Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 31E (Integrable Systems)

Set $u=-2\nu {\partial }_x\log \varphi$. Direct differentiation givesThus the Cole-Hopf transformation maps every nonvanishing solution of the heat equation to a solution of the viscous Burgers equation. Conversely the Burgers equation makes the ratio inside the derivative a function of time alone; a time-dependent rescaling of $\varphi$, which leaves $u$ unchanged, removes that function and gives the heat equation exactly.

The initial data require $\Phi (\xi )=\exp [-(2\nu {)}^{-1}{\int }_{x_{\ast }}^{\xi }U(s)ds]$, up to a positive constant that cancels from $u$. Substituting this into the heat kernel formula gives the exponent $-G/(2\nu )$ withDifferentiate the kernel integral in $x$ and apply the Cole-Hopf transformation. Its constant prefactor cancels, leavingThe choice of $x_{\ast }$ adds an irrelevant constant to $G$; the usual convergence assumptions on the initial data justify the integrals and differentiation.

Balancing the time derivative, cubic nonlinearity and third spatial derivative gives $x\sim t^{1/3}$ and $u\sim t^{-1/3}$. Choose $\zeta =x/(3t{)}^{1/3}$ and $u=(3t{)}^{-1/3}w(\zeta )$. ThenThus the similarity solution isThe extra spherical term changes the coefficient of $w$. This third-order equation cannot be integrated directly into the standard second-order Painlevé II equation of the ordinary modified KdV reduction.