Quiescent optical solitons with nonlinear Schrödinger’s equation having Kerr law of self-phase modulation and intensity-dependent dispersion

Quiescent optical solitons provide stationary localized states in media where nonlinear dispersion can suppress ordinary wave translation. Motivated by the need to understand this mechanism, we study a Kerr-type nonlinear Schrödinger equation containing two intensity-dependent dispersive corrections. The objective is to determine the admissible real-profile wave reduction and construct its exact stationary solutions. A Lie-symmetry analysis shows that, when the envelope-curvature correction is active, a nonconstant localized real profile cannot sustain a nonzero carrier wavenumber or propagation velocity; consequently, the admissible localized structures are quiescent. The resulting nonlinear ordinary differential equation is solved using the enhanced direct algebraic method. Under the closure condition associated with the adopted auxiliary-function ansatz, bright, dark kink-type, singular, Jacobi elliptic, Weierstrass elliptic, and straddled solution families are derived. Their limiting connections are established, demonstrating how elliptic and straddled profiles recover standard hyperbolic solitons.