# The Bäcklund Transformations and Abundant Exact Explicit Solutions for a General Nonintegrable Nonlinear Convection-Diffusion Equation

Authors: Yong Huang; Yadong Shang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The Bäcklund transformations and abundant exact explicit solutions for a class of nonlinear wave equation are obtained by the extended homogeneous balance method. These solutions include the solitary wave solution of rational function, the solitary wave solutions, singular solutions, and the periodic wave solutions of triangle function type. In addition to rederiving some known solutions, some entirely new exact solutions are also established. Explicit and exact particular solutions of many well-known nonlinear evolution equations which are of important physical significance, such as Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system, are obtained as special cases.


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## Body

## 1. Introduction

The existence of solitary wave solutions and periodic wave solutions is an important question in the study of nonlinear evolution equations. The methods of finding such solutions for integrable equations are well known: the solitary wave solutions can be found by inverse scattering transformation [1] and the Hirota bilinear method [2], and the periodic solutions can be represented by sums of equally spaced solitons represented by sech-function [3, 4]. Weiss et al. developed the singular manifold method to introduce the Painlevé property in the theory of partial differential equations [5]. The beauty of the singular manifold method is that this expansion for a nonlinear PDE contains a lot of information about this PDE. For an equation that possesses the Painlevé property the singular manifold method leads to the Bäcklund transformation, the Lax pair, and Miura transformations and makes connections to the Hirota bilinear method, Laplace-Darboux transformations [6]. Most nonlinear nonintegrable equations do not possess the Painlevé property; that is, they are not free from “movable” critical singularities. For some nonintegrable nonlinear equations it is still possible to obtain single-value expansions by putting a constraint on the arbitrary function in the Painlevé expansion. Such equations are said to be partially integrable, and Weiss [7] conjectured that these systems can be reduced to integrable equations. Another treatment of the partially integrable systems was offered by Hietarinta [8] by the generalization of the Hirota bilinear formalism for nonintegrable systems. He conjectured that all completely integrable PDEs can be put into a bilinear form. There are also nonintegrable equations that can be put into the bilinear form and then the partial integrability is associated with the levels of integrability defined by the number of solitons that can be combined to an N-soliton solution. Partial integrability then means that the equation allows a restricted number of multisoliton solutions. In [9] Berloff and Howard suggested joining these treatments of the partial nonintegrability and using the Painlevé expansion truncated before the “constant term” level as the transform for reducing a nonintegrable PDE to a multilinear equation.The Bäcklund transformation is not only a useful tool to obtain exact solutions of some soliton equation from a trivial “seed” but also related to infinite conservation laws and inverse scattering method [1]. In [10–12], Wang Mingliang proposed the homogeneous balance method—an effective method solving nonlinear partial differential equations. Fan and Zhang extended the homogeneous balance method and proposed an approach to obtain Bäcklund transformation for the nonlinear evolution equations [13]. In a recent paper [14], Shang obtained the Bäcklund transformation, a Lax pair, and some new explicit exact solutions of Hirota-Satsuma SWW equation (2.3) by means of the Bäcklund transformations and the extension of the hyperbolic function method presented in [15].In this paper we investigate a general nonintegrable nonlinear convection-diffusion equation(1.1)ut-uxx+αuux+βu+γu2+δu3=0,
where α, β, γ, and δ are arbitrary real constants. Equation (1.1) include many well-known nonlinear equations that are with applied background as special examples, such as Burgers equation, Kolmogorov-Petrovskii-Piskunov equation, FitzHugh-Nagumo equation, Burgers-Huxley equation, Chaffee-Infante reaction-diffusion equation, Newell-Whitehead equation, Fisher equation, Fisher-Burgers equation, and an isothermal autocatalytic system. The extended homogeneous balance method is applied for a reliable treatment of the nonintegrable nonlinear equation (1.1). Some Bäcklund transformations and abundant explicit exact particular solutions of the nonintegrable nonlinear equation (1.1) are obtained by means of the extended homogeneous balance method. Some explicit exact solutions obtained here have more general form than some known solutions, and some explicit exact solutions obtained here are entirely new solutions.


## 2. Bäcklund Transformations for the Nonintegrable Nonlinear Wave Equation

According to the extended homogeneous balance method, we suppose that the solution of (1.1) is of the form(2.1)u(x,t)=f′(ϕ)ϕx+u1(x,t),
where f, ϕ are two functions to be determined and u1(x,t) is a solution of (1.1).From (2.1), we have(2.2)ut=f′′(ϕ)ϕxϕt+f′(ϕ)ϕxt+u1t,(2.3)ux=f′′(ϕ)ϕx2+f′(ϕ)ϕxx+u1x,(2.4)uxx=f′′′(ϕ)ϕx3+3f′′(ϕ)ϕxϕxx+f′(ϕ)ϕxxx+u1xx,(2.5)u2=(f′)2(ϕ)ϕx2+2f′ϕxu1(x,t)+u12(x,t),u3=(f′)3(ϕ)ϕx3+3(f′)2ϕx2u1(x,t)+3f′ϕxu12(x,t)+u13(x,t).
Substituting (2.1)–(2.5) into the left side of (1.1) and collecting all terms with ϕx3, we obtain(2.6)ut-uxx+αuux+βu+γu2+δu3=(αf′′f′-f′′′+δ(f′)3)ϕx3+[f′′ϕxϕt-3f′′ϕxϕxx+αf′′ϕx2u1+α(f′)2ϕxϕxx+γ(f′)2ϕx2+3δ(f′)2ϕx2u1(x,t)]+f′[ϕxt-ϕxxx+αϕxxu1+αϕxu1x+βϕx+2γϕxu1+3δϕxu12]+[u1t-u1xx+αu1u1x+βu1+γu12+δu13]=0.
Setting the coefficient of ϕx3 in (2.6) to be zero, we obtain an ordinary differential equation for f(2.7)αf′′f′-f′′′+δ(f′)3=0,
which has a solution(2.8)f(ϕ)=λln⁡(ϕ),
where λ=(α±α2+8δ)/2δ. And then(2.9)(f′)2=(-λ)f′′.
By virtue of (2.7)–(2.9), (2.6) becomes(2.10)ut-uxx+αuux+βu+γu2+δu3=f′′[ϕxϕt-3ϕxϕxx+αϕx2u1-αλϕxϕxx-γλϕx2-3δλϕx2u1(x,t)]+f′[ϕxt-ϕxxx+αϕxxu1+αϕxu1x+βϕx+2γϕxu1+3δϕxu12]+[u1t-u1xx+αu1u1x+βu1+γu12+δu13]=0.
Setting the coefficients of f′′,f′,f0 to be zero, respectively, it is easy to see from (2.10) that(2.11)ϕt+(αu1-γλ-3δλu1)ϕx-(3+αλ)ϕxx=0,(2.12)ϕxt-ϕxxx+αϕxxu1+αϕxu1x+βϕx+2γϕxu1+3δϕxu12=0,(2.13)u1t-u1xx+αu1u1x+βu1+γu12+δu13=0.
Substituting (2.8) into (2.1), we obtain a Bäcklund transformation(2.14)u(x,t)=λϕxϕ+u1(x,t),
where λ=(α±α2+8δ)/2δ,ϕ,u1 satisfy (2.11)–(2.13). Substituting a seed solution u1(x,t) of (1.1) into linear equations (2.11) and (2.12), then solving (2.11) and (2.12), we can get a new solution of (1.1) from (2.14). Thus we can obtain infinite solutions of (1.1) by the Bäcklund transformation (2.14) and (2.11)-(2.12) from a seed solution of (1.1).Takingu1=0, by (2.11)–(2.14), we obtain a transformation(2.15)u(x,t)=λϕxϕ,
that transforms (1.1) into linear equations(2.16)ϕt-γλϕx-(3+αλ)ϕxx=0,ϕt-ϕxx+βϕ=E,
where λ=(α±α2+8δ)/2δ, E is an arbitrary constant.Takingu1=(-γ±Δ)/2δ, from (2.11)–(2.14) we obtain another transformation(2.17)u(x,t)=-γ±Δ2δ+λϕxϕ.
Equation (1.1) can be solved by solving two linear equations
(2.18)ϕt+(αu1-γλ-3δλu1)ϕx-(3+αλ)ϕxx=0,ϕxt-ϕxxx+αϕxx+βϕx+2γϕxu1+3δϕxu12=0,
where u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ.


## 3. Exact Explicit Solutions to (1.1)

In this section we want to obtain abundant exact explicit particular solutions of (1.1) from the Bäcklund transformation (2.14) and a trivial solution of (1.1).Noting the homogeneous property of (2.16) we can expect that ϕ in (2.16) is of the form(3.1)ϕ(x,t)=Asinh⁡(kx+ωt+ξ0)+Bcosh⁡(kx+ωt+ξ0)+C
with A, B, C, k, ω, and ξ0 constants to be determined. Substituting (3.1) into (2.16), one gets a set of nonlinear algebraic equation(3.2)Aω-γλAk-(3+αλ)Bk2=0,Bω-γλBk-(3+αλ)Ak2=0,Aω-Bk2+βB=0,Bω-Ak2+βA=0,βC=E.
Solving (3.2), we have the following.Case 1.
A=B, C=E/β, ω=k2-β, and k is a root of second-order algebraic equation (2+αλ)k2+γλk+β=0.Case 2.
A=-B, C=E/β, ω=β-k2, and k is a root of second-order algebraic equation (2+αλ)k2-γλk+β=0.Thus we obtain the following explicit exact solutions of (1.1) given by(3.3)u(x,t)=λkexp⁡(kx+ωt+ξ0)exp⁡(kx+ωt+ξ0)+C,
where λ=(α±α2+8δ)/2δ, ω=k2-β, k is a root of second-order algebraic equation (2+αλ)k2+γλk+β=0, C≠0, and ξ0 are arbitrary constants.We can also obtain the following explicit exact solutions of (1.1) given by(3.4)u(x,t)=λk1Cexp⁡(kx+ωt+ξ0)-1,
where λ=(α±α2+8δ)/2δ, ω=β-k2, k is a root of second-order algebraic equation (2+αλ)k2-γλk+β=0, C≠0, and ξ0 are arbitrary constants.By direct computation, we readily obtain the following two useful formulas:(3.5)exp⁡(ξ)C+exp⁡(ξ)={1,forC=0,12[tanh12(ξ-ln⁡C)+1],forC>0,12[coth⁡12(ξ-ln⁡(-C))+1],forC<0,(3.6)1Cexp⁡(ξ)-1={-1,forC=0,12[coth⁡12(ξ+ln⁡C)-1],forC>0,12[tanh12(ξ+ln⁡(-C))-1],forC<0,
where C is arbitrary.Thanks to the two formulas (3.5) and (3.6), we can assert.The solutions (3.3) ((3.4), resp.) are soliton solutions of kink type in the case of C>0 (C<0, resp.).The solutions (3.3) ((3.4), resp.) are soliton-like solutions of singular type in the csae of C<0 (C>0, resp.).Analogously, we assume thatϕ in (2.16) is of the form(3.7)ϕ(x,t)=Asin⁡(kx+ωt+ξ0)+Bcos⁡(kx+ωt+ξ0)+C
with A, B, C, k, ω, and ξ0 constants to be determined. Substituting (3.7) into (2.16), one gets a set of nonlinear algebraic equation(3.8)Aω-γλAk+(3+αλ)Bk2=0,-Bω+γλBk+(3+αλ)Ak2=0,Aω+Bk2+βB=0,-Bω+Ak2+βA=0,βC=E.
Solving (3.8), we have the following.Case 1.
A=Bi, C=E/β, ω=(k2+β)i, and k is a root of second order algebraic equation (2+αλ)k2-γλki-β=0, i=-1.Case 2.
A=-Bi, C=E/β, ω=-i(k2+β), and k is a root of second order algebraic equation (2+αλ)k2+γλki-β=0, i=-1.According to the result of Case1, from (2.15) and (3.7), we obtain the exact explicit solutions of (1.1) given by(3.9)u(x,t)=λkiexp⁡(iξ)exp⁡(iξ)+C,
where λ=(α±α2+8δ)/2δ, ξ=kx+ωt+ξ0, ω=i(k2+β), k is a root of second-order algebraic equation (2+αλ)k2-γλki-β=0, i=-1.By the result of Case2 and (2.15), (3.7), we can obtain the following exact explicit solutions of (1.1) given by(3.10)u(x,t)=(-λki)11+Cexp⁡(iξ),
where λ=(α±α2+8δ)/2δ, ξ=kx+ωt+ξ0, ω=(-i)(k2+β), k is a root of second-order algebraic equation (2+αλ)k2+γλki-β=0, i=-1.Analogously, we have the following two useful formulas:(3.11)exp⁡i(ξ)C+exp⁡i(ξ)={1,forC=0,12[itan12(ξ+iln⁡C)+1],forC>0,12[-icot12(ξ+iln⁡(-C))+1],forC<0,(3.12)1Cexp⁡i(ξ)+1={1,forC=0,12[1-itan12(ξ-iln⁡C)],forC>0,12[1+icot12(ξ-iln⁡(-C))],forC<0.Due to the formula (3.11), we have from(3.9)(3.13)u(x,t)={-λk2tan[12(kx+ωt+ξ0+iln⁡(C))]+λki2,forC>0,λk2cot[12(kx+ωt+ξ0+iln⁡(-C))]+λki2,forC<0,
where λ=(α±α2+8δ)/2δ, ξ=kx+ωt+ξ0, ω=i(k2+β), k is a root of second-order algebraic equation (2+αλ)k2-γλki-β=0, i=-1.Owing to the formula (3.12), we have from  (3.10)(3.14)u(x,t)={-λk2tan[12(kx+ωt+ξ0-iln⁡(C))]-λki2,forC>0,λk2cot[12(kx+ωt+ξ0-iln⁡(-C))]-λki2,forC<0,
where λ=(α±α2+8δ)/2δ, ξ=kx+ωt+ξ0, ω=(-i)(k2+β), k is a root of second-order algebraic equation (2+αλ)k2+γλki-β=0, i=-1.By virtue of the homogeneous property of (2.18), we can expect that ϕ is of the linear function form(3.15)ϕ(x,t)=kx+ωt+ξ0,
with k and ω, ξ0 constants to be determined. Substituting (3.15) into (2.18), we find that (3.15) satisfies (2.18), provided that k and ω satisfy the following algebraic equations:(3.16)ω+(αu1-γλ-3δλu1)k=0,βk+2γku1+3δku12=0,
where u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ. Solving (3.16), we obtain that(3.17)ω=α∓α2+8δ4δγk,k=arbitraryconstant,u1=-γ2δ,
provided that coefficients β, γ, and δ of (1.1) satisfy condition γ2=4βδ.Substituting (3.15) with (3.17) into (2.17), we obtain the exact particular solutions of  (1.1)(3.18)u(x,t)=-γ2δ+α±α2+8δ2δ1x+((α∓α2+8δ)/4δ)γt+ξ0.Now we suppose that (2.18) has solutions of the form (3.1) substituting (3.1) into (2.18), one gets a set of algebraic equations:
(3.19)Aω+(αu1-γλ-3δλu1)Ak-(3+αλ)Bk2=0,Bω+(αu1-γλ-3δλu1)Bk-(3+αλ)Ak2=0,Akω-Bk3+αAk2+βBk+2γu1Bk+3δu12Bk=0,Bkω-Ak3+αBk2+βAk+2γu1Ak+3δu12Ak=0.
In order to obtain nontrivial solutions of (1.1), we need to require that  k, ω are all nonzero constants. Solving (3.19), one gets the following solutions.Case 1.
One has(3.20)A=B,C=arbitraryconstant,ω=k2-αk-β-2γu1-3δu12,orω=(3+αλ)k2+(γλ+3δλu1-αu1)k,
where k is a root of second-order algebraic equation (2+αλ)k2+(γλ+3δλu1+α-αu1)k+β+2γu1+3δu12=0, u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ.Case 2.
One has(3.21)A=-B,C=arbitraryconstant,ω=β+2γu1+3δu12-k2-αk,orω=(γλ+3δλu1-αu1)k-(3+αλ)k2,
where k is a root of second-order algebraic equation (2+αλ)k2+(αu1-γλ-3δλu1-α)k+β+2γu1+3δu12=0, u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ.By Case1, we obtain the exact solutions of the (1.1) from (2.17),  (3.1)(3.22)u(x,t)=-γ±Δ2δ+λkexp⁡(kx+ωt+ξ0)exp⁡(kx+ωt+ξ0)+C,
where λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ, ω=k2-αk-β-2γu1-3δu12, k is a root of second-order algebraic equation (2+αλ)k2+(γλ+3δλu1+α-αu1)k+β+2γu1+3δu12=0, u1=(-γ±Δ)/2δ, ξ0, C≠0 are arbitrary constants.According to the result of Case2 and (2.17), (3.1), one obtain the other exact solutions(3.23)u(x,t)=-γ±Δ2δ+λkexp⁡(kx+ωt+ξ0)Cexp⁡(kx+ωt+ξ0)-1,
where λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ, ω=β+2γu1+3δu12-k2-αk, k is a root of second-order algebraic equation (2+αλ)k2+(αu1-γλ-3δλu1-α)k+β+2γu1+3δu12=0, u1=(-γ±Δ)/2δ, ξ0, C≠0 are arbitrary constants.According to formulas (3.5), (3.6), we can get multiple new soliton solutions of kink type and multiple new soliton-like solutions of singular type from (3.22) and (3.23).Analogously, we assume that (2.18) has solutions of the form (3.7); substituting (3.7) into (2.18), one gets a set of algebraic equations(3.24)Aω+(αu1-γλ-3δλu1)Ak+(3+αλ)Bk2=0,-Bω-(αu1-γλ-3δλu1)Bk+(3+αλ)Ak2=0,-Akω-Bk3-αAk2-βBk-2γu1Bk-3δu12Bk=0,-Bkω+Ak3-αBk2+βAk+2γu1Ak+3δu12Ak=0.
In order to obtain a nontrivial solution of (1.1), we also need to assume that k,ω are all nonzero constants. Solving (3.24), we obtain the following.Case 1.
One has(3.25)A=Bi,C=arbitraryconstant,ω=ik2-αk+i(β+2γu1+3δu12),orω=(3+αλ)ik2+(γλ+3δλu1-αu1)k,
where k is a root of second-order algebraic equation (2+αλ)ik2+(γλ+3δλu1+α-αu1)k-i(β+2γu1+3δu12)=0, u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ.Case 2.
One has(3.26)A=-Bi,C=arbitraryconstant,ω=-ik2-αk-i(β+2γu1+3δu12),orω=-(3+αλ)ik2+(γλ+3δλu1-αu1)k,
where k is a root of second-order algebraic equation (2+αλ)ik2-(γλ+3δλu1+α-αu1)k-i(β+2γu1+3δu12)=0, u1=(-γ±Δ)/2δ, λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ.Collecting (2.17), (3.7), (3.25), and (3.26), we obtain the following explicit exact periodic traveling wave solutions(3.27)u(x,t)=-γ±Δ2δ+iλkexp⁡(iξ)exp⁡(iξ)+C,
where λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ, ξ=kx+ωt+ξ0, ω=ik2-αk+i(β+2γu1+3δu12), k is a root of second-order algebraic equation (2+αλ)ik2 + (γλ+3δλu1 + α-αu1)k-i(β+2γu1 + 3δu12)=0, u1=(-γ±Δ)/2δ, ξ0, C≠0 are arbitrary constants,(3.28)u(x,t)=-γ±Δ2δ-iλkCexp⁡(iξ)exp⁡(iξ)+1,
where λ=(α±α2+8δ)/2δ, Δ=γ2-4βδ, ξ=kx+ωt+ξ0, ω=-ik2-αk-i(β+2γu1+3δu12), k is a root of second order algebraic equation (2+αλ)ik2-(γλ+3δλu1+α-αu1)k-i(β+2γu1+3δu12)=0, u1=(-γ±Δ)/2δ, ξ0, C≠0 are arbitrary constants.By using of formulas (3.11) and (3.12), we can obtain multiple new periodic wave solutions in form tanξ and cotξ.Choosing the solutions (3.3) ((3.4), (3.13), (3.14), (3.18), (3.22), (3.23), (3.27) and (3.28), resp.) as a new “seed” solution u1(x,t) and solving the linear PDEs (2.11), (2.12), one gets a quasisolution ϕ(x,t). Then substituting the quasisolution ϕ(x,t) and u1(x,t) chosen above into (2.14), we can obtain more and more new exact particular solutions of (1.1). Taking C=1 in solutions (3.3), (3.4), (3.22), and (3.23), we can obtain shock wave solutions and singular traveling wave solutions of (1.1). Putting C=1 in solutions (3.9), (3.10), (3.27), and (3.28), we can obtain periodic wave solutions in form tanξ and cotξ.


## 4. Conclusion

It is worthwhile pointing out that the exact solutions obtained in this paper have more general form than some known solutions in previous studies. In addition to rederiving all known solutions in a systematic way, several entirely new exact solutions can also be obtained. Specially, choosingα=0 in the all solutions above, one can obtain abundant explicit and exact solutions to the Kolmogorov-Petrovskii-Piskunov equation [16]. Setting α=0, γ=0, β=-δ in the all solutions above, one can get abundant explicit and exact solutions to the Chaffee-Infante reaction diffusion equation [17]. We can also obtain abundant explicit and exact solutions to the Burgers-Huxley equation [18] by taking α≠0, β=δη, γ=-(1+η)δ, η arbitrary in the all solutions above. Go a step further, taking α=0, β=η, γ=-(1+η), η arbitrary in the all solutions above, we also obtain abundant explicit and exact solutions to the FitzHugh-Nagumo equation [19]. We can obtain abundant explicit exact solutions to the Newell-Whitehead equation when taking α=0, β=-1, γ=0, δ=1 in the all solutions above [17]. Putting α=0, β=1-(3η/2), γ=(5η/2)-2, δ=1-η in the all solutions above, we can obtain abundant explicit and exact solutions to an isothermal autocatalytic system [20].



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