KDV PDE: energy constant in time

Show that if u solves the KDV equation
$u_t + u_{xxx} + 6uu_x = 0$ for $x \in \mathbb{R}$, $t > 0$
then the energy
$\int_{-\infty}^{\infty} \frac{1}{2} u_x ^2 – u^3 \,dx$
is constant in time.

Attempt: The usual idea is to differentiate under the integral and then maybe do integration by parts, but I couldn’t find anything nice. Any hints? Thanks in advance.


Source: S06_Final_Exam-L.Evans.pdf

Actual Attempt: Put $e(t) = \int_{-\infty}^{\infty} \frac{1}{2} u_x ^2 – u^3 \,dx$. Differentiating under the integral and then integrating by parts, we have $e'(t) = \int_{-\infty}^{\infty} u_x u_{xt} – 3u^2 u_t \,dx = u_x u_t |^{+\infty}_{-\infty} – \int_{-\infty}^{\infty} u_{xx} u_t \,dx – \int_{\infty}^{\infty} 3u^2 u_t \,dx$.

Assuming u has compact support, the first term vanishes, so we’re left with
$e'(t) = -\int_{-\infty}^{\infty}u_t(u_{xx} + 3u^2)\,dx$.

The only reason why I think this may be useful is the fact that we can rewrite the KDV equation as $u_t + (u_{xx})_x + (3u^2)_x = 0$ and integrate. But I don’t see how to finish the problem from here, which is why I’m looking for other suggestions.

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