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

We give a simple proof of the existence of solutions of the dispersion management and diffraction management equations for a 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 the existence of a maximizer is rather direct and avoids the use of Lions’ concentration compactness argument or Ekeland’s variational principle. The existence of diffraction managed solitons in the discrete case is shown under the weakest possible assumptions on the diffraction profile, and the existence of dispersion managed solitons in the continuous case is shown under very mild conditions on the dispersion profile, which cover all physically relevant cases.