Quiescent Solitons for the Resonant Nonlinear Schrödinger’s Equation in Quantum Optics and Quantum Fluids

This work constructs exact quiescent solitons of the resonant nonlinear Schrödinger equation with nonlinear chromatic dispersion and nine distinct self-phase-modulation laws. Using an enhanced direct algebraic method, we derive bright, dark, singular, and straddled solitons and classify their existence domains via explicit parameter constraints. For the Kerr law, the model supports bright and singular solitons with amplitudes determined by dispersion parameters; for the power-law case, bright and singular families appear with characteristic hyperbolic profiles; and for elliptic-function constructions, Jacobian and Weierstrass forms reduce to solitons in the modulus-one limit. The analysis also yields the algebraic constraints required for physical realizability. Collectively, these results delineate when nonlinear chromatic dispersion, together with generalized self-phase modulation, produces stationary localized structures in quantum-optical and quantum-fluid settings.