  The two most commonly used models for passively modelocked lasers with fast
saturable absorbers are the Haus modelocking equation (HME) and the
cubic-quintic modelocking equation (CQME). The HME corresponds to a special
limit of the CQME in which only a cubic nonlinearity in the fast saturable
absorber is kept in the model. Here, we use singular perturbation theory to
demonstrate that the CQME has a stable high-energy solution for an arbitrarily
small but non-zero quintic contribution to the fast saturable absorber. As a
consequence, we find that the CQME predicts the existence of stable modelocked
pulses when the cubic nonlinearity is orders of magnitude larger than the value
at which the HME predicts that modelocked pulses become unstable. This
intrinsically larger stability range is consistent with experiments. Our
results suggest a possible path to obtain high-energy and ultrashort pulses by
fine tuning the higher-order nonlinear terms in the fast saturable absorber.
