We give a Bäcklund transformation in a unified form for each member in the Burgers hierarchy. By applying the Bäcklund transformation to the trivial solutions, we generate some solutions of the Burgers hierarchy.

1. Introduction

Let

P-1=1,P0(u)=u,
and for j≥1, define the differential expressions Pj(u,…,∂xju) recursively as follows:

Pj(u,…,∂xju)=(u+∂x)Pj-1(u,…,∂xj-1u).
Then the Burgers hierarchy is defined by

ut=∂xPj(u,…,∂xju),j≥1.
The first few members of the hierarchy (1.3) are

ut=2uux+uxx,ut=3u2ux+3ux2+3uuxx+uxxx,ut=4u3ux+12uux2+6u2uxx+10uxuxx+4uuxxx+uxxxx,
with (1.4) being just the Burgers equation.

There is much literature on the Burgers hierarchy. Olver [1] derived the hierarchy (1.3) from the point of view of infinitely many symmetries. The work in [2] showed that the Cole-Hopf transformation

w↦u=wxw
transforms solutions of the linear equation

wt=∂xj+1w
to that of (1.3). With the help of the Cole-Hopf transformation (1.9), Taflin [3] and Tasso [4] showed, respectively, that the Burgers equation (1.4) and the second member (1.5) of the hierarchy (1.3) can be written in the Hamiltonian form. More recently, Talukdar et al. [5] constructed an appropriate Lagrangian by solving the inverse problem of variational calculus and then Hamiltonized (1.5) to get the relevant Poisson structure. Furthermore, they pointed out that their method is applicable to each member of (1.3). Pickering [6] proved explicitly that each member of (1.3) passes the Weiss-Tabor-Carnevale Painlevé test.

This paper is devoted to the study of Bäcklund transformation for the Burgers hierarchy. Bäcklund transformation was named after the Swedish mathematical physicist and geometer Albert Victor Bäcklund(1845-1922), who found in 1883 [7], when studying the surfaces of constant negative curvature, that the sine-Gordon equation

uxt=sinu
has the following property: if u solves (1.9), then for an arbitrary non-zero constant λ, the system on v

vx=ux-2λsinu+v2,vt=-ut+2λsinu-v2
is integrable; moreover, v also solves (1.9). So (1.10) gives a transformation u↦v, now called Bäcklund transformation, which takes one solution of (1.9) into another. For example, substituting the trivial solution u(x,t)≡0 into (1.10) yields one-soliton solution:

v(x,t)=4arctanexp(α-λx-1λt),
where α is an arbitrary constant. By repeating this procedure one can get multiple-soliton solutions. Some other nonlinear partial differential equations (PDEs), such as KdV equation [8]

ut=6uux+uxxx,
modified KdV equation [9]

ut=u2ux+uxxx,
Burgers equation (1.4) [10], and a generalized Burgers equation [11]

ut+b(t)uux+a(t)uxx=0,
also possess Bäcklund transformations. Now Bäcklund transformation has become a useful tool for generating solutions to certain nonlinear PDEs. Much literature is devoted to searching Bäcklund transformations for some nonlinear PDEs (see, e.g., [12–15]). In this paper, we give a Bäcklund transformation for each member in the Burgers hierarchy. As an application, by applying our Bäcklund transformation to the trivial solutions, we generate some new solutions of (1.3).

2. Bäcklund Transformation

First, the differential expressions Pj have the following property.

Theorem 2.1.

For an arbitrary constant λ, let
u=v+vxλ+v.
Then
Pj(u,…,∂xju)=λPj(v,…,∂xjv)+Pj+1(v,…,∂xj+1v)λ+v,j≥1.

Proof.

We use induction to prove (2.2).

First, for j=1,
P1(u,ux)=u2+ux=(v+vxλ+v)2+vx-vx2(λ+v)2+vxxλ+v=λ(v2+vx)+v3+3vvx+vxxλ+v=λP1(v,vx)+P2(v,vx,vxx)λ+v.
So (2.2) is true for j=1.

Next, fix a k>1, and assume that (2.2) is true for j=k-1. Then
Pk(u,…,∂xku)=(v+vxλ+v+∂x)Pk-1(u,…,∂xk-1u)=(v+vxλ+v+∂x)λPk-1(v,…,∂xk-1v)+Pk(v,…,∂xkv)λ+v=(v+∂x)(λPk-1(v,…,∂xk-1v)+Pk(v,…,∂xkv))λ+v=λPk(v,…,∂xkv)+Pk+1(v,…,∂xk+1v)λ+v;
that is, (2.2) is valid for j=k.

Therefore, (2.2) is always true for j≥1.

Now we state our main result.

Theorem 2.2.

If u is a solution of (1.3), then the system on vvx=(λ+v)(u-v),vt=(λ+v)∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u))
is integrable; moreover, v also satisfies (1.3). Therefore, (2.5) defines a Bäcklund transformation u↦v, in a unified form, for each member of the Burgers hierarchy (1.3).

Proof.

By (1.3) and (2.5) we have
vxt=(λ+v)(u-v)∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u))+(λ+v)∂xPj(v,…,∂xjv)-(λ+v)2∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u)),vtx=(λ+v)(u-v)∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u))+(λ+v)∑k=0j(-λ)j-k∂xPk(v,…,∂xjv)-v(λ+v)∑k=0j-1(-λ)j-1-k∂xPk(v,…,∂xjv)-(λ+v)2(u-v)∑k=0j(-λ)j-kPk-1(u,…,∂xku)=(λ+v)(u-v)∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u))+(λ+v)∂xPj(v,…,∂xjv)-(λ+v)2∑k=0j-1(-λ)j-1-k∂xPk(v,…,∂xjv)-(λ+v)2(u-v)∑k=0j(-λ)j-kPk-1(u,…,∂xku)=(λ+v)(u-v)∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u))+(λ+v)∂xPj(v,…,∂xjv)-(λ+v)2∑k=0j(-λ)j-k(Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u));
therefore vxt=vtx; that is, (2.5) is an integrable system associated with (1.3).

From the first equation of (2.5) we have
u=v+vxλ+v.
So
P0(u)-v=vxλ+v=∂xP0(v)λ+v.
On the other hand, by (2.2)
Pk(u,…,∂xku)-vPk-1(u,…,∂xk-1u)=λ∂xPk-1(v,…,∂xk-1v)+∂xPk(v,…,∂xkv)λ+v,k≥1.
Substituting (2.9) and (2.10) into the second equation of (2.5) yields
vt=∂xPj(v,…,∂xjv);
that is, v also satisfies the Burgers hierarchy (1.3).

3. Exact Solutions

In this section we always assume that λ is an arbitrary nonzero constant.

From a known solution u of (1.3), the first equation of (2.5) gives

v(x,t)=e∫(λ+u)dx-λ∫e∫(λ+u)dxdx-λc(t)∫e∫(λ+u)dxdx+c(t),
with the “integration constant" c(t) satisfying a first-order ordinary differential equation determined by the second equation of (2.5).

Example 3.1.

Take the trivial solution u(x,t)≡1 of (1.3). Then from (1.2) we have
Pj(u,…,∂xju)≡1forj≥1.
So (2.5) becomes
vx=(λ+v)(1-v),vt=(λ+v)(1-v)(1-(-λ)j+1)1+λ.
Solving (3.3) gives the following solution of (1.3):
v(x,t)=e(1+λ)x+(1+(-1)jλj+1)t+λece(1+λ)x+(1+(-1)jλj+1)t-ec,
where c is an arbitrary constant.

Note that (3.4) is a traveling wave solution.

Example 3.2.

By the Cole-Hopf transformation (1.7),
u(x,t)=1x
is a solution of (1.3). Then from (1.2) we have
Pj(u,…,∂xju)≡0forj≥1.
So (2.5) becomes
vx=(λ+v)(1x-v),vt=(λ+v)((-λ)j(1x-v)-(-λ)j-1vx).
Solving (3.7) gives the following solution of (1.3):
v(x,t)=λeλ(x+(-λ)jt)+λec(-1+λx)eλ(x+(-λ)jt)-ec.

Note that (3.8) is not a traveling wave solution.

Example 3.3.

By the Cole-Hopf transformation (1.7),
u(x,t)=2x
is a solution of (1.3) for j≥2. Then from (1.2) we have
P1(u,ux)=2x2,Pj(u,…,∂xju)≡0forj≥2.
So (2.5) becomes
vx=(λ+v)(2x-v),vt=(λ+v)((-λ)j(2x-v)+2(-λ)j-1(1x2-vx)-2(-λ)j-2vx2).
Solving (3.11) gives the following solution of (1.3) for j≥2:
v(x,t)=2λ(-1+λx)eλ(x+(-λ)jt)+λec(2-2λx+λ2x2)eλ(x+(-λ)jt)-ec.

Note that (3.12) is not a traveling wave solution.

Example 3.4.

By the Cole-Hopf transformation (1.7),
u(x,t)=3x
is a solution of (1.3) for j≥3. Then from (1.2) we have
P1(u,ux)=6x2,P2(u,ux,uxx)=6x3,Pj(u,…,∂xju)≡0forj≥3.
So (2.5) becomes
vx=(λ+v)(3x-v),vt=(λ+v)((-λ)j(3x-v)+3(-λ)j-1(2x2-vx)+6(-λ)j-2(1x3-vx2)-6(-λ)j-3vx3).
Solving (3.15) gives the following solution of (1.3) for j≥3:
v(x,t)=3λ(2-2λx+λ2x2)eλ(x+(-λ)jt)+λec(-6+6λx-3λ2x2+λ3x3)eλ(x+(-λ)jt)-ec.

Note that (3.16) is not a traveling wave solution.

Remark 3.5.

In general, for an arbitrary positive integer k,
u(x,t)=kx
is a solution of (1.3) for j≥k. Substituting (3.17) into (2.5) gives the following solution of (1.3) for j≥k:
v(x,t)=(∂f(x,x2,…,xk)/∂x)eλ(x+(-λ)jt)+λecf(x,x2,…,xk)eλ(x+(-λ)jt)-ec,
where
f(x,x2,…,xk)=(-1)kk!+(-1)k-1k!λx+(-1)k-2k!2!λ2x2+⋯-kλk-1xk-1+λkxk.

Acknowledgment

This work is supported by the National Natural Science Foundation of China through the Grant no. 10571149.

OlverP. J.Evolution equations possessing infinitely many symmetriesChoodnovskyD. V.ChoodnovskyG. V.Pole expansions of nonlinear partial differential equationsTaflinE.Analytic linearization, Hamiltonian formalism, and infinite sequences of constants of motion for the Burgers equationTassoH.Hamiltonian formulation of odd Burgers hierarchyTalukdarB.GhoshS.DasU.Inverse variational problem and canonical structure of Burgers equationsPickeringA.The Weiss-Tabor-Carnevale Painlevé test and Burgers' hierarchyRogersC.ShadwickW. F.WahlquistH. D.EstabrookF. B.Bäcklund transformation for solutions of the Korteweg-de Vries equationLambG. L.Jr.Bäcklund transformations for certain nonlinear evolution equationsLüZ.-S.An explicit Bäcklund transformation of Burgers equation with applicationsHongW.-P.On Bäcklund transformation for a generalized Burgers equation and solitonic solutionsConteR.MusetteM.GrundlandA. M.Bäcklund transformation of partial differential equations from the Painlevé-Gambier classification. II. Tzitzéica equationGrundlandA. M.LeviD.On higher-order Riccati equations as Bäcklund transformationsLeviD.Toward a unification of the various techniques used to integrate nonlinear partial differential equations: Bäcklund and Darboux transformations vs. dressing methodWuH.On Bäcklund transformations for nonlinear partial differential equations