Research Article | Open Access
A Bäcklund Transformation for the Burgers Hierarchy
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.
and for , define the differential expressions recursively as follows:
Then the Burgers hierarchy is defined by
The first few members of the hierarchy (1.3) are
with (1.4) being just the Burgers equation.
transforms solutions of the linear equation
to that of (1.3). With the help of the Cole-Hopf transformation (1.9), Taflin  and Tasso  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.  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  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 , when studying the surfaces of constant negative curvature, that the sine-Gordon equation
has the following property: if solves (1.9), then for an arbitrary non-zero constant , the system on
is integrable; moreover, also solves (1.9). So (1.10) gives a transformation , now called Bäcklund transformation, which takes one solution of (1.9) into another. For example, substituting the trivial solution into (1.10) yields one-soliton solution:
where 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 
modified KdV equation 
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 have the following property.
Theorem 2.1. For an arbitrary constant , let Then
Now we state our main result.
Theorem 2.2. If is a solution of (1.3), then the system on is integrable; moreover, also satisfies (1.3). Therefore, (2.5) defines a Bäcklund transformation , in a unified form, for each member of the Burgers hierarchy (1.3).
Proof. By (1.3) and (2.5) we have
therefore ; that is, (2.5) is an integrable system associated with (1.3).
From the first equation of (2.5) we have So On the other hand, by (2.2) Substituting (2.9) and (2.10) into the second equation of (2.5) yields that is, also satisfies the Burgers hierarchy (1.3).
3. Exact Solutions
In this section we always assume that is an arbitrary nonzero constant.
with the “integration constant" satisfying a first-order ordinary differential equation determined by the second equation of (2.5).
Note that (3.4) is a traveling wave solution.
Note that (3.8) is not a traveling wave solution.
Note that (3.12) is not a traveling wave solution.
Note that (3.16) is not a traveling wave solution.
This work is supported by the National Natural Science Foundation of China through the Grant no. 10571149.
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Copyright © 2010 Xifang Cao and Chuanyou Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.