Quantum Fluctuations around Bistable Solitons in the Cubic-Quintic   nonlinear Schr\"{o}dinger equation
  Small quantum fluctuations in solitons described by the cubic-quintic nonlinear Schr\"{o}dinger equation (CQNLSE) are studied with the linear approximation. The cases of both self-defocusing and self-focusing quintic term are considered (in the latter case, solitons may be effectively stable, despite the possibility of collapse). The numerically implemented back-propagation method is used to calculate the optimal squeezing ratio for the quantum fluctuations vs. the propagation distance. In the case of the self-defocusing quintic nonlinearity, opposite signs in front of the cubic and quintic terms make the fluctuations around bistable pairs of solitons (which have different energies for the same width) totally different. The fluctuations around nonstationary Gaussian pulses in the CQNLSE model are studied too. 
Ray-Kuang Lee, Yinchieh Lai, and Boris A. Malomed