Existence of multi-soliton solution for Zakharov–Rubenchik system

In this study, we investigate the Zakharov–Rubenchik system, focusing on the existence of multi-soliton solutions. For a given set of N solitons {Rk}j=1N within this system, we establish the existence of a multi-soliton solution that, in the energy space, asymptotically approaches the sum ∑j=1NRk. The proof of this result builds upon and extends previous work on the nonlinear Schrödinger equation and generalized Klein–Gordon equations by incorporating the complexities introduced by quadratic and cubic nonlinearities in this system. Given the structure of the system, we adopt an alternative approach that avoids the traditional modulation arguments. In this paper, we focus on obtaining uniform estimates for the mass associated with the system. Specifically, we will demonstrate that for sufficiently large values of the relative velocity between the solitons, it is possible to control certain key estimates, which will allow us to guarantee the existence of multi-soliton solutions. Subsequently, leveraging the system’s coercivity property and these estimates, we will proceed to establish a bootstrap result in energy spaces, which will enable us to construct the solution that is the main object of our study.