How dispersive term in KdV PDE causes smoothing

Question

In KdV equation, $$u_t+u_{xxx}-6uu_x=0,$$ $uu_x$ is the nonlinear term which cause blow up, and $u_{xxx}$ is the dispersive term, I am wondering how the dispersive term smooth the solution, otherwise solitary wave solution won't exist.

Answer 1

Dispersion makes different spectral components travel at different speeds. This can be illustrated by the dispersion relation $\omega = -\kappa^3$ valid for small $|u| \ll 1$, which links the temporal frequency $\omega$ to the spatial frequency (wavenumber) $\kappa$. It follows that the phase velocity is a function of the wavenumber. Now consider a waveform with various frequencies inside. If the spectral frequencies separate, then high frequencies (fast oscillations) will separate from lower frequencies (slow oscillations). This 'filtering' makes the low-frequency waveform look smoother, as fast oscillations have been removed.