# A Class of PDEs with Nonlinear Superposition Principles

Authors: Li Peng; Liu Keying; Pan Zuliang; Zhong Weizhou
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Through assuming that nonlinear superposition principles (NLSPs) are embedded in a Lie group, a class of 3rd-order PDEs is derived from a general determining equation that determine the invariant group. The corresponding NLSPs and transformation to linearize the nonlinear PDE are found, hence the governing PDE is provedC-integrable. In the end, some applications of the PDEs are explained, which shows that the result has very subtle relations with linearization of partial differential equation.


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## Body

## 1. Introduction

Construction of new solutions by superposition of known ones is a familiar tool in nonlinear partial differential equations. The idea of superpositions for nonlinear differential operators originated in 1893 by Vessiot [1]. It was immediately generalized by Guldberg [2]. Marius Sophus Lee pointed out that these are special cases of his own theory of “Fundamental Solutions of Differential Equations” [3, 4]. The problem was reconsidered in 1960 by Temple [5] who showed the existence of nonlinear d.e. and the general solution cannot be obtained as a finite number of particular solutions. In 1965, Inselberg showed that there exist classes of equations, involving nonlinear operators, where it is possible to “compose” two solutions to obtain a different solution [6]. This is different than the above works where the general solution is sought. In 2000, Ibragimov gives specific examples of the Vessiot-Guldberg-Lie algebra applied to some partial differential equations [7]. Jones and Ames introduce the definition “connecting function” to describe nonlinear superposition in [8].Generally, these methods for finding NLSPs can be classified into two categories: one is based on ad hoc methods [9–12]; the other is Lie’s classical symmetry algorithm [7, 13–15]. In contrast with the ad hoc methods, Lie’s classical symmetry algorithm is systematic and automatic.Following the definition of nonlinear superposition introduced by in [8, 15], formally, the simplest form of a superposition principle is an operation(1.1)F(u,v)=u*v,F:V*V⟶W,
(where V and W are function spaces) which preserves some governing equations, so that if(1.2)f(x,u,uxi,uxixj,…)=0,f(x,v,vxi,vxixj,…)=0
then(1.3)f(x,w,wxi,wxixj,…)=0,
where w=u*v. Of course, it is a simple matter to extend this concept to any n-ary operation that constructs new solutions from n old solutions.Goard and Broadbridge have investigated how to use Lie symmetry algebras to find general superposition principles for nonlinear PDEs [15] and they have given the NLSPs of some first-order and second-order PDEs in two independent variables and linearized the PDEs. However, for higher-order nonlinear PDEs, its linearization and NLSPs have not been fully investigated. It is the purpose of the paper to demonstrate how to find the superposition principles for a class higher-order nonlinear PDEs.We assume that for any pair of solutionsv(x) and z(x) to the governing PDE, there exists a parameter ϵ so that we have a one-parameter solution(1.4)u̅=F(v,z,ϵ).
In further, suppose that the NLSP is surjective in the sense that, for any two solutions u(x) and z(x), there exists a solution v(x) such that(1.5)u=F(v,z,0).
From (1.4) then(1.6)u̅=u+ϵ∂F∂ϵ(v,z,ϵ)|ϵ=0+O(ϵ2)=u+ϵU¯(v,z)+O(ϵ2).
From (1.5), if Fv≠0, the implicit function theorem allows us to regard v as a function of u and z. Hence we write(1.7)U¯=h(z,u)≡h(z(x,y),u),
which we find by solving the classical linear determining equations of the PDE. Then solving(1.8)du¯dϵ=U¯(z(x,y),u¯)|u¯=u,ϵ=0,
we find the NLSP for the PDE.Goard and others have shown in their work that for first- and second-order PDEs of two independent variables, this will actually imply that the PDE will have a generator of the form(1.9)A¯(x,y)B(u)∂∂u,orA¯(z)B(u)∂∂u,
where A¯(x,y) is the general solution of a linear homogeneous PDE. Hence the PDE satisfies the necessary condition for the existence of a transformation to a linear PDE.In the paper, we will start from a general third-order PDE and find the third-order PDE class with NLSPs. Furthermore, a discussion for higher-order PDEs is given.


## 2. A Class of Third-Order PDEs with NLSPs

Firstly, consider a general third-order determining equation:(2.1)α1(x,y,u)Uuuu+α2(x,y,u)Uuux+α3(x,y,u)Uuxx+α4(x,y,u)Uxxx+α5(x,y,u)Uuu,α6(x,y,u)Uux+α7(x,y,u)Uxx+α8(x,y,u)Uu+α9(x,y,u)Ux+α10(x,y,u)Uy+α11(x,y,u)Uuy+α12(x,y,u)Uyy+α13(x,y,u)Uxy+α14(x,y,u)Uuuy+α15(x,y,u)Uuyy+α16(x,y,u)Uyyy+α17(x,y,u)Uuxy+α18(x,y,u)Uxyy+α19(x,y,u)Uxxy=α20(x,y,u)U.
Substitution of U=h(z(x,y),u) into (2.1) gives(2.2)huuu[α1(x,y,u)]+huuz[α2(x,y,u)zx+α14(x,y,u)zy]+huzz[α3(x,y,u)zx2+α15(x,y,u)zy2+α17(x,y,u)zxzy]+huu[α11(x,y,u)]+huz[α6(x,y,u)zx+α11(x,y,u)zy+α3(x,y,u)zxx+α17(x,y,u)zxy+α15(x,y,u)zyy]+hzz[α7(x,y,u)zx2+α12(x,y,u)zy2+α13(x,y,u)zxzy+3α4(x,y,u)zxzxx+3α16(x,y,u)zyzyy+α18(x,y,u)zxzyy+2α18(x,y,u)zyzxy+2α19(x,y,u)zxzxy+α19(x,y,u)zyzxx[α7(x,y,u)zx2+α12(x,y,u)zy2+α13(x,y,u)zxzy[α2(x,y,u)zx+α14(x,y,u)zy]]+hz[α9(x,y,u)zx+α10(x,y,u)zy+α7(x,y,u)zxxα12(x,y,u)zyy+α13(x,y,u)zxy+α16(x,y,u)zyyyα4(x,y,u)zxxx+α18(x,y,u)zxyy+α19(x,y,u)zxxy]=α20(x,y,u)h.
To obtain a solution of (2.2) where h depends on z,u alone, the coefficients of (2.2) need to be in z and u alone. Hence we consider the possible way in which (2.2) can be written as an equation in z and u. It is shown that the determining equation admits separable solutions(2.3)h(z,u)=A(z)B(u),
or, more generally,(2.4)U¯(x,y,u)=A¯(x,y)B(u).
In the case, at least one of α4(x,y,u),α16(x,y,u),α18(x,y,u),α19(x,y,u) is not zero. (We note that at least one of these coefficients must be nonzero for a third-order governing PDE.) In addition, we suppose(2.5)α1(x,y,u)=α¯(x,y)γ1(u),α2(x,y,u)=0,α3(x,y,u)=0,α4(x,y,u)=c0(x,y)α(x,y)β(u),α5(x,y,u)=α¯(x,y)γ5(u),α6(x,y,u)=0,α7(x,y,u)=c1(x,y)α(x,y)β(u),α8(x,y,u)=α¯(x,y)γ8(u),α9(x,y,u)=c2(x,y)α(x,y)β(u),α10(x,y,u)=c3(x,y)α(x,y)β(u),α11(x,y,u)=0,α12(x,y,u)=c4(x,y)α(x,y)β(u),α13(x,y,u)=c5(x,y)α(x,y)β(u),α14(x,y,u)=0α15(x,y,u)=0,α16(x,y,u)=c6(x,y)α(x,y)β(u),α17(x,y,u)=0,α18(x,y,u)=c7(x,y)α(x,y)β(u),α19(x,y,u)=c8(x,y)α(x,y)β(u),α20(x,y,u)=α¯(x,y)γ20(u)
with β(u)≠0, then from (2.2) we let(2.6)hzz=k1(z)hz,hzzz=k2(z)hz
for some two functions k1,k2 of z, and (2.2) becomes(2.7)hzα(x,y)β(u)[[α2(x,y,u)zx+α14(x,y,u)zy]c0(x,y)k2(z)zx3+3c0(x,y)k1(z)zxzxx+c0(x,y)zxxx+c1(x,y)k1(z)zx2+c1(x,y)zxx+c2(x,y)xx+c3(x,y)zy+c4(x,y)k1(z)zx2+c4(x,y)zxx+c5(x,y)k1(z)zx2+c5(x,y)zxx+c6(x,y)k2(z)zy3+3c6(x,y)k1(z)zyzy+c6(x,y)zyyy+c7(x,y)k2(z)zxzy2+c7(x,y)k1(z)(zxzyy+2zyzxy)+c7(x,y)zxyyc8(x,y)k2(z)zx2zy+c8(x,y)k1(z)(zyzxx+2zxzxy)+c8zxxy[[α2(x,y,u)zx+α14(x,y,u)zy]c0(x,y)k2(z)zx3+3c0(x,y)k1(z)zxzxx+c0(x,y)zxxx[α2(x,y,u)zx+α14(x,y,u)zy]]+α¯(x,y)γ8(u)hu+α¯(x,y)γ5huu+α¯(x,y)γ1(u)huuu=α¯(x,y)γ20(u)h.
Then if we let(2.8)c0(x,y)k2(z)zx3+3c0(x,y)k1(z)zxzxx+c0(x,y)zxxx+c1(x,y)k1zx2+c1(x,y)zxx+c2(x,y)zx+c3(x,y)zy+c4(x,y)k1(z)zx2+c4(x,y)zxx+c5(x,y)k1(z)zx2+c5(x,y)zxy+c6(x,y)k2(z)zy3+3c6(x,y)k1(z)zyzyy+c6(x,y)zyyy+c7(x,y)k2(z)zxzy2+c7(x,y)k1(z)(zxzyy+2zyzxy)+c7(x,y)zxyy+c8(x,y)k2(z)zx2zy+c8(x,y)k1(z)(zyzxx+2zxzxy)+c8(x,y)zxxy=g(z),(2.2) becomes(2.9)α(x,y)hzg(z)+α¯(x,y)f1(u)hu+α¯(x,y)f2(u)huu+α¯(x,y)f3(u)huuu=α¯(x,y)f4(u)h,
where(2.10)f1(u)=γ8(u)β(u),f2(u)=γ5(u)β(u),f3(u)=γ1(u)β(u),f4(u)=γ20(u)β(u),
and notice that, from (2.6),(2.11)h(z,u)=A(z)B(u)+R(u),
where A(z)=∫exp(∫k(z)dz)dz.We consider the following possibilities for (2.7).(i)
g(z)=0, α¯(x,y)≠0, not all fi(u)=0, i=1,2,3,4. From (2.9) we have(2.12)f1(u)hu+f2(u)huu+f3(u)huuu=f4(u)h.
Equation (2.12) admits the separable solution (2.3), where B(u) satisfies(2.13)f1(u)B′(u)+f2(u)B′′(u)+f3(u)B′′′(u)=f4(u)B(u).The linear PDE whichA¯(x,y) satisfies is(2.14)c0(x,y)A¯xxx+c1(x,y)A¯x+c2(x,y)A¯y+c4(x,y)A¯xx+c5(x,y)A¯xx+c6(x,y)A¯yyy+c7(x,y)A¯xyy+c8A¯xxy=0.(ii)
g(z)≠0, α(x,y)≠0, fi(u), i=1,2,3,4. We then require α(x,y)=α¯(x,y) in (2.9).Equation (2.9) admits separable solutions where(2.15)A′(z)g(z)=λA(z)
and B(u) satisfies(2.16)f4(u)B(u)-f1(u)B′(u)-f2(u)B′′(u)-f3(u)B′′′(u)=λB(u),
for some nonzero constant λ.From (2.15), we have(2.17)k(z)=λ-g′(z)g(z)A¯(x,y) satisfies(2.18)c0(x,y)A¯xxx+c1(x,y)A¯xc2(x,y)A¯y+c4(x,y)A¯xx+c5(x,y)A¯xx+c6(x,y)A¯yyy+c7(x,y)A¯xyy+c8A¯xxy=λA¯.So, we obtain (2.8) which has symmetries with generator(2.19)A¯(x,y)B(u)∂∂u.
The PDEs are linearizable with(2.20)ψ=∫1B(u)du
(by [14, Theorem  6.4, 2-2]). Equation (2.8) represents a class of third-order PDEs with various choices of ci(x,y)(i=0,1,…,8), k(u) and F(u). A class of general third-order PDEs in two independent variables with NLSPs of (1.1) embedded in a Lie group are identified in (2.8).


## 3. Symmetry Analysis of (2.8)

We rewrite (2.8) in the following formulation:(3.1)c0(x,y)k2(u)ux3+3c0(x,y)k1(u)uxuxx+c0(x,y)uxxx+c1(x,y)k1(u)ux2+c1(x,y)uxx+c2(x,y)ux+c3(x,y)uy+c4(x,y)k1(u)uy2+c4(x,y)uyy+c5(x,y)k1(u)uxuy+c5(x,y)uxy+c6(x,y)k2(u)uy3+3c6(x,y)k1(u)uyuyy+c6(x,y)uyyy+c7(x,y)k2(u)uxuy2+c7(x,y)k1(u)(uxuyy+2uyuxy)+c7(x,y)uxyy+c8(x,y)k2(u)ux2uy+c8(x,y)k1(u)(uyuxx+2uxuxy)+c8uxxy=F(u),
where for F≠0, k(u)=(λ-F′(u))/F(u), λ is constant, and for F=0, k(u) is arbitrary.Equation (3.1) has classical symmetry with generator(3.2)Γ=U(x,y,u)∂∂u
with its third-order prolongation(3.3)Γ*=U∂∂u+DxU∂∂ux+DyU∂∂uy+DxxU∂∂uxx+DxyU∂∂uxy+DyyU∂∂uyy+DxxxU∂∂uxxx+DxxyU∂∂uxxy+DxyyU∂∂uxyy+DyyyU∂∂uyyy,
where U satisfies(3.4)c0(x,y)Uxxx+c1(x,y)Uxx+c2(x,y)Ux+c3(x,y)Uy+c4(x,y)Uyy+c5(x,y)Uxy+c6(x,y)Uyyy+c7(x,y)Uxyy+c8(x,y)Uxxy+Uug(u)+[Uuuu+3k1(u)Uuu+k′2(u)U+2k2(u)Uu]*[c0(x,y)ux3+c8(x,y)ux2uy+c7(x,y)uxuy2+c6(x,y)uy3]+[Uuu+k′1(u)U+k1(u)Uu]*[3c0(x,y)uxuxx+c8(x,y)(2uxuxy+uxuxx)+c7(x,y)(2uyuxy+uxuyy)+3c6(x,y)uyuyy]+[Uuux+2k1(u)Uux+k2(u)Ux]*[3c0(x,y)ux2+2c8(x,y)uxuy+c7(x,y)uy2]+[Uuuy+2k1(u)Uuy+k2(u)Uy]*[c8(x,y)ux2+2c7(x,y)uxuy+3c6(x,y)uy2]+[Uuu+k′1(u)U+k1(u)Uu]*[c1(x,y)ux2+c4(x,y)uxuy+c5(x,y)uy2]+[Uuxx+k1(u)Uxx]*[3c0(x,y)ux+c8(x,y)uy]+[Uuxy+k1(u)Uxy]*[2c8(x,y)ux+2c7(x,y)uy]+[Uuyy+k1(u)Uyy]*[c7(x,y)ux+3c6(x,y)uy]+[Uux+k1(u)Ux]*[3c0(x,y)uxx+2c8(x,y)uxy+c7(x,y)uyy+2c1(x,y)ux+c5(x,y)uy]+[Uuy+k1(u)Uy]*[c8(x,y)uxx+2c7(x,y)uxy+3c6(x,y)uyy+c5(x,y)ux+2c4(x,y)uy]=F′(u)U.
We require(3.5)Uuuu+3k1(u)Uuu+k′2(u)U+2k2(u)Uu=0,Uuu+k′1(u)U+k1(u)Uu=0,Uuux+2k1(u)Uux+k2(u)Ux=0,Uuuy+2k1(u)Uuy+k2(u)Uy=0,Uuxx+k1(u)Uxx=0,Uuxy+k1(u)Uxy=0,Uuyy+k1(u)Uyy=0,Uux+k1(u)Ux=0,Uuy+k1(u)Uy=0,
where k1(u)=k(u),k2(u)=k2(u)+k′(u). Let(3.6)Uu+k(u)U=0,
then,(3.7)Uuuu+3k1(u)Uuu+k′2(u)U+2k2(u)Uu=Uuuu+3k(u)Uuu+(2k(u)k′(u)+k′′(u))U+2(k2(u)+k′(u))Uu=Uuuu+k′′(u)+2k′(u)Uu+k(u)Uuu+2k(u)(Uuu+k′(u)U+k(u)Uu)=Uuu(Uu+k(u)U)+2k(u)Du(Uu+k(u)U)=0.
It is easy to verify other equations if (3.6) is satisfied. Then, (3.4) degenerates into(3.8)Uu+k(u)U=0,c0(x,y)Uxxx+c1(x,y)Uxx+c2(x,y)Ux+c3(x,y)Uy+c4(x,y)Uyy+c5(x,y)Uxyy+c6(x,y)Uyyy+c7(x,y)Uxyy+c8(x,y)Uxxy=UF′(u)-UuF(u).
Letting U¯=h(z(x,y),u) and substituting U¯ into (3.8) give(3.9)h=A(z)exp⁡(-∫k(u)du),
where A(z) needs to satisfy(3.10)c0(x,y)[Azzzzx3+3Azzzxzxx+Azzxxx]+c1(x,y)[Azzzx2+Azzxx]+c2(x,y)[Azzx]+c3(x,y)[Azzx]+c4(x,y)[Azzzy2+Azzyy]+c5(x,y)[Azzzxzy+Azzxy]+c6(x,y)[Azzzzy3+3Azzzyzyy+Azzyyy]+c7(x,y)[Azzzzxzy2+Azz(zxzyy+2zyzxy)+Azzxyy]+c8(x,y)[Azzzzxxzy+Azz(zyzxx+2zxzxy)+Azzxxy]=λA,
where λ=0 if F=0. Hence we choose A(z)=∫exp(∫k(z)dz)dz as in (2.11), and we can find that A(z) make (3.10) satisfied by substituting it into (3.10).Then solving (1.8), that is,(3.11)du¯dϵ=∫exp⁡(∫k(z)dz)dzexp⁡(-∫k(u¯)du¯)|u¯=u,ϵ=0,
gives the NLSP for (3.4) as(3.12)u¯=P-1[ϵP(z)+P(u)],
where(3.13)P(u)=∫exp⁡(∫k(u)du)du.If we let(3.14)U¯=h(x,y,u)=A¯(x,y)B(u)
and substitute it into (3.8), then(3.15)B(u)=exp⁡(-∫k(u)du),
and A¯ satifies(3.16)c0(x,y)A¯xxx+c1(x,y)A¯xc2(x,y)A¯y+c4(x,y)A¯xx+c5(x,y)A¯xx+c6(x,y)A¯yyy+c7(x,y)A¯xyy+c8A¯xxy=λA¯,
where again λ=0 when F=0. By solving(3.17)du¯dϵ=A¯(x,y)B(u¯)|u¯=u,ϵ=0,
we obtain(3.18)u¯=P-1[ϵA¯(x,y)+P(u)],
where P is given in (3.13). By using (2.20) and (3.15), the transformation to linearize (3.1) is obtained as(3.19)v=P(u)=∫exp⁡(∫k(u)du)du,
which transforms (3.1) into the linear equation(3.20)c0(x,y)vxxx+c1(x,y)vxx+c2(x,y)vx+c3(x,y)vy+c4(x,y)vyy+c5(x,y)uxy+c6(x,y)vyyy+c7(x,y)vxyy+c8vxxy=λv+α
for some constant α and where λ=0 if F=0. It is indicated that the governing PDE (3.1) is C-integrable [16].It is noticed that there is no explicit general solution for the linearized equation except few special cases. Thus, it is generally hard to obtain explicit general solution of the original nonlinear PDE, although the NLSP and the transformation to linearise the system are explicit. Equation (3.1) can also be used to determine whether some equations are C-integrable. For example, the kdv equation is only S-integrable [16], but not C-integrable, since it is not included in the class. Many numerical methods (such as Finite Difference Method, Finite Element Method) can be used to solve the linearized equation, and the solution of (3.1) can be obtained through the corresponding variable transformation.


## 4. Some Explanations of the Result

By explicitly constructing the full class of PDEs with a Lie group of NLSPs, we arrive at a single expression (3.1) for a class of third-order variable-coefficient equations.


### 4.1. Relations with ODE

First of all, we should point out that when lettingc0(x,y)=c6(x,y)=c7(x,y)=c8(x,y)=0, (3.1) is second-order PDE, which is (2.44) in [15](4.1)c1(x,y)k1(u)ux2+c1(x,y)uxx+c2(x,y)ux+c3(x,y)uy+c4(x,y)k1(u)uy2+c4(x,y)uyy+c5(x,y)k1(u)uxuy+c5(x,y)uxy=F(u).Second, lettingc0(x,y)=c3(x,y)=c4(x,y)=c5(x,y)=c6(x,y)=c7(x,y)=c9(x,y)=0, c1(x,y), c2(x,y) be only related to variable x, an ODE is obtained(4.2)c1(x)k1(u)ux2+c1(x)uxx+c2(x)ux=F(u)
which is a special case of the Painleve´ equation [17](4.3)y′′+P(x,y)y′2+Q(x,y)y′+R(x,y)=0.
The special situation of Painleve´ equation has the NLSPs and can be transformed into linear equation.Furthermore, we can get a class of third-order ODEs that have NLSPs(4.4)c0(x)k2(u)ux3+3c0(x)k1(u)uxuxx+c0(x)uxxx+c1(x)k1(u)ux2+c1(x)uxx+c2(x)ux=F(u).


### 4.2. Relations with PDEs Whose Potential Equations Are in the Form (3.1)

LettingF(u)=0, k1(u)=k2(u)=1, ci(x,y)=0, i=4,5,6,7,8, and differentiate it with variable x, we get(4.5)∂∂x{[ϵA¯(x,y)+P(u)]c0(x,y)ux3+3c0(x,y)uxuxx+c0(x,y)uxxx+c1(x,y)ux2+c1(x,y)uxx+c2(x,y)ux+c3(x,y)uy[ϵA¯(x,y)+P(u)]}=0.
Making the replacement ux→v, we get a class nonlinear PDEs whose potential equations can be linearized. Here we only give out the form with that coefficients are constants for simplicity(4.6)c0(3v2v2+3vx2+3vvxx)+c1(2vvx+vxx)+c2vx+c3vy=0.
The transformation to linearize the PDE is(4.7)w=exp⁡(u)=exp⁡(∫vdx).
In fact, it is the Cole-Hopf transform [18, 19](4.8)v=(ln⁡w)x=wxw.
It is easy to calculate that the NLSP is(4.9)∂∂x(ln⁡(ϵexp⁡(∫zdx)+exp⁡(∫vdx))).Whenc0=c2=0 the potential Burgers equation is included in (3.1) [15]. And at the same time, we can also get the Burgers equation from(4.6)(4.10)c1(2vvx+vxx)+c3vy=0.


### 4.3. Some Examples of NLSPs and Transformations to Linearize Nonlinear PDEs

Some more concrete examples are given below. Lettingc0(x,y)=c6(x,y)=1, ci(x,y)=0, i≠0,6 in (3.1), then(4.11)k2(u)ux3+3k1(u)uxuxx+uxxx+k2(u)uy3+3k1(u)uyuyy+uyyy=F(u).
We discuss the two following cases.Case 1.
In the case ofF(u)=u2, let λ=0, then k1(u)=-2/u, k2(u)=6/u2 and
(4.12)6ux3-6uuxuxx+u2uxxx+6uy3-6uuyuyy+u2uyyy=u4.
In this case, P(u)=-1/u, hence the NLSP is
(4.13)u¯=-1-ϵ(1/z)-1/u.
The transformation to linearize the PDE is
(4.14)v=-1u.
The corresponding linearized PDE is
(4.15)vxxx+vyyy=1.Case 2.
In the case ofF(u)=0, let k1(u)=1,k2(u)=1, then
(4.16)ux3+3uxuxx+uxxx+uy3+3uyuyy+uyyy=0.
In the case P(u)=exp(u), the NLSP is
(4.17)u¯=ln⁡[ϵexp⁡(u)+exp⁡(z)].
The transformation to linearize the PDE is
(4.18)v=exp⁡(u).
The corresponding linearized PDE is
(4.19)vxxx+vyyy=0.Letk1(u)=u,k2(u)=u2+1(4.20)(u2+1)ux3+3uuxuxx+uxxx+(u2+1)uy3+3uuyuyy+uyyy=0.
In the case P(u)=∫exp(u2/2)du, the NLSP is(4.21)u¯={2ln⁡[ddu[ϵ∫exp⁡(u22)du+∫exp⁡(z22)dz]]}1/2.
The transformation to linearize the PDE is(4.22)v=∫exp⁡(u22)du.
The corresponding linearized PDE is(4.23)vxxx+vyyy=0.


## 4.1. Relations with ODE

First of all, we should point out that when lettingc0(x,y)=c6(x,y)=c7(x,y)=c8(x,y)=0, (3.1) is second-order PDE, which is (2.44) in [15](4.1)c1(x,y)k1(u)ux2+c1(x,y)uxx+c2(x,y)ux+c3(x,y)uy+c4(x,y)k1(u)uy2+c4(x,y)uyy+c5(x,y)k1(u)uxuy+c5(x,y)uxy=F(u).Second, lettingc0(x,y)=c3(x,y)=c4(x,y)=c5(x,y)=c6(x,y)=c7(x,y)=c9(x,y)=0, c1(x,y), c2(x,y) be only related to variable x, an ODE is obtained(4.2)c1(x)k1(u)ux2+c1(x)uxx+c2(x)ux=F(u)
which is a special case of the Painleve´ equation [17](4.3)y′′+P(x,y)y′2+Q(x,y)y′+R(x,y)=0.
The special situation of Painleve´ equation has the NLSPs and can be transformed into linear equation.Furthermore, we can get a class of third-order ODEs that have NLSPs(4.4)c0(x)k2(u)ux3+3c0(x)k1(u)uxuxx+c0(x)uxxx+c1(x)k1(u)ux2+c1(x)uxx+c2(x)ux=F(u).


## 4.2. Relations with PDEs Whose Potential Equations Are in the Form (3.1)

LettingF(u)=0, k1(u)=k2(u)=1, ci(x,y)=0, i=4,5,6,7,8, and differentiate it with variable x, we get(4.5)∂∂x{[ϵA¯(x,y)+P(u)]c0(x,y)ux3+3c0(x,y)uxuxx+c0(x,y)uxxx+c1(x,y)ux2+c1(x,y)uxx+c2(x,y)ux+c3(x,y)uy[ϵA¯(x,y)+P(u)]}=0.
Making the replacement ux→v, we get a class nonlinear PDEs whose potential equations can be linearized. Here we only give out the form with that coefficients are constants for simplicity(4.6)c0(3v2v2+3vx2+3vvxx)+c1(2vvx+vxx)+c2vx+c3vy=0.
The transformation to linearize the PDE is(4.7)w=exp⁡(u)=exp⁡(∫vdx).
In fact, it is the Cole-Hopf transform [18, 19](4.8)v=(ln⁡w)x=wxw.
It is easy to calculate that the NLSP is(4.9)∂∂x(ln⁡(ϵexp⁡(∫zdx)+exp⁡(∫vdx))).Whenc0=c2=0 the potential Burgers equation is included in (3.1) [15]. And at the same time, we can also get the Burgers equation from(4.6)(4.10)c1(2vvx+vxx)+c3vy=0.


## 4.3. Some Examples of NLSPs and Transformations to Linearize Nonlinear PDEs

Some more concrete examples are given below. Lettingc0(x,y)=c6(x,y)=1, ci(x,y)=0, i≠0,6 in (3.1), then(4.11)k2(u)ux3+3k1(u)uxuxx+uxxx+k2(u)uy3+3k1(u)uyuyy+uyyy=F(u).
We discuss the two following cases.Case 1.
In the case ofF(u)=u2, let λ=0, then k1(u)=-2/u, k2(u)=6/u2 and
(4.12)6ux3-6uuxuxx+u2uxxx+6uy3-6uuyuyy+u2uyyy=u4.
In this case, P(u)=-1/u, hence the NLSP is
(4.13)u¯=-1-ϵ(1/z)-1/u.
The transformation to linearize the PDE is
(4.14)v=-1u.
The corresponding linearized PDE is
(4.15)vxxx+vyyy=1.Case 2.
In the case ofF(u)=0, let k1(u)=1,k2(u)=1, then
(4.16)ux3+3uxuxx+uxxx+uy3+3uyuyy+uyyy=0.
In the case P(u)=exp(u), the NLSP is
(4.17)u¯=ln⁡[ϵexp⁡(u)+exp⁡(z)].
The transformation to linearize the PDE is
(4.18)v=exp⁡(u).
The corresponding linearized PDE is
(4.19)vxxx+vyyy=0.Letk1(u)=u,k2(u)=u2+1(4.20)(u2+1)ux3+3uuxuxx+uxxx+(u2+1)uy3+3uuyuyy+uyyy=0.
In the case P(u)=∫exp(u2/2)du, the NLSP is(4.21)u¯={2ln⁡[ddu[ϵ∫exp⁡(u22)du+∫exp⁡(z22)dz]]}1/2.
The transformation to linearize the PDE is(4.22)v=∫exp⁡(u22)du.
The corresponding linearized PDE is(4.23)vxxx+vyyy=0.


## 5. Conclusion

We have given out a class of nonlinear PDEs with NLSPs by using the idea suggested by Goard and Broadbridge in [15]. By explicitly constructing the full class of PDEs with a Lie group of NLSPs, we arrived at a single expression (3.1) for a class of third-order variable-coefficient equations that includes previously second-order PDEs obtained in [15]. The class of PDEs includes the Painleve´ type equations that can be linearized. We derive Cole-Hopf transformation of (4.3) whose potential equations are included in (3.1). PDE (3.1) is C-integrable since it can be linearized.It is should be pointed out that higher-order nonlinear PDEs with NLSPs can be obtained in the same way, although much effort for symbolic computation is needed. For example, a fourth-order PDE with NLSP is in the following:(5.1)k3(u)ux4+6k2(u)ux2uxx+3k1(u)uxx2+6k1(u)uxuxxx+uxxxx=F(u).
Here k1(u)=k(u), k2(u)=k2(u)+k′(u), k3(u)=k′′′(u)+3k(u)k′(u)+k3(u), k(u)=λ-F′(u)/F(u)(F(u)≠0), λ is constant. When F(u)=0, k(u) is arbitrary. Here the C-integrable fourth-order ODE (5.1) can be obtained by the proposed Lie algorithm method relative easily on the computer.



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*Source: 346824-2012-04-05.xml*