Optical wave thermalization is a process characterized by an irreversible evolution of the field toward a thermodynamic equilibrium state. We study the influence of third-order dispersion on the thermalization of an optical wave propagating in a single mode optical fiber (1D nonlinear SchrÃÂ¨odinger equation). The optical field exhibits a process of anomalous thermalization: It relaxes toward an equilibrium state of fundamental different nature than the thermodynamic equilibrium state. The spectrum of the wave is shown to exhibit a highly asymmetric deformation characterized by a lateral spectral shoulder and the subsequent formation of a constant spectral pedestal. The wave turbulence theory reveals that this anomalous thermalization is due to the existence of a local invariant in frequency space, which explains in detail the essential properties of the highly asymmetric spectral evolution of the wave.© (2010) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.