The propagation of partially coherent nonlinear optical waves is a topic of growing interest in different fields of investigations, such as, e.g., wave propagation inhomogeneous or periodic media, waveguides, cavity systems [1-5]. Here we study the temporal evolution of a partially coherent wave that propagates in a nonlinear medium characterized by a non-instantaneous (Raman-like) response function. In this case, the incoherent wave can exhibit two interesting behaviors, (i) it can self-organize into a spectral incoherent soliton, i.e., an incoherent soliton that cannot be identified in the spatio-temporal domain, but solely in the spectral domain , (ii) the spectrum of the incoherent wave can exhibit singular behaviors, such as incoherent dispersive shock waves (IDSW) or incoherent collapse singularities (ICS), depending on the form of the response function .
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