# Incoherent Shock and Collapse Singularities in Non-Instantaneous Nonlinear Media
## 1 Introduction  
Shockwaves are known to play a key role in many different branches of physics [1; 2; 3]. When dissipative effects can be neglected, the formation of shockwaves is regularized, owing to dispersion, through the onset of rapidly oscillating nonstationary structures, the so-called dispersive shockwaves (DSWs) or undular bores. Originally observed in plasmas [4], water surface waves [5] and fiber optics [6], DSWs have become the subject of intense theoretical and experimental studies in different areas and specifically in non-linear optics where they have regained great interest [7; 8; 9; 10; 11; 12], in particular to study hydrodynamic analogues [13], non-local non-linearities [14; 15; 16; 17; 18] or the impact of a structural disorder of the non-linear material [19; 20].  
These previous studies on DSWs have been reported for purely coherent, i.e., deterministic, amplitudes of the waves. From a different perspective, a rather recent work predicted the existence of incoherent DSWs, which manifest themselves as a wave breaking process (\"gradient catastrophe\") in the spectral dynamics of the incoherent wave evolving in a non-instantaneous non-linear environment [21; 22]. We note that such a DSW behavior solely occurs in the spectral domain: The shock singularity cannot be identified in the spatio-temporal domain, where the incoherent wave exhibits fluctuations that are statistically stationary in time.  
Our aim in this work is to study the development of a shock singularity with a highly non-instantaneous non-linearity starting from an initial incoherent wave whose envelope profile is localized in time, i.e., starting from an incoherent optical pulse that propagates, e.g., in an optical fiber. On the basis of a long-range Vlasov-like formalism, we show that the incoherent wave exhibits ashock singularity that is of different nature than conventional DSW [1; 2; 3]. The analysis reveals that the optical wave exhibits a _global collective behavior in which it is the incoherent pulse as a whole that exhibits a shock singularity_.  
This type of collective incoherent behavior is of the same form as the incoherent shocks [23] identified in the _spatial domain_ in the presence of a highly non-local spatial response [14; 15; 16; 24; 25; 26; 27]. In particular, on the basis of a temporal version of the long-range Vlasov formalism [28], here we show that the incoherent pulse exhibits the CSCS behavior: In addition to the expected shock singularity for the evolution of the momentum, the incoherent pulse also exhibits a _collapse-like singularity_ for the evolution of the intensity of the random wave. In contrast with the previous study reported in the spatial domain [23], here in the temporal domain the response function is constrained by the causality condition, a distinguished feature which confers an _asymmetric temporal dynamics_ to the evolution of the wave. As a consequence, the intensity of the incoherent pulse as a whole exhibits a temporally drifted collapse behavior characterized by a high-intensity peak--a feature that may be interpreted in analogy with an incoherent rogue wave phenomenon. In this work we provide physical insights into the nature of the CSCS by using the method of the characteristics to solve in explicit form a reduced hydrodynamic model derived from the temporal Vlasov equation. In addition, we study the subsequent post-shock and post-collapse behaviors through the analysis of the phase-space dynamics of the incoherent pulse.  
Besides its fundamental interest, optical fibers and waveguides [29] turn out to be ideal testbeds for the experimental verification of our predictions, thanks to the easily tailorable Raman response function, as well as other recently investigated mechanisms involving liquid or gas-filled photonic crystal fibers (PCF), as well as surface plasmon polariton systems [30; 31; 32; 33; 34; 35; 36; 37].