We prove a threshold phenomenon for the existence/non-existence of energy
minimizing solitary solutions of the diffraction management equation for
strictly positive and zero average diffraction. Our methods allow for a large
class of nonlinearities, they are, for example, allowed to change sign, and the
weakest possible condition, it only has to be locally integrable, on the local
diffraction profile. The solutions are found as minimizers of a nonlinear and
nonlocal variational problem which is translation invariant. There exists a
critical threshold ?cr such that minimizers for this variational problem exist
if their power is bigger than ?cr and no minimizers exist with power less than
the critical threshold. We also give simple criteria for the finiteness and
strict positivity of the critical threshold. Our proof of existence of
minimizers is rather direct and avoids the use of Lions' concentration
compactness argument.
Furthermore, we give precise quantitative lower bounds on the exponential
decay rate of the diffraction management solitons, which confirm the physical
heuristic prediction for the asymptotic decay rate. Moreover, for ground state
solutions, these bounds give a quantitative lower bound for the divergence of
the exponential decay rate in the limit of vanishing average diffraction. For
zero average diffraction, we prove quantitative bounds which show that the
solitons decay much faster than exponentially. Our results considerably extend
and strengthen the results of [15] and [16].Comment: 49 pages, no figure
