Literature

• A. Scott, ”Nonlinear Science” (Sec. 3.3).
• Sulem & Sulem, "The nonlinear Schroedinger equation".

Waves and dispersion relation (Video)

The wave equation has solutions

$$ \phi(x,t) = A\,e^{i(kx-\omega t)},\quad \omega(k) = ck. $$

A general solution is an integral over Fourier components

$$ \Phi(x,t) = \int F(k)\,e^{i[kx-\omega(k) t]}\, dk $$

where $F(k)$ are amplitudes depending on the initial condition.

Quadratic dispersion

Dispersion relation. A wave function has wave solutions with angular frequency $\omega$ and wave number $k$ with $\omega = \omega(k)$, but this will, in general, not be the linear relation of the wave equation.

Consider a localized wave form with main frequency $\omega_0$ and dispersion, for $\omega$ close to $\omega_0$,

$$ \omega = \omega_0 + b_1 (k-k_0) + b_2 (k-k_0)^2 + \ldots. $$

The wave form is

$$ \Phi(x,t) = e^{i(k_0 x - \omega_0 t)} \int F(k)\,e^{i[(k-k_0)x-b_1(k-k_0)t - b_2(k-k_0)^2 t]}\, dk $$

where $e^{i(k_0 x - \omega_0 t)}$ is a carrier wave.

We are interested in the envelope wave