Nonlinear wave structures in a higher-order NLSE with dispersion, Kerr nonlinearity, and Raman effects

The generalized higher-order nonlinear Schrödinger equation (GHO-NLSE) provides an important framework for describing nonlinear wave propagation in optical fibers and other dispersive media where higher-order effects become significant. The considered model incorporates second- and third-order dispersion together with Kerr nonlinearity (KN), self-steepening (SSt), and Raman-related nonlinear effects (RrNLE). In this work, six families of analytical travelling-wave solutions (TWS) are constructed through an amplitude-phase formulation combined with hyperbolic, trigonometric, and Jacobi elliptic-function (JEF) ansätze. The resulting structures include a localized dipole-type solitary wave (SW), singular csch- and sec-type waves, and bounded JEF cn–dn, sn, and cn periodic wavetrains. Explicit algebraic relations among the wave amplitude, velocity, frequency, wavenumber, inverse width, elliptic modulus, and governing coefficients are derived together with the corresponding existence conditions. The limiting behavior of the JE solutions further connects the periodic wavetrains with trigonometric, kink-type, sech, and sech 2 profiles. The finite-distance numerical robustness of the analytical solutions is investigated by introducing small additive random perturbations and propagating the perturbed profiles using a split-step Fourier scheme (SSFS) with pseudo-spectral evaluation of the derivative nonlinearities. Numerical indicators based on relative power variation, peak-intensity variation, central intensity, and intensity-profile deviation show that the nonsingular localized and periodic solutions retain their principal structural features over the considered propagation interval, whereas the singular solutions exhibit stronger sensitivity to numerical regularization near their poles. The Vakhitov-Kolokolov slope (VKS) condition is considered as an analytical stability indicator only for applicable localized finite-power profiles. The obtained analytical solutions provide useful benchmark profiles for investigating higher-order nonlinear wave dynamics and for validating numerical methods for NLS systems.