Thresholds for Existence of Dispersion Management Solitons for General Nonlinearities

We prove a threshold phenomenon for the existence of solitary solutions of the dispersion management equation for positive and zero average dispersion for a large class of nonlinearities. These solutions are found as minimizers of nonlinear and nonlocal variational problems which are invariant under a large noncompact group. There exists a threshold such that minimizers exist when the power of the solitons is bigger than the threshold. Our proof of existence of minimizers is rather direct and avoids the use of Lions' concentration compactness argument. The existence of dispersion-managed solitons is shown under very mild conditions on the dispersion profile and the nonlinear polarization of an optical active medium, which cover all physically relevant cases for the dispersion profile and a large class of nonlinear polarizations; for example, they are allowed to change sign.