Consider the Poisson structure

$$\{F, G\}=\int_{\mathbb{R}} \frac{\delta F}{\delta u(x)} \frac{\partial}{\partial x} \frac{\delta G}{\delta u(x)} d x$$

where $F, G$ are polynomial functionals of $u, u_{x}, u_{x x}, \ldots$. Assume that $u, u_{x}, u_{x x}, \ldots$ tend to zero as $|x| \rightarrow \infty$.

(i) Show that $\{F, G\}=-\{G, F\}$.

(ii) Write down Hamilton's equations for $u=u(x, t)$ corresponding to the following Hamiltonians:

$$H_{0}[u]=\int_{\mathbb{R}} \frac{1}{2} u^{2} d x, \quad H[u]=\int_{\mathbb{R}}\left(\frac{1}{2} u_{x}^{2}+u^{3}+u u_{x}\right) d x$$

(iii) Calculate the Poisson bracket $\left\{H_{0}, H\right\}$, and hence or otherwise deduce that the following overdetermined system of partial differential equations for $u=u\left(x, t_{0}, t\right)$ is compatible:

$$\begin{gathered} u_{t_{0}}=u_{x} \\ u_{t}=6 u u_{x}-u_{x x x} \end{gathered}$$

[You may assume that the Jacobi identity holds for (1).]

(iv) Find a symmetry of (3) generated by $X=\partial / \partial u+\alpha t \partial / \partial x$ for some constant $\alpha \in \mathbb{R}$ which should be determined. Construct a vector field $Y$ corresponding to the one parameter group

$$x \rightarrow \beta x, \quad t \rightarrow \gamma t, \quad u \rightarrow \delta u,$$

where $(\beta, \gamma, \delta)$ should be determined from the symmetry requirement. Find the Lie algebra generated by the vector fields $(X, Y)$.