We present a long-wavelength theory for the stability of periodic traveling wave solutions to equations of KdV type.
\[
u_t + (f(u))_x = u_{xxx}
\]
for essentially arbitrary $f$. Some examples are $f(u) = u^3 + a u^2$ (MKdV), which governs internal waves, and $f(u) = u^2 + a u^{3/2}$,  which arises in plasmas. The stability theory for solitary waves is well-developed but the analogous periodic problem is much  less well understood.

We give a rigorous construction for the spectrum of the linearized operator in a neighborhood of the origin in the spectral plane, and construct two stability indices. The first of these detects instability to perturbations of the same period, while the second detects instability to long-wavelength perturbations (a modulational instability). These stability indices can be expressed in terms of Jacobians of the map between the constants of integration of the traveling wave ODE and the conserved quantities of the PDE. This is, in essence, a rigorous Whitham modulation theory for the spectrum of the linearized operator.

This is joint work with Mathew Johnson.