Dispersion relation

The dispersion relation $\omega =\omega (k)$ of a constant coefficient linear evolution equation determines how time oscillations $e_{}^{i\omega t}$ are linked to spatial oscillations $e_{}^{ik\cdot x}$ of wave number $k$. In other words, the dispersion relation is the function for which the plane waves $e^{ik\cdot x}e^{i\omega (k)t}$ solve the equation. For instance:

• For the transport equation $u_{t}+v\cdot u_{x}=0$, the dispersion relation is $\omega (k)=-v$.
• For the Airy equation, the dispersion relation is $\omega (k)=k^{3}$.
• For the free Schrodinger equation, the dispersion relation is $\omega (k)=-|k|^{2}$. (It is common to adjust the constants in this equation to change the dispersion relation to $\omega (k)=+|k|^{2}$ or $\omega (k)=-|k|^{2}/2$.)
• For the free wave equation, the dispersion relation is $\omega (k)=\pm |k|$.
• For the Klein-Gordon equation, the dispersion relation is $\omega (k)=\pm \langle k\rangle$.

The principle of stationary phase implies that waves of spatial frequency $k$ will propagate with group velocity $-\nabla _{k}\omega (k)$. This should be compared with the phase velocity $-k\omega (k)/|k|^{2}$.

For semilinear equations, we define the dispersion relation by using the dispersion relation of the linear component of the equation. The relationship between spatial oscillation, time oscillation, and velocity should now be considered only as being heuristic (which in general only tends to be accurate in the semi-classical (high-frequency) limit). Note however that the tool of X^s,b spaces can be used to capture this relationship more rigorously.

For variable coefficient or quasilinear equations, the dispersion relation can now depend on the position variable $x$ and time variable $t$. The relationship between position, frequency, and velocity then becomes one of Hamilton's equations of motion (after identifying frequency with momentum).