Soliton solutions and self-steepening in the photon-conserving nonlinear Schrödinger equation

"We have recently introduced a new modeling equation for the propagation of pulses in optical waveguides, the photon-conserving Nonlinear Schrödinger Equation (pcNLSE) which, unlike the canonical NLSE, guarantees strict conservation of both the energy and the number of photons for any arbitrary frequency-dependent nonlinearity. In this paper, we analyze some properties of this new equation in the familiar case where the nonlinear coefficient of the waveguide does not change sign. We show that the pcNLSE effectively adds a correction term to the NLSE proportional to the deviation of the self-steepening (SS) parameter from the photon-conserving condition in the NLSE. Furthermore, we describe the role of the self-steepening parameter in the context of the conservation of the number of photons and derive an analytical expression for the relation of the SS parameter with the time delay experienced by pulses upon propagation. Finally, we put forth soliton-like solutions of the pcNLSE that, unlike NLSE solitons, conserve the number of photons for any arbitrary SS parameter. "