# Incoherent Shock and Collapse Singularities in Non-Instantaneous Nonlinear Media
## 4 Hydrodynamic-Like Model
### Evolution Along the Characteristics  
The singular behavior of the random wave can be described theoretically by solving Equations (3) and (4) by the method of the characteristics [42]. We define \(w(z)=\Omega(T(z),z)\), \(\xi(z)=\partial_{z}\Omega\big{(}T(z),z\big{)}\), \(\tau(z)=\partial_{t}\Omega\big{(}T(z),z\big{)}\), and \(\phi(z)=N\big{(}T(z),z\big{)}\), which can be shown to satisfy the set of coupled ordinary differential equations (ODE):\[\dot{T}(z) = \sigma w(z),\quad T(0)=t_{0}, \tag{5}\] \[\dot{w}(z) = \xi(z)+\sigma w(z)\tau(z),\quad w(0)=0,\] (6) \[\dot{\xi}(z) = -\partial_{zt}^{2}V(T(z),z)-\sigma\tau(z)\xi(z),\;\xi(0)=-\partial _{t}V(t_{0},0)\] (7) \[\dot{\tau}(z) = -\partial_{t}^{2}V(T(z),z)-\sigma\tau^{2}(z),\quad\tau(0)=0,\] (8) \[\dot{\phi}(z) = -\sigma\tau(z)\phi(z),\quad\phi(0)=N(t_{0},0), \tag{9}\]  
where the dots denote the derivatives with respect to the evolution variable, \(\dot{T}(z)=\partial_{z}T(z)\). The method of the characteristics allows for a major simplification of the problem since the partial difference Equations (3) and (4) are reduced to a small number of ODEs (5)-(9) without approximations. A significant advantage of the ODEs (3) and (4) is that they show that the PDE system (5)-(9) exhibits finite 'time' singularities for the gradient of the momentum and the intensity. In addition, the ODEs (5)-(9) specify _how_ such singularities develop during the time evolution of the system, see Equation (11) below. The equations for the characteristics (5)-(9) look similar to those considered in the spatial domain for the propagation of two-dimensional speckle beams [23]. There is however an important difference with the spatial case: In the temporal case considered here, because of the causality condition inherent to the temporal response function, the _derivatives of \(R(t)\) are not continuous at \(t=0\)_. This aspect becomes relevant by noting that the whole dynamics is driven by the gradient of the envelope of the incoherent pulse, i.e., the variable \(\tau(z)\) in Equation (8). In contrast with the spatial case, _the effective potential \(V(t,z)\) and its derivatives are not uniformly bounded_. As a consequence, the first source term in the equation for \(\tau(z)\) takes the general form \(-2\pi\partial_{t}^{2}V(t,z)=\bar{R}^{\prime}(0)N(t,z)-\bar{R}(0)\partial_{t}N (t,z)+\int_{0}^{\infty}\bar{R}^{\prime\prime}(\theta)N(t+\theta)d\theta\). Considering the continuity of the response function (\(\bar{R}(0)=0\)) [43], we can write the evolution of the potential along the characteristic \(T(z)\)  
\[-2\pi\partial_{t}^{2}V(T(z),z)=R^{\prime}(0)\phi(z)+\int_{0}^{\infty}R^{\prime \prime}(\theta)N(T(z)+\theta,z)d\theta. \tag{10}\]  
The second term in the right-hand side is bounded (because \(\bar{R}^{\prime\prime}(t)\) is bounded and \(\int N(t,z)dt=2\pi N\) is constant). We anticipate that the first term is smaller than \(\tau^{2}(z)\) in Equation (8) (see Equation (11)), so that it cannot prevent from the catastrophic singularity. We then obtain the singular behaviors of \(\tau(z)\) and \(\phi(z)\) just before the singularity at \(z=z_{\infty}\):  
\[\phi(z)\simeq\frac{1}{z_{\infty}-z},\qquad\tau(z)\simeq\frac{-\sigma}{z_{ \infty}-z}. \tag{11}\]  
Please note that the behavior of \(\phi(z)\) is obtained by remarking that, irrespective of the dispersion regime (normal or anomalous), we have from Equation (9) \(\phi(z)=N(t_{0},0)\exp\big{(}|\int_{0}^{z}\tau(s)ds|\big{)}\). To complete our study we have solved by numerical integration the complete set of ODE (5)-(9)--note in this respect that the first term in the right-hand side of (7) involves partial derivatives of the effective potential \(V(t,z)\) taken along the characteristics \(T(z)\), so that the numerical integration requires the explicit evolution of \(V(t,z)\) which is obtained by solving the hydrodynamic Equations (3) and (4).