Re-writing a PDE in Hamiltonian form

Question

I am reading a paper about Camassa-Holm equation, which is given by $$ u_t-u_{txx}=-2\kappa u_x-3uu_x+2u_xu_{xx}+uu_{xxx}, \qquad t,x\in\mathbb{R}, $$ where $u(t,x)$ is a real-valued function and $\kappa\geq0$ is a fixed parameter. This equation has (among others) the following conservation laws: $$ E(u):=\int_{\mathbb{R}} u^2(t,x)+u_x^2(t,x)dx, $$ $$ F(u):=\int_{\mathbb{R}} u^3(t,x)+u(t,x)u_x^2(t,x)+2\kappa u^2(t,x)dx. $$ My question is the following, according to the author this equation can be re-written in Hamiltonian form as $$ \partial_tE'(u)=-\partial_x F'(u). $$ My problem is that I do not know how to compute $E'(u)$ and $F'(u)$, so I cannot verify that sentence. Could someone explain to me how to compute any of them? I would appreciate it. If anyone has any short reading on Hamiltonian PDEs, I would appreciate it very much. I hope something not too long, I do not intend to do a whole course on Hamiltonian PDEs (it's not my goal).

Answer 1

Briefly, the derivative $E'(u)$ is a linear functional on an appropriate function space. Let $v\in C^\infty_c(\mathbb{R})$, then $$ E'(u)(v)=\lim_{t\to 0}\frac{E(u+tv)-E(u)}{t} $$ So \begin{align*} E'(u)(v)&=\int_{\mathbb{R}} [2u(x,t)v(x,t)+2u_x(x,t)v_x(x,t)]\,\mathrm{d}x\\ &=\int_{\mathbb{R}} 2(u(x,t)-u_{xx}(x,t))v(x,t)\,\mathrm{d}x \end{align*} So abusing notation, we identify $E'(u)$ with the integration kernel $2(u-u_{xx})$. Similarly, $F'(u)=3u^2-u_x^2-2uu_{xx}+4\kappa u$.

Hence we check $\partial_t E'(u)=-\partial_x F'(u)$ by differentiating \begin{align*} \partial_t E'(u)&=2(u_t-u_{txx})\\ \partial_x F'(u) &=6uu_x-4u_xu_{xx}-2uu_{xxx}+4\kappa u_x\\ &=2(2\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}) \end{align*} by the PDE for $u$, we get $\partial_t E'(u)=-\partial_x F'(u)$.