Quiescent Optical Solitons for Cubic–Quintic Nonlinear Schrödinger’s Equation with Intensity-Dependent Dispersion and Weak Nonlocality

This study examined quiescent optical structures in a cubic–quintic nonlinear Schrödinger model that combines intensity-dependent dispersion with a weakly nonlocal intensity-curvature response. A real phase–amplitude reduction establishes the compatibility conditions for stationary localized profiles. The enhanced direct algebraic method then produces regular bright and dark states, singular hyperbolic states, and Jacobi and Weierstrass elliptic families. Spatially shifted formulas are identified as translated representatives rather than new orbit types. Each family is validated through the auxiliary equation, the stationary residual, explicit reality and nondegeneracy restrictions, and an independent first-integral formulation. The quintic response changes the dominant balance and the admissible coefficient manifolds relative to the corresponding Kerr-only setting. Parameter continuations illustrate distinct roles of cubic, quintic, dispersive, and weakly nonlocal effects without implying unconstrained one-parameter dynamics. For a representative regular bright state, refined Chebyshev collocation locates no persistent unstable eigenvalue and direct Fourier propagation under simultaneous amplitude and phase perturbations remains bounded. The numerical conclusion is restricted to the tested branch and finite propagation interval.