Multi-soliton solutions of Klein-Gordon-Zakharov system

In this study, we investigate the Klein-Gordon-Zakharov system with a focus on identifying multi-soliton solutions. Specifically, for a given number $N$ of solitons, we demonstrate the existence of a multi-soliton solution that asymptotically converges, in the energy space, to the sum of these solitons. Our proof extends and builds upon the previous results in \cite{cote, cotem, IA} concerning the nonlinear Schr\"{o}dinger equation and the generalized Klein-Gordon equation. In contrast to the method used in \cite{cotem} to establish the existence of multi-solitons for the Klein-Gordon equation, where the difficulty arises from the directions imposed by the coercivity property, requiring the identification of eigenfunctions of the coercivity operator to derive new control estimates, the structure of the present system allows for a more refined result. Specifically, the directional constraints can be eliminated by employing orthogonality arguments derived from localization and modulation techniques.