A dispersive partial differential equation is one where solutions tend to spread out, due to waves with different frequencies traveling at different phase velocities. As a result, finite energy solutions tend to decay in amplitude, and solutions can be frequently more regular than the initial data. Examples of such equations include Schrödinger's equation, the Klein-Gordon equation, the Korteweg de-Vries equation (and many other equations governing surface waves on fluid interfaces), and the wave equation in dimensions > 1.