# Incoherent Shock and Collapse Singularities in Non-Instantaneous Nonlinear Media
## 2 NLS Simulations  
A non-instantaneous non-linear response of the medium arises in several problems of radiation-matter interaction. A typical example in one-dimensional systems is provided by the Raman effect in optical fibers, which finds its origin in the delayed molecular response of the material. We consider the standard one-dimensional NLS equation accounting for a non-instantaneous non-linear response function  
\[i\partial_{z}\psi+\frac{\sigma}{2}\partial_{tt}\psi+\psi\int R(t-t^{\prime}) \ |\psi|^{2}(z,t^{\prime})\ dt^{\prime}=0. \tag{1}\]  
For convenience, we normalized the equation with respect to the 'healing time' \(\tau_{0}=\sqrt{|\beta_{2}|/(\gamma\rho)}\) and the non-linear length scale \(L_{0}=1/(\gamma\rho)\), where \(\rho\) is the power, \(\beta_{2}\) the dispersion coefficient (\(\sigma=-\text{sign}(\beta_{2})\)) and \(\gamma(>0)\) the non-linear coefficient. The time \(\tau_{0}\) plays a key role: It refers to the time scale for which linear and non-linear effects are of same order of magnitude (e.g., \(\tau_{0}\) is the typical modulational instability period in the limit of an instantaneous non-linearity [28]). The variables can be recovered in real units through the transformations \(t\to t\tau_{0}\), \(z\to zL_{0}\), \(\psi\rightarrow\psi\sqrt{\rho}\) and \(R(t)\rightarrow\tau_{0}R(t)\). Hence \(\mathcal{N}/T_{0}=T_{0}^{-1}\int|\psi|^{2}\,dt=1\), where \(T_{0}\) is the temporal numerical window (\(T_{0}=10^{4}\) in our simulations). The response function \(R(t)\) is constrained by the causality condition. In the following we use the convention that \(t>0\) corresponds to the leading edge of the pulse, so that the causal response will be on the trailing edge of a pulse, i.e., \(R(t)=0\) for \(t>0\) (clearly, the physical phenomena we are going to present do not depend on the choice of the convention). We will write the response function in the form \(R(t)=H(-t)\bar{R}(-t)\), where \(\bar{R}(t)\) is a smooth function from \([0,\infty)\) to \((-\infty,\infty)\), while the Heaviside function \(H(-t)\) ensures the causality property. In the following we focus the presentation on the normal dispersion regime, where a priori one would not expect the formation of a collapse-like behavior of the wave. The formation of a shock-like singularity is known to require a strong non-linear interaction, i.e., a regime in which non-linear effects dominate linear dispersion effects \(\tau_{0}\ll t_{c}\), where \(t_{c}\) denotes the time correlation of the initial partially coherent wave--note that this condition is analogous to (\(L_{d}=t_{c}^{2}/|\beta_{2}|\) being the dispersion length in dimensional units). If we consider a quasi-instantaneous non-linear regime \(\tau_{R}\sim\tau_{0}\), then since the response time is smaller than the correlation time (\(\tau_{R}\ll t_{c}\)), each individual fluctuation in the initial pulse will lead to the formation of a dispersive shockwave. This quasi-instantaneous non-linear regime is in some sense similar to what occurs for a purely coherent initial pulse since the fluctuations of the incoherent pulse evolve (at least initially) independently of each other for \(t_{c}\gg\tau_{R}\). It is important to note that in this regime the intensity does not exhibit a collapse-like behavior, a feature that will be discussed below. The situation is fundamentally different in the regime characterized by a highly non-instantaneous non-linear response time, \(\tau_{R}\gg t_{c}\). In contrast with the quasi-instantaneous non-linear regime, here the incoherent pulse exhibits a global collective behavior, in which it is the incoherent pulse as a whole that exhibits a shock singularity.