KdV equation as an integrable system

Example

The Korteweg–De Vries equation, written here as $$u_t + u_{xxx} + 6 u u_x = 0$$ for a real function $u(x, t)$ that decays fast as $\abs{x} \to \infty$, is an infinite-dimensional integrable system.

It is Hamiltonian: with $$\mathcal{H}(u) = \int_{\R} \left( \tfrac{1}{2} u_x^2 - u^3 \right) dx,$$ the variational derivative is $\delta \mathcal{H} / \delta u = -u_{xx} - 3 u^2$, and the equation reads $u_t = \partial_x \left( \delta \mathcal{H} / \delta u \right)$.

It has infinitely many conserved quantities. The first ones are $\int u \, dx$, $\int u^2 \, dx$ and $\int \left( u^3 - \tfrac{1}{2} u_x^2 \right) dx$. For the third one, integration by parts gives $\frac{d}{dt} \int u^3 \, dx = -3 \int u_x^3 \, dx$ and $\frac{d}{dt} \int \tfrac{1}{2} u_x^2 \, dx = -3 \int u_x^3 \, dx$, so the difference is constant.

It has a Lax pair: with the Schrödinger operator $L = -\partial_x^2 - u$, there is a third-order differential operator $B$ such that the equation is equivalent to $L_t = [B, L]$. Hence the spectrum of $L$ does not change in time.