The multiphase Whitham modulation equations with N phases have 2N
characteristics which may be of hyperbolic or elliptic type. In this paper a
nonlinear theory is developed for coalescence, where two characteristics change
from hyperbolic to elliptic via collision. Firstly, a linear theory develops
the structure of colliding characteristics involving the topological sign of
characteristics and multiple Jordan chains, and secondly a nonlinear modulation
theory is developed for transitions. The nonlinear theory shows that coalescing
characteristics morph the Whitham equations into an asymptotically valid
geometric form of the two-way Boussinesq equation. That is, coalescing
characteristics generate dispersion, nonlinearity and complex wave fields. For
illustration, the theory is applied to coalescing characteristics associated
with the modulation of two-phase travelling-wave solutions of coupled nonlinear
Schr\"odinger equations, highlighting how collisions can be identified and the
relevant dispersive dynamics constructed.Comment: 40 pages, 2 figure
