Burgers' Equation

For Burgers' equation

$$u_t+uu_x=\sigma u_{xx}\ ,\eqno (3.1)$$

we assume that

$$u=\phi^{\alpha}\sum_{j=0}^{\infty} u_j\phi^j\ ,\eqno (3.2)$$

where

$$\phi=\phi(x,t)\ ,\qquad u_j=j_j(x,t)$$

are analytic functions of $(x,t)$ near $M=\bigl\{(x,t)\colon\ \phi(x,t) =0\bigr\}$.

If $M$ is a singularity manifold, it is readily found that

$$\alpha=-1\eqno (3.3)$$

by a leading order analysis. The recursion relations for $u_j$ are found to be

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...phi_{xx}+(j-1)(j-2)u_j\phi_x^2\bigr]\ .\cr\crcr}}\,\eqno (3.4)$$

Collecting terms involving $u_j$, it is found that

$$\sigma\phi_x^2(j-2)(j+1)u_j=F(u_{j-1},\ldots,u_0,\phi_t,\phi_x, \phi_{xx},\ldots)\eqno (3.5)$$

for $j=0,1,2,\ldots$ .

We note that the recursion relations (3.5) are not defined when $j=-1,2$. These values of $j$ are called the ``resonances" [4] of the recursion relation and correspond to points where arbitrary functions of $(x,t)$ are introduced into the expansion. $j=-1$ corresponds to the arbitrary (undefined) singularity manifold $(\phi=0)$. On the other hand, the resonance at $j=2$ introduces an arbitrary function $u_2$ and a ``compatability condition" on the functions $(\phi,u_0,u_1)$ that requires the right-hand side of (3.5) vanish identically.

For Burgers' equation, we find from (3.4)

truecm $j=0\ ,\qquad u_0=-2\sigma\phi_x\ ,$ (3.6)

truecm $j=1\ ,\qquad \phi_t+u_1\phi_x=\sigma \phi_{xx}\ ,$(3.7)

truecm $j=2\ ,\qquad\partial_x(\phi_t+u_1\phi_x -\sigma\phi_{xx})=0\ .$(3.8)

By (3.7) the compatability condition (3.8) at $j=2$ is satisfied identically. Thus, Burgers' equation possesses the Painlevé property. Furthermore, if we set the arbitrary function $u=2$ equal to zero,

$$u_2=0\ ,\eqno (3.9)$$

and require that

$$u_{1t}+u_1u_{1x}=\sigma u_{1xx}\ ,\eqno (3.10)$$

then

$$u_j=0\ ,\qquad \ge 2\ .$$

In this case, we find the following Bäcklund transform [10] for Burgers' equation,

$$u=-2\sigma\phi_x/\phi+u_1\ ,\eqno (3.11)$$

where $(u,u_1)$ satisfy Burgers' equation and

$$\phi_t+u_1\phi_x=\sigma\phi_{xx}\ .\eqno (3.12)$$

When $u_1=0$, the Cole-Hopf transform [7] is obtained, and when $u_1=\phi$ it is found that

$$u=-2\sigma\phi_x/\phi+\phi\ ,\eqno (3.13)$$

where

$$\phi_t+\phi\phi_x=\sigma\phi_{xx}\ .\eqno (3.14)$$

The Bäcklund transform (3.13) and (3.14) was discovered by Fokas [11], using the method of Lie symmetries. The general form of the Bäcklund transform, (3.11) and (3.12), appears to be a new result.