A higher order system of some coupled nonlinear Schrödinger and Korteweg-de Vries equations

We prove the existence and multiplicity of bound and ground state solutions, under appropriate conditions on the parameters, for a bi-harmonic stationary system coming from a system of coupled nonlinear Schrödinger–Korteweg-de Vries equations. We arrive at that stationary system looking for “standing-traveling” wave solutions. We first show the existence of a semi-trivial solution of the form (0, V2), where V2 is a ground state of Δ2v+λ2v=12||v||v. This semi-trivial solution will have the lowest energy among all the semi-trivial solutions. Moreover, depending on the coupling parameter, this semi-trivial solution will be a strict local minimum or a saddle point. Furthermore we show the existence of a global minimum on the Nehari manifold with energy below the energy of the semi-trivial solution, for some values of the coupling parameter. In addition, by applying the mountain-pass theorem, we find another critical point for certain values of the parameters. All of this is obtained constraining the functionals to the appropriate Nehari manifolds and, in the high-dimensional case, restricted to radial framework. This analysis is supported by some numerical evidence finding the profiles of some solutions