Resonant

From DispersiveWiki


Given a constant coefficient linear operator L, a plane wave is resonant with respect to L if it is annihilated by L (i.e. it lies in the kernel of L). The frequency of this plane wave is then a resonant frequency. One thus expects these frequencies to give an extremely large contribution to Duhamel's formula, which is after all a formula which attempts to invert L.

If L is a linear dispersive operator such as , , or , then the collection of resonant frequencies is the graph of the dispersion relation.

A nonlinearity is resonant if resonant frequency inputs generate resonant (or nearly resonant) frequency outputs, and non-resonant otherwise. Using the Fourier transform, one can often usefully decompose a given nonlinearity into resonant and nonresonant components. One expects the former component to be the strongest (unless null forms are present in the nonlinearity). For the local wellposedness theory one can quantify this intuition via X^{s,b} spaces.

One can also attempt to use transforms to eliminate (or at least attenuate) nonresonant portions of the nonlinearity. The method of normal forms is particularly well suited to this task.