# Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

Authors: Baoyong Guo; Huanhe Dong; Yong Fang
Journal: Complexity
 (2020)

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## Abstract

In this paper, the time-fractional Fujimoto–Watanabe equation is investigated using the Riemann–Liouville fractional derivative. Symmetry groups and similarity reductions are obtained by virtue of the Lie symmetry analysis approach. Meanwhile, the time-fractional Fujimoto–Watanabe equation is transformed into three kinds of reduced equations and the third of which is based on Erdélyi–Kober fractional integro-differential operators. Furthermore, the conservation laws are also acquired by Ibragimov’s theory.


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## Body

## 1. Introduction

Nowadays, nonlinear partial differential equations (NPDEs) have become more and more significant in fluid mechanics, mathematical physics, oceanography, and so on [1–4]. As we know, NPDEs are usually of integer order and researchers have proposed abundant methods to obtain solutions of NPDEs, including inverse scattering transformation [5], Riemann–Hilbert method [6–10], Hirota direct method [11, 12], Darboux transformation [13, 14], Bäcklund transformation [15], Frobenius integrable decompositions [16, 17], and so on [18–24]. As a generalization, the notions of fractional derivatives are put forward and the classical are Riemann–Liouville and Caputo fractional derivatives. Nonlinear fractional differential equations (NFDEs) are also introduced and have a large number of applications in mathematical physics and automation. The exploration of solutions for NFDEs is a crucial aspect. A variety of methods are presented, for instance, the first integral method [25], functional variable method [26], auxiliary equation method [27], and exponential function method [28].The Lie symmetry method also provides a way to seek solutions for NPDEs and NFDEs [29–32]. This is a way of using known (old) solutions to find new ones. If we obtain a solution for NPDEs or NFDEs, then by using group transformations, the new solutions can be derived. It means that if a NPDE or NFDE has a solution, then it will actually have infinitely many solutions. This method is so effective. Researchers obtain analytical solutions and conservation laws to equations with the help of this method, for example, seventh-order time-fractional Sawada–Kotera–Ito equation, time-fractional fifth-order modified Sawada–Kotera equation, Burridge–Knopoff equation, and so on [33–37].The time-fractional Fujimoto–Watanabe equation [38] is(1)Dtγu=u3uxxx+3u2uxuxx+3u2ux,where Dtγu is the Riemann–Liouville fractional derivative of u=ux,t with respect to time variable t.The Fujimoto–Watanabe equation is one important equation and applied in some fields [38]. Its analytical solutions are obtained, and these solutions can reveal many different natural phenomena [39, 40]. For instance, its traveling wave solutions describe the propagation status of water waves in mathematical physics and oceanography. In geography, specialists can predict natural disasters with the help of its solutions. In fluid mechanics, researchers acquire its period solutions and study its dynamical behaviors [41].This paper is organized as follows. In Section2, we introduce basic concepts and properties about the Riemann–Liouville fractional derivative. In Section 3, symmetry groups are obtained with the help of the Lie symmetry analysis approach. In Section 4, similarity reductions are derived and the time-fractional Fujimoto–Watanabe equation is transformed into three kinds of reduced equations. In Section 5, based on Ibragimov’s theory, the conservation laws of the time-fractional Fujimoto–Watanabe equation are constructed. In Section 6, some conclusions are given.


## 2. Basic Concept and Properties of the Riemann–Liouville Fractional Derivative

Definition 1.
(Riemann–Liouville fractional derivative) (see [42]). Assuming f=fx,t is a real-valued function, where x is the space variable and t is the time variable, then the Riemann–Liouville fractional derivative of f of order γγ>0 is defined as follows:(2)Dtγfx,t=∂γf∂tγ=1Γn−γ∂n∂tn∫0tt−sn−γ−1fx,sds,n−1<γ<n,∂nf∂tn,γ=n,where Γx is the gamma function.
The Riemann–Liouville fractional derivative has many properties, for instance,(3)aDtγtη=Γη+1tη−γΓη−γ+1,η>0,bDtγfgt=fggt⋅Dtγgt=Dgγfgt⋅gtγ,cDtγft⋅gt=Dtγft⋅gt+ft⋅Dtγgt,where fandg are real-valued functions, fg=df/dg, and gt=dg/dt
From Definition1, we find that the Riemann–Liouville fractional derivative is a generalized form of the ordinary integer-order derivative. Property (b) is the composite rule of the Riemann–Liouville fractional derivative. Property (c) is the Leibniz rule of the Riemann–Liouville fractional derivative.


## 3. Symmetry Group of the Time-Fractional Fujimoto–Watanabe Equation

In this section, we seek symmetry groups for the time fractional Fujimoto–Watanabe equation with the help of the Lie symmetry analysis method.The general time-fractional differential equation is as follows:(4)Dtγu=Lx,t,u,ux,uxx,uxxx,…,where u=ux,t, the subscripts represent partial derivatives, i.e., ux=∂u/∂x,uxx=∂2u/∂x2,uxxx=∂3u/∂x3,…, and the corresponding one-parameter transformations are(5)x¯=x+ελx,t,u+Oε2,t¯=t+εμx,t,u+Oε2,u¯=u+εφx,t,u+Oε2,Dt¯γu¯=Dtγu+εφγx,t,u+Oε2,∂u¯∂x¯=∂u∂x+εφxx,t,u+Oε2,∂2u¯∂x¯2=∂2u∂x2+εφxxx,t,u+Oε2,∂3u¯∂x¯3=∂3u∂x3+εφxxxx,t,u+Oε2,…where ε≪1 is an infinitesimal parameter and λ=λx,t,u, μ=μx,t,u, and φ=φx,t,u are real-valued infinitesimal functions with respect to variables x,t, and u. φγ=φγx,t,u, φx=φxx,t,u, φxx=φxxx,t,u, and φxxx=φxxxx,t,u are extended infinitesimal functions. These extended infinitesimal functions can be determined using the following expressions:(6)φγ=Dtγφ+λDtγux−Dtγλux+DtγDtμu−Dtγ+1μu+μDtγ+1u,(7)φx=Dxφ−uxDxλ−utDxμ,φxx=Dxφx−uxxDxλ−uxtDxμ,φxxx=Dxφxx−uxxxDxλ−uxxtDxμ,…where Dtγ represents the total Riemann–Liouville fractional derivative with respect to t, Dx represents the total derivative with respect to x, and Dt represents the total derivative with respect to t, i.e.,(8)Dx=∂∂x+ux∂∂u+uxx∂∂ux+uxxx∂∂uxx+⋯,Dt=∂∂t+ut∂∂u+uxt∂∂ux+uxxt∂∂uxx+⋯.In order to obtain the symmetry groups, let the infinitesimal generator be as follows:(9)V=λx,t,u∂∂x+μx,t,u∂∂t+φx,t,u∂∂u,where V is sometimes called vector field.The third-order prolongation ofV is(10)pr3V=λ∂∂x+μ∂∂t+φ∂∂u+φγ∂∂utγ+φx∂∂ux+φxx∂∂uxx+φxxx∂∂uxxx,where utγ=Dtγu.Assume(11)Δx,t,u,u3=Dtγu−Lx,t,u,ux,uxx,uxxx,…=Dtγu−u3uxxx−3u2uxuxx−3u2ux.In order to satisfy the invariance condition of Lie symmetry,pr3V needs to meet the following identity:(12)pr3VΔΔ=0=0.By direct calculation, we obtain(13)ϕγ−u3ϕxxx−3u2uxϕxx−3u2uxx+3u2ϕx−3u2uxxx+6uuxuxx+6uuxϕΔ=0=0.From equation (7), we have(14)φx=φx+φuux−λxux−λuux2−μxut−μuuxut,φxx=φxx+2ϕxu−λxxux+φuu−2λxuux2−λuuux3+φu−2λxuxx−3λuuxuxx−μxxut−2μxuxt−2μxuuxut−2μuuxuxt−μuuxxut−μuuu2ut,φxxx=φxxx+3φxxu−λxxxux+3φxuu−λxxuux2+φuuu−3λxuuux3−λuuuux4+3φxu−λxxuxx+3φuu−3λxuuxuxx−6λuuux2uxx+φu−3λxuxxx−3λuuxx2−μxxxut−3μxxuuxut−3μxxuxt−6μxuuxuxt−3μxuxxt−3μxuuux2ut−3μxuuxxut−3μuuux2uxt−3μuuxxuxt−3μuuxuxxt−3μuuuxuxxut−μuuxxxut−μuuuux3ut−4λuuxuxxx.Definition 2.
(Generalized Leibniz rule of the Riemann–Liouville fractional derivative). Assumingu=ut and v=vt are real-valued functions, then the generalized Leibniz rule is(15)Dtγut⋅vt=∑n=0∞γnDtγ−nut⋅Dtnvt,γ>0,where(16)γn=−1n−1γΓn−γΓ1−γΓn+1.Definition 3.
(Generalized composite (chain) rule). Assumingu=ut and v=vt are real-valued functions, then the generalized composite rule is(17)dnuvtdtn=∑i=0n∑j=0iij1i!−vtjdndtnvti−jdiuvdvi.
Because of equation (11), equation (6) can be rewritten as(18)φγ=Dtγφ−γDtμ⋅∂γu∂tγ−∑n=1∞γnDtnλ⋅Dtγ−nux−∑n=1∞γn+1Dtn+1μ⋅Dtγ−nu.
According to equations (11) and (13), we obtain(19)Dtγφ=∂γφ∂tγ+φu∂γu∂tγ−u∂γφu∂tγ+∑n=1∞γn∂γφu∂tγ⋅Dtγ−nu+ϕ,where(20)ϕ=∑n=2∞∑i=2n∑j=2i∑kj−1γnnijk1j!tn−γΓn+1−γ−uk∂iφ∂tiuj−k∂n−i+jφ∂tn−i∂uj.
According to equations (3)b and (3)c, equation (14) can be rewritten as(21)φγ=∂γφ∂tγ+φu−γDtμ∂γu∂tγ−u∂γφu∂tγ+ϕ+∑n=1∞γn∂nφu∂tn−γn+1Dtn+1μDtγ−nu−∑n=1∞γnDtnλ⋅Dtγ−nux.
Substituting equations (14) and (21) into equation (13) and equating the coefficients of all powers of partial derivatives of u to 0, we obtain a set of determining equations as follows:(22)λt=λu=0,μx=μu=0,φuu=0,φu−2λx=0,3λx−γμt−3φu=0,∂γφ∂tγ−u∂γφu∂tγ−u3φxxx−3u2φx=0,γn∂nφu∂tn−γn+1Dtn+1μ=0.
Solving equation (22), we acquire the solution as follows:(23)λx,t,u=γC1x+C2,μx,t,u=−3C1t,φx,t,u=2γC1u,where C1andC2 are arbitrary constants.
Based on the above results, the infinitesimal generator can be rewritten as(24)V=γC1x+C2∂∂x−3C1t∂∂t+2γC1u∂∂u.
If we let(25)V1=γx∂∂x−3t∂∂t+2γu∂∂u,V2=∂∂x,then V can also be rewritten as(26)V=C1V1+C2V2.
Introducing Lie bracket operation, i.e., for arbitrary vector fieldsA and B, A,B=AB−BA.
From Table1, we can find V1andV2 are closed obviously. Consequently, the symmetry groups of the time-fractional Fujimoto–Watanabe equation can be spanned by V1,V2.Table 1
Lie bracket operation ofV1 and V2.
V
i
,
V
j
V
1
V
2
V
1
0
−
γ
V
2
V
2
γ
V
2
0


## 4. Similarity Reductions for the Time-Fractional Fujimoto–Watanabe Equation

In this section, we investigate the similarity reductions for the time-fractional Fujimoto–Watanabe equation. Thus, we can obtain reduced equations.Because the symmetry groups are spanned byV1 and V2, we need to discuss in two cases:Case 1: forV=∂/∂x, we need to solve the following system of equations:(27)dx1=dt0=du0.From equation (27), we arrive at t and u are similarity variables. We can assume the solution of equation (1) has the form u=ft.Substitutingu=ft into equation (1), we have the following reduced equation:(28)Dtγu=0.Solving equation (28), we obtain the group invariant solutions:(29)u=ux,t=ft=ltγ−1,wherel is an arbitrary constant.Case 2 (method 1): forV=γx∂/∂x−3t∂/∂t+2γu∂/∂u, similar to Case 1, we also need to solve the following system of equations:(30)dxγx=dt−3t=du2γu.From equation (30), we arrive at similarity variables as follows:(31)η=tx3/γ,f=ux−2.We can assume the solution of equation (1) has the form(32)u=x2fη=x2ftx3/γ.Substituting equation (32) into equation (1), we have the following reduced equation:(33)Dηγfη=12f4+6f3+27γ3+81γ2+78γηf3f′+81γ3+81γ2η2f3f″+27γ3η3f3f″′+81γ3+81γ2η2f2f′2+81γ3η3f2f′f″+9γηf2f′,whereη=tx3/γ.For Case 2, we have another method to obtain the reduced equation. We need to use Erdélyi–Kober fractional integro-differential operators.Case 2 (method 2): forV=γx∂/∂x−3t∂/∂t+2γu∂/∂u, we also have the following similarity variables:(34)η=xtγ/3,f=ut2/3γ.Then, the solution of equation (1) has the form(35)u=t−2/3γfη=t−2/3γfxtγ/3.Assumingn−1<γ<n and substituting equation (35) into Definition 1, we have(36)Dtγux,t=∂γu∂tγ=∂n∂tn1Γn−γ∫0tt−ξn−γ−1ξ−2/3γfxξγ/3dξn−1<γ<n.Introducing variable transformation,(37)ζ=tξ.Thus,(38)dξ=−tζ2dζ.Substituting equations (37) and (38) into equation (36), we derive(39)Dtγux,t=∂γu∂tγ=∂n∂tntn−5/3γ1Γn−γ∫1∞ζ−1n−γ−1ζ−n−5/3γ+1fζ−γ/3ηdζ=∂n∂tntn−5/3γK−3/γ1−2/3γ,n−γfη=∂n−1∂tn−1∂∂ttn−5/3γK−3/γ1−2/3γ,n−γfη=∂n−1∂tn−1tn−5/3γ−1n−53γ+γ3ηddηK−3/γ1−2/3γ,n−γfη,where the definition of the Erdélyi–Kober fractional integral operator is as follows:(40)Kki,jfη=1Γj∫1∞ξ−1j−1ξ−i+jfξ1/kηdξ,j>0,fη,j=0.Repeating the same proceduren−1 times, we derive(41)Dtγux,t=∂γu∂tγ=∂n−1∂tn−1tn−5/3γ−1n−53γ+γ3ηddηK−3/γ1−2/3γ,n−γfη=…=t−5/3γ∏l=0n−11−53γ+l+γ3ηddηK−3/γ1−2/3γ,n−γfη=t−5/3γP−3/γ1−5/3γ,γfη,where the definition of the Erdelyi–Kober fractional differential operator is as follows:(42)Pki,jfη=∏l=0n−1i+l−1kηddηKki+j,n−jfη,n=j+1,j∉ℕ,j,j∈ℕ.Consequently, the time-fractional Fujimoto–Watanabe equation is transformed into the following fractional ordinary differential equation:(43)P−3/γ1−5/3γ,γfη=f3f″′+3f2f′f″+3f2f′,where η=xtγ/3.


## 5. Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation

Conservation laws have always been one significant aspect of the investigation on NPDEs and NFDEs. In this section, we construct conservation laws for the time-fractional Fujimoto–Watanabe equation. In this process, Ibragimov’s theory plays a key role [43].Assume(44)vx,t=Dtγu−u3uxxx−3u2uxuxx−3u2ux.Introduce a new dependent variableξ=ξx,t and a formal Lagrangian:(45)ℱ=ξv=ξDtγu−u3uxxx−3u2uxuxx−3u2ux.We can present action integral of equation (45) in the following form:(46)∫0T∫baℱx,t,u,ξ,Dtγu,ux,uxx,uxxxdxdt.The corresponding adjoint equation of equation (44) is obtained as follows:(47)v∗=δℱδu=Dtγ∗ξ+6uux2ξx+6u2uxξxx+3u2ξx+6u2uxxξx+u3ξxxx=0,where(48)δδu=∂∂u+Dtγ∗∂∂utγ+∑n=1∞−1nDi1,…,Din∂∂ui1,…,in=∂∂u+Dtγ∗∂∂utγ−Dx∂∂ux+Dx2∂∂uxx−Dx3∂∂uxxx,D
t
γ
∗ is the adjoint operator of Dtγ, and it is defined as follows:(49)Dtγ∗=LTn−γDtn,where(50)LTn−γux,t=−1nΓn−γ∫tTux,ζξ−t1+γ−ndζ,n−1<γ<n.In order to construct conservation laws, we need to find conservation vectorV♯=Vx,Vt which satisfies the following conservation law equation:(51)DxVx+DtVt1=0,where Dx and Dt are the total derivatives with respect to x and t.According to Ibragimov’s theory, conservation vectors can be acquired as follows:(52)Vt=μℱ+∑l=0n−1−1lDtγ−1−lℛDtl∂ℱ∂utγ−−1nJℛ,Dtn∂ℱ∂utγ,Vx=λℱ+ℛ∂ℱ∂ux−Dx∂ℱ∂uxx+DxDx∂ℱ∂uxxx+Dxℛ∂ℱ∂uxx−Dx∂ℱ∂uxxx+DxDxℛ∂ℱ∂uxxx,where n−1<γ<n and ℛ=φ−λux−μut s.t. infinitesimal generator V=λ∂/∂x+μ∂/∂t+φ∂/∂u, and J operator is defined by(53)Ju,v=1Γn−γ∫0t∫tTux,ζvx,λλ−ζ1+γ−ndλdζ.Case 1: infinitesimal generatorV1=γx∂/∂x−3t∂/∂t+2γu∂/∂u, λ=γx, μ=−3t, φ=2γu, and ℛ1=2γu−γxux+3tut. Then, we have(54)Vt=−3tℱ+∑l=0n−1−1lDtγ−1−lℛ1Dtl∂ℱ∂utγ−−1nJℛ1,Dtn∂ℱ∂utγ=∑l=0n−1−1lDtγ−1−l2γu−γxux+3tutDtlξ−−1nJ2γu−γxux+3tut,Dtnξ,Vx=γxℱ+ℛ1∂ℱ∂ux−Dx∂ℱ∂uxx+DxDx∂ℱ∂uxxx+Dxℛ1∂ℱ∂uxx−Dx∂ℱ∂uxxx+DxDxℛ1∂ℱ∂uxxx=2γu−γxux+3tut−3u2uxxξ−3u2ξ−Dx−3u2uxξ+Dxx−u3ξ+Dx2γu−γxux+3tut−3u2uxξ−Dx−u3ξ+DxDx2γu−γxux+3tut−u3ξ=−2γu−γxux+3tut3u2ξ+3u2uxξx+3u2uxxξ+u3ξxx+γuxx−γxuxx+3tuxtu3ξx+γxuxxx−3tuxxtu3ξ.Case 2: infinitesimal generatorV2=∂/∂x, λ=1, μ=0, φ=0, and ℛ2=−ux. Then, we have(55)Vt=∑l=0n−1−1lDtγ−1−lℛ2Dtl∂ℱ∂utγ−−1nJℛ2,Dtn∂ℱ∂utγ=∑l=0n−1−1l+1Dtγ−1−luxDtlξ−−1n+1Jux,Dtnξ,Vx=ℱ+ℛ2∂ℱ∂ux−Dx∂ℱ∂uxx+DxDx∂ℱ∂uxxx+Dxℛ2∂ℱ∂uxx−Dx∂ℱ∂uxxx+DxDxℛ2∂ℱ∂uxxx=−ux−3u2uxxξ−3u2ξ−Dx−3u2uxξ+Dxx−u3ξ+Dx−ux−3u2uxξ−Dx−u3ξ+DxDx−ux−u3ξ=ux3u2ξ+3u2uxξx+3u2uxxξ+u3ξxx−uxxu3ξx+uxxxu3ξ=3u2uxξ+3u2uxuxxξ+3u2ux2ξx+u3uxξxx−u3uxxξx+u3uxxxξ.


## 6. Conclusions

In this investigation, we explore the time-fractional Fujimoto–Watanabe equation in the perspective of the Riemann–Liouville derivative. We acquire symmetry groups and similarity reductions by means of the Lie symmetry approach. Meanwhile, three kinds of reduced equations have been obtained. For the second infinitesimal generator, we use two distinct methods to derive two different reduced equations, one of which is based on fractional integral operators and fractional differential operators. At last, with the help of Ibragimov’s theory, the conservation laws are constructed. These results reveal this approach is very effective to obtain reduced equations for the fractional differential equations.



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