The Korteweg-de Vries equation is $$\partial_t\phi+\partial^3_x\phi-6\phi\partial_x\phi=0 $$
Apparently, it comes from the Lagrangian $$L=\frac{1}{2}\partial_t\psi\partial_x\psi+(\partial_x\psi)^3-\frac{1}{2}(\partial^2_x\psi)^2 $$
where $\partial_x\psi=\phi$ (as derived in the Wikipedia article).
This Lagrangian is non-degenerate (as per the requirement of the Ostrogradsky theorem) since $\partial^2_{\psi_{xx}}L\neq 0$. Therefore, this system ought to suffer from the Ostrogradsky instability.
Normally, that is interpreted to mean that it has no minimum energy (which is considered unphysical), but that interpretation relies on the Lagrangian depending on the second time derivative of the generalized coordinate (i.e. $\partial^2_t\psi$). Since this Lagrangian depends on the second derivative with respect to position, it could mean that the Ostrogradsky theorem doesn't apply or that the system is unbounded in momentum rather than energy or something else entirely - I just don't know.
Does this system suffer from the Ostrogrady instability, and if so, what does that mean for the behaviour of the system?