This tag is for questions relating to dispersive partial differential equation or dispersive PDE. Informally, “dispersion” will refer to the fact that different frequencies in this equation will tend to propagate at different velocities, thus dispersing the solution over time.
Definition: An evolution partial differential equation is dispersive if, when no boundary conditions are imposed, its wave solutions spread out in space as they evolve in time.
Example : Consider $~iu_t + u_{xx} = 0~$. If we try a simple wave of the form $~u(x, t) = Ae^{i(kx−ωt)}~$, we see that it satisfies the equation if and only if $~ω = k^2~$ . This is called the dispersive relation and shows that the frequency is a real valued function of the wave number. If we denote the phase velocity by $~v = \frac ωk~$ we can write the solution as $~u(x, t) = Ae^{ik(x−v(k)t)}~$ and notice that the wave travels with velocity $~k~$. Thus the wave propagates in such a way that large wave numbers travel faster than smaller ones.
For more details see
$1.~$ https://en.wikipedia.org/wiki/Dispersive_partial_differential_equation
$2.~$ "Nonlinear dispersive equations: local and global analysis" by Terence Tao
$3.~$ "AN INTRODUCTION TO DISPERSIVE PARTIAL DIFFERENTIAL EQUATIONS" by NIKOLAOS TZIRAKIS
