We numerically investigate transverse stability and instability of so-called
cnoidal waves, i.e., periodic traveling wave solutions of the Korteweg-de Vries
equation, under the time-evolution of the Kadomtsev-Petviashvili equation. In
particular, we find that in KP-I small amplitude cnoidal waves are stable (at
least for spatially localized perturbations) and only become unstable above a
certain threshold. In contrast to that, KP-II is found to be stable for all
amplitudes, or, equivalently, wave speeds.

An efficient algorithm for computing the branching structure of a compact
Riemann surface defined via an algebraic curve is presented. Generators of the
fundamental group of the base of the ramified covering punctured at the
discriminant points of the curve are constructed via a minimal spanning tree of
the discriminant points. This leads to paths of minimal length between the
points, which is important for a later stage where these paths are used as
integration contours to compute periods of the surface.

The small dispersion limit of solutions to the Camassa-Holm (CH) equation is
characterized by the appearance of a zone of rapid modulated oscillations. An
asymptotic description of these oscillations is given, for short times, by the
one-phase solution to the CH equation, where the branch points of the
corresponding elliptic curve depend on the physical coordinates via the Whitham
equations. We present a conjecture for the phase of the asymptotic solution. A
numerical study of this limit for smooth hump-like initial data provides strong
evidence for the validity of this conjecture.