Modelling relativistic solitary wave interactions in over-dense plasmas:   a perturbed nonlinear Schr\"{o}ndinger equation framework
  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. 
E. Siminos, G. S\'anchez-Arriaga, V. Saxena, I. Kourakis
physics.plasm-ph nlin.PS