Weak turbulence is the phenomenon that solutions, even when they exist globally, tend to shift their energy into increasingly high frequencies over time. One way to detect this is by computing higher Sobolev norms (e.g. the norm) and observing that they grow at some suitable rate (logarithmic, polynomial, or exponential).
Weak turbulence is incompatible with scattering, and is also usually incompatible with complete integrability due to the numerous conservation laws available in that case. It has been observed numerically in periodic non-integrable equations, though rigorous results establishing weak turbulence are extremely rare.