(i) Using the Cole-Hopf transformation

$$u=-\frac{2 \nu}{\phi} \frac{\partial \phi}{\partial x}$$

map the Burgers equation

$$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=\nu \frac{\partial^{2} u}{\partial x^{2}}$$

to the heat equation

$$\frac{\partial \phi}{\partial t}=\nu \frac{\partial^{2} \phi}{\partial x^{2}}$$

(ii) Given that the solution of the heat equation on the infinite line $\mathbb{R}$ with initial condition $\phi(x, 0)=\Phi(x)$ is given by

$$\phi(x, t)=\frac{1}{\sqrt{4 \pi \nu t}} \int_{-\infty}^{\infty} \Phi(\xi) e^{-\frac{(x-\xi)^{2}}{4 \nu t}} d \xi$$

show that the solution of the analogous problem for the Burgers equation with initial condition $u(x, 0)=U(x)$ is given by

$$u=\frac{\int_{-\infty}^{\infty} \frac{x-\xi}{t} e^{-\frac{1}{2 \nu} G(x, \xi, t)} d \xi}{\int_{-\infty}^{\infty} e^{-\frac{1}{2 \nu} G(x, \xi, t)} d \xi}$$

where the function $G$ is to be determined in terms of $U$.

(iii) Determine the ODE characterising the scaling reduction of the spherical modified Korteweg-de Vries equation

$$\frac{\partial u}{\partial t}+6 u^{2} \frac{\partial u}{\partial x}+\frac{\partial^{3} u}{\partial x^{3}}+\frac{u}{t}=0$$