FluidPDEs
We would like to study the surface waves formed by incompressible fluids governed by the Euler equation. Dispersive equations generalize the Schrödinger equation:
The equation in the Fourier domain (w.r.t. ) is
The solution is given by the convolution of with a Gaussian: or equivalently integrating in the original domain:
If , as the integral is bounded and so the solution is bounded. This is called linear dispersion and is generalized by the following relation: with general solution For a periodic function , so that
We next consider a wavepacket, i.e. a function concentrated around some frequency , for example . Since it is concentrated around a point, a local Taylor approximation would be accurate leading to the solution
the first term is the Fourier transform of a wavepacket, the second term is a translation in frequency space modulation i.e. a modulation in the space domain: .