Optical Solitons in a Cascaded Quadratic Medium by the Lie Symmetry Method and a Generalized G′/G-Expansion Approach

Optical solitary waves in a cascaded quadratic medium are investigated using the Lie symmetry analysis and a generalized G′/G-expansion approach. The governing nonlinear evolution equation incorporates group-velocity dispersion, self-phase modulation, and two nonlinear derivative terms representing self-steepening contributions. Its real-valued formulation admits a four-dimensional non-Abelian symmetry algebra comprising phase rotation, translations, and a combined scaling and phase transformation. A traveling-wave reduction establishes compatibility conditions linking the wave velocity and self-steepening coefficients and yields a second-order nonlinear ordinary differential equation. The generalized expansion produces two parameter branches of exact hyperbolic wave solutions, including combo-bright-singular, dark, and singular structures subject to explicit parameter and denominator conditions. The analytical solution families are further examined through direct residual verification, linear spectral analysis, and modulation instability analysis, providing independent assessments of their validity and the stability of the nonzero background supporting the dark-soliton solution. For a representative admissible parameter set, the selected dark-soliton profile exhibits a localized intensity notch on a finite background and translates uniformly with the propagation coordinate. Three-dimensional surfaces, two-dimensional profiles, and contour plots of the intensity, modulus, and real and imaginary components distinguish envelope translation from carrier-phase evolution. The results provide a systematic characterization of the admissible solution branches, their regularity conditions, and their stability properties, while offering analytical reference profiles for subsequent numerical propagation and perturbation studies.