# A Note on the Fractional Generalized Higher Order KdV Equation

Authors: Yongyi Gu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We obtain exact solutions to the fractional generalized higher order Korteweg-de Vries (KdV) equation using the complex method. It has showed that the applied method is very useful and is practically well suited for the nonlinear differential equations, those arising in mathematical physics.


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## Body

## 1. Introduction

Nonlinear fractional differential equations (NFDEs) are universally applied in signal processing, electrical networks, acoustics, fluid dynamics, biology, chemistry, physics, etc. For example, the singular behaviours [1–9] and impulsive phenomena [10–19] often exhibit some blow-up properties [20–25] which occur in a lot of complex physical processes. NFDEs have been attracted extensive attention and have been widely investigated [26–38]. Exact solutions of NFDEs play an important role in the study of mathematical physics phenomena. Therefore, seeking exact solutions of NFDEs is an interesting and significant subject.The fractional generalized higher order KdV equation is a useful model. Applying the generalized exp(-Φ(ζ))-expansion method, Luet al. [39] obtained exact solutions of this equation. In this article, we would like to utilize the complex method [40–43] to seek exact solutions to the fractional generalized higher order KdV equation.


## 2. Preliminaries

Letf:R→R,  x→f(x) be a continuous function and ν>0 denote a constant discrete span. Define the operator FW(ν) as follows:(1)FWνfx≔fx+ν,then the fractional difference of f(x) of order μ can be expressed as(2)Δμfx=FW-1μfx=∑n=0∞-1nμnfx+μ-nν,where 0<μ≤1, and its fractional derivative of order μ can be expressed as(3)fμx=limν→0∑n=0∞-1nμnfx+μ-nννμ.The above is expressed as(4)1Γ-μ∫0xx-z-μ-1fzdz,μ<0,1Γ1-μddx∫0xx-z-μfz-f0dz,0<μ<1,fμ-nxn,n≤μ≤n+1,n≥1.Further, Jumarie’s modified Riemann-Liouville derivative [44, 45] is given by(5)Dxμxγ=Γ1+γΓ1+γ-μxγ-μ,γ>0,then its related NFDE is given by(6)fu,ut,ux,Dtμu,Dxμu,Dtγu,Dxγu,…=0,0<μ≤1.Let m∈N≔{1,2,3,…}, rj∈{0,1,2,,…}, j=0,1,…,m, r=(r0,r1,…,rm), and(7)Krwz≔∏j=0mwjzrj,then the degree of Kr[w] is denoted by d(r)≔∑j=0mrj. We define the differential polynomial as(8)Fw,w′,…,wm≔∑r∈HarKrw,in which H is a finite index set and ar are constants. The degree of F(w,w′,…,w(m)) can be denoted by deg⁡F(w,w′,…,w(m))≔maxr∈H{d(r)}.The ordinary differential equation (ODE) is given by(9)Fw,w′,…,wm=cwn+d,where c≠0,d are constants, n∈N.Suppose that the meromorphic solutionsw of (9) have at least one pole. Let p,q∈N and insert the Laurent series(10)wz=∑τ=-q∞βτzτ,β-q≠0,q>0,into (9); if it is determined p different Laurent singular parts:(11)∑τ=-q-1βτzτ,then (9) is said to satisfy the weak p,q condition.Give two complex numbersν1,ν2 such that Imν1/ν2>0, let L be the discrete subset L[2ν1,2ν2]≔{ν∣ν=2c1ν1+2c2ν2,c1,c2∈Z}, and L is isomorphic to Z×Z. Let the discriminant Δ=Δ(b1,b2)≔b13-27b22 and(12)hn≔hnL≔∑ν∈L∖01νn.A meromorphic function℘(z)≔℘(z,g2,g3) with double periods 2ν1,2ν2, which satisfies the equation(13)℘′z2=4℘3z-g2℘z-g3,in which Δ(g2,g3)≠0, g2=60h4, and g3=140h6, is called the Weierstrass elliptic function.If a meromorphic functionς is a rational function of z, an elliptic function, or a rational function of eαz,α∈C, then we say that ς belongs to the class W.In 2009, Eremenkoet al. [46] considered the following m-order Briot-Bouquet equation (BBEq):(14)Fw,wm=∑j=0nFjwwmj=0,where m∈N and Fj(w) are constant coefficient polynomials. For the m-order BBEq, we have the lemma as follows.Lemma 1 (see [41, 47, 48]).
Letm,s,n,p∈N, and deg⁡F(w,w(m))<n, and the m-order BBEq(15)Fw,wm=cwn+dsatisfies weak 〈p,q〉 condition, then the meromorphic solutions w(z) belong to the class W. Suppose that for some values of parameters such solutions w exist, then other meromorphic solutions should form one-parametric family (z-z0), z0∈C. Then each elliptic solution with a pole at z=0 can be expressed as(16)wz=∑i=1s-1∑j=2q-1jβ-ijj-1!dj-2dzj-214℘′z+Ci℘z-Di2-℘z+∑i=1s-1β-i12℘′z+Ci℘z-Di+∑j=2q-1jβ-sjj-1!dj-2dzj-2℘z+β0,in which β-ij are determined by (10), ∑i=1sβ-i1=0, and Ci2=4Di3-g2Di-g3.
Each rational function solutionR(z) can be expressed as(17)Rz=∑i=1s∑j=1qβijz-zij+β0,and it has s(≤p) distinct poles of multiplicity q.
Each simply periodic solutionR(η) is a rational function of η=eαz(α∈C) and can be expressed as(18)Rη=∑i=1s∑j=1qβijη-ηij+β0,and it has s(≤p) distinct poles of multiplicity q.Lemma 2 (see [48, 49]).
Weierstrass elliptic functions℘(z) have the following addition formula:(19)℘z-z0=-℘z0-℘z+14℘′z+℘′z0℘z-℘z02.When g2=g3=0, it can be degenerated to rational functions according to(20)℘z,0,0=1z2.When Δ(g2,g3)=0, it can also be degenerated to simple periodic functions according to(21)℘z,3d2,-d3=2d-3d2coth23d2z.


## 3. Main Results

The fractional generalized higher order KdV equation [39] is given as(22)Dtμ+uux-uuxxx+uxxxxx=0.Substituting traveling wave transform(23)ux,t=wz,z=kx+λtμΓ1+μ,into (22), we get(24)kλw′+kww′-k3ww′′′+k5w5=0,Integrating (24) yields(25)λw+w22+k2ww′′-k22w′2+k4w4+δ=0,where k and λ are constants and δ is the integral constant.Theorem 3.
Ifk≠0, then the meromorphic solutions w(z) of (25) have the following forms.
( I ) The rational function solutions(26)wrz=-30k2z-z02+52,where λ=-5/2, δ=25/8, and z0∈C.
( I I ) The simply periodic solutions(27)wsz=-15k2α22coth2αz-z02+10k2α2+52,where λ=3k4α4-5/2, δ=30k6α6-55k4α4+25/8, and z0∈C.
( I I I ) The elliptic function solutions(28)wdz=-30k2-℘z+14℘′z+D℘z-C2+30k2C+52,where C2=4D3-g2D-g3, g2=2λ+5/36k4, and g3=-100+55λ+12δ/9720k6.Proof.
Substituting (10) into (25) we have p=1,  q=2,  β-2=-30k2,  β-1=0,  β0=5/2,  β1=0,  β2=-2λ+5/24k2, and β3 is an arbitrary constant.
Therefore, (25) satisfies the weak 〈1,2〉 condition. In the following, we will show the meromorphic solutions of (25).
By (17), we infer that the indeterminant rational solutions of (25) are(29)R1z=β11z2+β12z+β10,with pole at z=0.
InsertingR1(z) into (25), we have(30)12β102+λβ10+δ+λβ11+β11β10z+2β12β10+β112+2λβ122z2+2k2β10β11+β12β11z3+3k2β112+12k2β10β12+β1222z4+24k4β11+6k2β11β12z5+120k4β12+4k2β122z6=0,then we get β12=-30k2,  β11=0, and  β10=5/2.
So we can determine that(31)R1z=-30k2z2+52,where λ=-5/2 and δ=25/8.
Therefore the rational solutions of (25) are(32)wrz=-30k2z-z02+52,where λ=-5/2, δ=25/8, and z0∈C.
Letη=eαz. To derive simply periodic solutions, we substitute w=R(η) into (25) to yield(33)k4α4R4η4+6R′′′η3+7R′′η2+R′η-k22αR′η2+k2α2RηR′+η2R′′+R22+λR+δ=0.Substituting(34)R2z=β21η-12+β22η-1+β20into (33), we obtain that(35)R2z=-30k2α2η-12-30k2α2η-1-5k2α22+52,where λ=3k4α4-5/2 and δ=30k6α6-55k4α4+25/8.
Substitutingη=eαz into (35), we achieve simply periodic solutions to (25) with pole at z=0(36)ws0z=-30k2α2eαz-12-30k2α2eαz-1-5k2α22+52=-30k2α2eαzeαz-12-5k2α22+52=-15k2α22coth2αz2+10k2α2+52,where λ=3k4α4-5/2 and δ=30k6α6-55k4α4+25/8.
Thurs, simply periodic solutions of (25) are(37)wsz=-15k2α22coth2αz-z02+10k2α2+52,where λ=3k4α4-5/2, δ=30k6α6-55k4α4+25/8, and z0∈C.
From (16) of Lemma 1, the elliptic solutions of (25) are expressed as(38)wd0z=β-2℘z+β30,with pole at z=0.
Puttingwd0(z) into (25), we get(39)wd0z=-30k2℘z+52,where g2=2λ+5/36k4 and g3=-100+55λ+12δ/9720k6.
So, the elliptic solutions of (25) are(40)wdz=-30k2℘z-z0+52,where z0∈C.
We can apply the addition formula to rewrite it as(41)wdz=-30k2-℘z+14℘′z+D℘z-C2+30k2C+52,where C2=4D3-g2D-g3, g2=2λ+5/36k4, and g3=-100+55λ+12δ/9720k6.
Substitute traveling wave transform into the meromorphic solutionsw(z) of (25) to get traveling wave exact solutions to the fractional generalized higher order KdV equation. So we obtain Theorem 4 as follows.Theorem 4.
Ifk≠0, then traveling wave solutions of (25) have the following forms.
( I ) The rational function solutions(42)wrx,t=wrkx-5ktμ2Γ1+μ,where λ=-5/2, δ=25/8, and z0∈C.
( I I ) The simply periodic solutions(43)wsx,t=wskx+3k5α4-5ktμ2Γ1+μ,where λ=3k4α4-5/2, δ=30k6α6-55k4α4+25/8, and z0∈C.
( I I I ) The elliptic function solutions(44)wdx,t=wdkx+kλtμΓ1+μ,where C2=4D3-g2D-g3, g2=2λ+5/36k4, and g3=-100+55λ+12δ/9720k6.


## 4. Conclusions

In this note, we have used the complex method to construct exact solutions to the mentioned NFDE. Although we do not show that the meromorphic solutions of the fractional generalized higher order KdV equation belong to the classW, we can still obtain the meromorphic solutions to this NFDE and then get its traveling wave exact solutions. The results demonstrate that the applied method is direct and efficient method, which allows us to do tedious and complicated algebraic calculation. We can utilize these ideas to other NFDEs.



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