Description of the Inelastic Collision of Two Solitary Waves for the BBM Equation

We prove that the collision of two solitary waves of the BBM equation is inelastic but almost elastic in the case where one solitary wave is small in the energy space. We show precise estimates of the nonzero residue due to the collision. Moreover, we give a precise description of the collision phenomenon (change of size of the solitary waves and shifts in their trajectories). To prove these results, we extend the method introduced in Martel and Merle (Description of two soliton collision for the quartic gKdV equation, submitted preprint. http://arxiv.org/abs/0709.2672;Commun Math Phys 286:39-79, 2009) for the generalized KdV equation, in particular in the quartic case. The main argument is the construction of an explicit approximate solution in the collision region.