x4.
The solutions of (1) can be presented as follows:(19)Ω41x,t=−x2x1−x4sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x1x2−x42x1−x4sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x2−x42+λ2/4,ϕ41x,t=x2x1−x4sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x1x2−x4x1−x4sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x2−x4−λ2,(20)Ω42x,t=−x4x2−x3sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x3x2−x42+λ2/4,x2−x3sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x2−x42,ϕ42x,t=x4x2−x3sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x3x2−x4x2−x3sn2x1−x3x2−x4/2x−λ/βt+1/Γββ−ζ0,m−x2−x4−λ2,where m2=x1−x4x2−x3/x1−x3x2−x4.Case 5.
WhenD2D3≥0,D4<0, Fω has two different real roots and two conjugate complex roots, that is,(21)ωζ12=ω−y1ω−y2ω−y32+y42,where y1>y2 and y3,y4 are real numbers.
Making the following transformation:(22)ω=μ1cosψ+μ2μ3cosψ+μ4,where μ1=1/2y1+y2μ3−1/2y1+y2μ4,μ2=1/2y1+y2μ4−1/2y1+y2μ3,μ3=y1−y3−y4/m1,μ4=y1−y3−y4m1,E=y42+y1−y3y2−y3/y4y1+y2,andm1=E±E2+1.
The solutions of (1) can be presented as follows:(23)Ω5x,t=−μ1cn−2y4m1y1−y2/2mm1x−λ/βt+1/Γββ−ζ0,m+μ22μ3cn−2y4m1y1−y2/2mm1x−λ/βt+1/Γββ−ζ0,m+μ42+λ24,ϕ5x,t=μ1cn−2y4m1y1−y2/2mm1x−λ/βt+1/Γββ−ζ0,m+μ2μ3cn−2y4m1y1−y2/2mm1x−λ/βt+1/Γββ−ζ0,m+μ4−λ2,where the solutions are the doubly-periodic solutions of elliptic function.Case 6.
WhenD2D3≤0,D4>0, Fω has two pairs of conjugate complex roots, that is,(24)ωζ12=ω−z12+s12ω−z22+s22,where z1,z2,s1,ands2 are real numbers and s1≥s2>0.
Making the following transformation:(25)ω=σ1tanψ+σ2σ3tanψ+σ4,where σ1=z1σ3+s1σ4,σ2=z1σ4−s1σ3,σ3=−s1−s2/m1,σ4=z1−z2,E=z1−z22+s12+s22/2s1s2,m1=E+E2−1,andm2=m12−1/m12.
The solutions of (1) can be presented as follows:(26)Ω6x,t=−σ1snυx−λ/βt+1/Γββ−ζ0,m+σ2cnυx−λ/βt+1/Γββ−ζ0,m2σ3snυx−λ/βt+1/Γββ−ζ0,m+σ4cnυx−λ/βt+1/Γββ−ζ0,m2+λ24,ϕ6x,t=σ1snυx−λ/βt+1/Γββ−ζ0,m+σ2cnυx−λ/βt+1/Γββ−ζ0,mσ3snυx−λ/βt+1/Γββ−ζ0,m+σ4cnυx−λ/βt+1/Γββ−ζ0,m−λ2,where υ=s2σ32+σ42m12σ32+σ42/σ32+σ42.


## 3. Conclusion

In the paper, we have obtained the traveling wave solution of the Kaup–Boussinesq system with beta derivative arising from water waves by using the complete discriminant system method of polynomial. The rational function solution, the trigonometric function solution, the exponential function solution, and the Jacobian function solution are obtained. Compared with the existing literature [1–6, 24, 25], the paper not only considers the fractional version of the Kaup–Boussinesq system, but also obtains more abundant solutions. It is worth noting that this paper draws three-dimensional and two-dimensional diagrams to explain the propagation of the Kaup–Boussinesq system with beta derivatives in water waves. The Kaup–Boussinesq system with beta derivative arising from water waves is a very important water wave equation. In future research, our work will focus on the dynamic behavior and traveling wave solution of the Kaup–Boussinesq system.



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