  We investigate the dynamics of localized solutions of the relativistic cold
fluid plasma model in the small but finite amplitude limit, for slightly
overcritical plasma density. Adopting a multiple scale analysis, we derive a
perturbed nonlinear Schr\"{o}ndinger equation that describes the evolution of
the envelope of circularly polarized electromagnetic field. Retaining terms up
to fifth order in the small perturbation parameter, we derive a self-consistent
framework for the description of the plasma response in the presence of
localized electromagnetic field. The formalism is applied to standing
electromagnetic solitary wave interactions and the results are validated by
simulations of the full cold-fluid model. To lowest order, a cubic nonlinear
Schr\"{o}ndinger equation with a focusing nonlinearity is recovered. Classical
quasiparticle theory is used to obtain analytical estimates for the collision
time and minimum distance of approach between solitary waves. For larger
solitary wave amplitudes the inclusion of the fifth order terms is essential
for a qualitatively correct description of solitary wave interactions. The
defocusing quintic nonlinearity leads to inelastic solitary wave collisions,
while bound states of solitary waves appear unstable with respect to
perturbations in the initial phase or amplitude.
