Gadi Fibich

Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension

The formation of singularities of self-focusing solutions of the nonlinear Schrodinger equation (NLS) in critical dimension is characterized by a delicate balance between the focusing nonlinearity ...

Conservative Visibility and Strong Occlusion for Viewspace Partitioning of Densely Occluded Scenes

Computing the visibility of out‐door scenes is often much harder than of in‐door scenes. A typical urban scene, for example, is densely occluded, and it is effective to precompute its visibility space, since from a given point only a small fraction of the scene is visible. The difficulty is that although the majority of objects are hidden, some parts might be visible at a distance in an arbitrary location, and it is not clear how to detect them quickly. In this paper we present a method to partition the viewspace into cells containing a conservative superset of the visible objects. For a given cell the method tests the visibility of all the objects in the scene. For each object it searches for a strong occluder which guarantees that the object is not visible from any point within the cell. We show analytically that in a densely occluded scene, the vast majority of objects are strongly occluded, and the overhead of using conservative visibility (rather than visibility) is small. These results are further supported by our experimental results. We also analyze the cost of the method and discuss its effectiveness.

Explicit Solutions of Optimization Models and Differential Games with Nonsmooth (Asymmetric) Reference-Price Effects

Models in marketing with asymmetric reference effects lead to nonsmooth optimization problems and differential games which cannot be solved using standard methods. In this study, we introduce a new method for calculating explicitly optimal strategies, open-loop equilibria, and closed-loop equilibria of such nonsmooth problems. Application of this method to the case of asymmetric reference-price effects with loss-aversive consumers leads to the following conclusions: (1) When the planning horizon is infinite, after an introductory stage the optimal price stabilizes at a steady-state price, which is slightly below the optimal price in the absence of reference-price effects. (2) The optimal strategy is the same as in the symmetric case, but with the loss parameter determined by the initial reference-price. (3) Competition does not change the qualitative behavior of the optimal strategy. (4) Adopting an appropriate constant-price strategy results in a minute decline in profits.

Multiple filamentation induced by input-beam ellipticity

We provide what is to our knowledge the first experimental evidence that multiple filamentation (MF) of ultra-short pulses can be induced by input beam ellipticity. Unlike noise-induced MF, which results in complete beam breakup, the MF pattern induced by small input beam ellipticity appears as a result of nucleation of annular rings surrounding the central filament. Moreover, our experiments show that input beam ellipticity can dominate the effect of noise (transverse modulational instability), giving rise to predictable and highly reproducible MF patterns. The results are explained with a theoretical model and simulations.

Self-Focusing with Fourth-Order Dispersion

We analyze self-focusing and singularity formation in the nonlinear Schrodinger equation (NLS) with high-order dispersion

Collapse dynamics of super-Gaussian Beams.

i \psi_t \pm \Delta^q \psi + |\psi|^{2 \sigma} \psi = 0,

Vectorial and random effects in self-focusing and in multiple filamentation

in the isotropic mixed-dispersion NLS

Control of multiple filamentation in air

i \psi_t + \Delta \psi +\epsilon \Delta^2 \psi + |\psi|^{2 \sigma} \psi = 0

Self-Focusing in the Damped Nonlinear Schrödinger Equation

, and in nonisotropic mixed-dispersion NLS equations which model propagation in fiber arrays.

Self-focusing of elliptic beams: an example of the failure of the aberrationless approximation

We investigate the self-focusing dynamics of super-Gaussian optical beams in a Kerr medium. We find that up to several times the critical power for self-focusing, super-Gaussian beams evolve towards a Townes profile. At higher powers the super-Gaussian beams form rings which break into filaments as a result of noise. Our results are consistent with the observed self-focusing dynamics of femtosecond laser pulses in air [1] in which filaments are formed along a ring about the axis of the initial beam where the initial beam did not form a ring.