QUIESCENT VECTOR OPTICAL SOLITONS WITH NONLINEAR DISPERSION

This paper investigates proportional vector quiescent solitons for a coupled nonlinear Schrödinger system with Kerr-type self-phase modulation, cross-phase modulation, and nonlinear dispersion.A proportional-amplitude reduction is introduced by assuming that the two components share a common envelope and phase while their relative amplitude remains fixed.Physically, this reduction selects an amplitude-and phase-locked vector mode with a fixed component intensity ratio.The reduction is admissible only when explicit algebraic compatibility relations connect the self-phase modulation, cross-phase modulation, and nonlinear dispersive coefficients.Under these constraints, the coupled model reduces to a scalar nonlinear Schrödinger equation with effective nonlinear dispersion.Within the proportional real-profile reduction, the nonlinear dispersive structure excludes nonzero-velocity localized waves and yields quiescent profiles.This zero-velocity conclusion is restricted to the common real-profile, constant-ratio, linear-phase ansatz and does not exclude moving waves with more general complex or nonproportional structures.The resulting stationary profile equation is solved by the enhanced direct algebraic method.Exact proportional vector quiescent wave families are obtained, including bright, kink-type, singular, Jacobi elliptic, Weierstrass elliptic, and straddled solutions.The limiting transitions among these families and the direct validation of the constructed profiles confirm the consistency of the analytical results.The full four-component perturbation operator is also derived.For a representative regular bright state, Chebyshev spectral computation places the spectrum on the imaginary axis to numerical tolerance, while direct Fourier-ETDRK4 propagation under independent component perturbations preserves the localized shape and vector locking over the simulated interval.These tests support numerical spectral stability and finite-time robustness for the tested bright branch only.