Nonlinear waves in optical fibers and photonic waveguides: theoretical advances and modern applications

Nonlinear wave phenomena in optical fibers and photonic waveguides, governed by the nonlinear Schrödinger-type equation, have expanded from soliton physics into topological photonics, non-Hermitian systems, and integrated quantum technologies. The balance of group-velocity dispersion, Kerr nonlinearity, and higher-order effects controls soliton propagation, modulation instability, breather formation, and rogue wave statistics. Extensions to multimode fibers, Parity-Time-symmetric configurations, and Floquet topological lattices introduce distinct dynamics that are often examined separately, despite shared mathematical underpinnings. This review proceeds from mathematical foundations to theoretical developments and then to applied devices. It first treats the nonlinear Schrödinger equation and its integrable structure, the inverse scattering transform, the nonlinear Fourier transform, modulation instability with breathers and rogue waves, and integrable turbulence with soliton gas kinetic theory. It then addresses extended systems: multimode fiber dynamics including Kerr beam self-cleaning and spatiotemporal mode-locking, parity-time symmetry and non-Hermitian photonics, and nonlinear topological photonics with edge solitons and corner-state lasers. It closes with applications: Kerr microcomb generation and supercontinuum, and dispersion engineering in silicon, silicon nitride, and lithium niobate. Real-time rogue wave detection via the dispersive Fourier transform, chip-scale dissipative Kerr soliton combs, and topological laser arrays are among the advances covered here. Open problems remain in near-integrable multimode theory, non-equilibrium soliton gas thermodynamics, quantum nonlinear photonics, and heterogeneous integration of chip-scale classical-quantum photonic systems.