Consider the Kirchhoff equation, given by
$$u\_{tt}-\left(1+\int\_{\mathbb{R}} u\_x^2\;dx\right)u\_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}\_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. How to find the conserved quantitie of this equation?



How to find the conserved quantities of the Kirchhoff equation?
Some of them:


* Multiply by $u\_x$ and integrate, you get
$$ \int u\_{tt} u\_x ~dx = 0 $$
so 
$$ \partial\_t \int u\_{t} u\_x ~dx - \int u\_t u\_{tx} ~dx = 0 $$
the second term integrates to zero.
* Multiply by $u\_t$ and integrate by parts you get
$$ \int u\_{tt} u\_t + (1 + \int u\_x^2 ~dx) u\_{xt} u\_x + uu\_t - u^{2r+1} u\_t ~dx = 0 $$
This you can rewrite as
$$ \partial\_t \int \frac12 u\_t^2 + \frac12 (1 + \frac12 \int u\_x^2~dx) u\_x^2 + \frac12 u^2 - \frac1{2r+2} u^{2r+2} ~dx = 0$$


