On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

We give a simple proof of existence of solutions of the dispersion management and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local variational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions’ concentration compactness argument or Ekeland’s variational principle.