On the existence of infinitely many modes of a nonlocal nonlinear Schrödinger equation related to dispersion-managed solitons

We present a comprehensive study of a nonlinear Schrodinger equation with additional quadratic potential and general, possibly highly nonlocal, cubic nonlinearity. In particular, this equation arises in a variety of applications and is known as the Gross--Pitaevskii equation in the context of Bose--Einstein condensates with parabolic traps or as a model equation describing average pulse propagation in dispersion-managed fibers. Both global and local bifurcation behavior is determined showing the existence of infinitely many symmetric modes of the equation. In particular, our theory provides a strict theoretical proof of the existence of a symmetric bi-soliton which recently was found by numerical simulations.