Dispersive partial differential equation

Remarks

Example: the Schrödinger equation $i u_t + \Delta u = 0$ has $\omega(\xi) = \abs{\xi}^2$, which is real, with Hessian $2I \ne 0$, so it is dispersive. The Airy equation $u_t + u_{xxx} = 0$, the linear part of the Korteweg–De Vries equation, has $\omega(\xi) = -\xi^3$, with $\omega''(\xi) = -6\xi$, so it is dispersive. Non-example: the transport equation $u_t + c u_x = 0$ has $\omega(\xi) = c\xi$, whose second derivative vanishes, so clause 2 fails; solutions move without changing shape. The heat equation $u_t = \Delta u$ has $\omega(\xi) = -i\abs{\xi}^2$, which is not real, so clause 1 fails; it is dissipative rather than dispersive. The one-dimensional wave equation, with $\omega = \pm c\xi$, is not dispersive. In dimension $n \ge 2$ the branches $\pm c\abs{\xi}$ have nonzero Hessian, so the wave equation meets clause 2, although it is often not called dispersive. By the Fourier transform, the solution is $\hat u(\xi, t) = e^{-i\omega(\xi) t}\hat u(\xi, 0)$ for a single branch, and a nondegenerate Hessian, as for the Schrödinger equation, is what yields decay estimates in $L^\infty$.