Multi-speed solitary waves for the Klein–Gordon–Schrödinger system with cubic interaction

Considered in this report is the existence of multi-speed solitary waves of a Klein–Gordon–Schrödinger system with cubic interaction. These solutions behave at large time as a couple of single solitary waves at different speeds. This type of solutions has been investigated for nonlinear Schrödinger system in [15] and [27]. It is obtained by solving the system backward in time around a sequence of approximate multi-speed solitary waves and showing convergence to a solution with the desired property. The new ingredients of the proof are coercivity of the Hessian of the action around each component of the multi-speed solitary waves, energy estimate and finite speed of propagation.