In this talk, we will be interested in standing wave solutions to a fourth order nonlinear Schrödinger equation having second and fourth order dispersion terms. This kind of equation naturally appears in nonlinear optics. In a first time, we will establish the existence of ground-state and renormalized solutions. We will then be interested in their qualitative properties, in particular their stability.

Joint works with Denis Bonheure, Ederson Moreira Dos Santos, Tianxiang Gou, Louis Jeanjean and Robson Nascimento.