To study the effect of boundaries on diffusion of new products, we introduce two novel analytic t... more To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.
Preprint Series of Department of Mathematics, Hokkaido University, 2007
We study analytically and numerically the stability of the standing waves for a nonlinear Schrödi... more We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blow-up in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.
We consider nonlinear bound states of the nonlinear Schrödinger equation i∂ z φ(z, x) = −∂ 2 x φ ... more We consider nonlinear bound states of the nonlinear Schrödinger equation i∂ z φ(z, x) = −∂ 2 x φ − (1 + m(N x))|φ| p−1 φ, in the presence of a nonlinear periodic microstructure m(N x). This equation models the propagation of laser beams in a medium whose nonlinear refractive index is modulated in the transverse direction, and also arises in the study of Bose-Einstein Condensation (BEC) in a medium with a spatially dependent scattering length. In the nonlinear optics context, N = r beam /r ms denotes the ratio of beam width to microstructure characteristic scale. We study the profiles of the nonlinear bound states using a multiple scale (homogenization) expansion for N ≫ 1 (wide beams), a perturbation analysis for N ≪ 1 (narrow beams) and numerical simulations for N = O(1). In the subcritical case p < 5, beams centered at local maxima of the microstructure are stable. Furthermore, beams centered at local minima of the microstructure are unstable to general (asymmetric) perturbations but stable relative to symmetric perturbations. In the critical case p = 5, a nonlinear microstructure can only stabilize narrow beams centered at a local maximum of the microstructure, provided that the microstructure also satisfies a certain local condition. Even in this case, the stability region is very small so that small (O(10 −2)) perturbations can destabilize the beam. Therefore, such beams are "mathematically" stable but "physically" unstable.
The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation ... more The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation NLS in critical dimension is characterized by a delicate balance between the focusing nonlinearity and di raction Laplacian, and is thus very sensitive to small perturbations. In this paper we i n troduce a systematic perturbation theory for analyzing the e ect of additional small terms on self focusing, in which the perturbed critical NLS is reduced to a simpler system of modulation equations that do not depend on the spatial variables transverse to the beam axis. The modulation equations can be further simpli ed, depending on whether the perturbed NLS is power conserving or not. We review previous applications of modulation theory and present several new ones that include: Dispersive saturating nonlinearities, self-focusing with Debye relaxation, the Davey Stewartson equations, self-focusing in optical ber arrays and the e ect of randomness. An important and somewhat surprising result is that various small defocusing perturbations lead to a generic form of the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. In the special case of the unperturbed critical NLS, modulation theory leads to a new adiabatic law for the rate of blowup which is accurate from the early stages of self-focusing and remains valid up to the singularity point. This adiabatic law preserves the lens transformation property of critical NLS and it leads to an analytic formula for the location of the singularity as a function of the initial pulse power, radial distribution and focusing angle. The asymptotic limit of this law agrees with the known loglog blowup behavior. However, the loglog behavior is reached only after huge ampli cations of the initial amplitude, at which point the physical basis of NLS is in doubt. We also include in this paper a new condition for blowup of solutions in critical NLS and an improved version of the Dawes-Marburger formula for the blowup location of Gaussian pulses.
Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this qu... more Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this question using the stochastic discrete Bass model in which consumers may differ in their individual external influence rates [Formula: see text] and in their individual internal influence rates [Formula: see text]. When the network is complete and the heterogeneity is only manifested in [Formula: see text] or only in [Formula: see text], it always slows down the diffusion, compared with the corresponding homogeneous network. When, however, consumers are heterogeneous in both [Formula: see text] and [Formula: see text], heterogeneity slows down the diffusion in some cases but accelerates it in others. Moreover, the dominance between the heterogeneous and homogeneous adoption levels is global in time in some cases but changes with time in others. Perhaps surprisingly, global dominance between two networks is not always preserved under “additive transformations”, such as adding an identical n...
We study private-value auctions with n risk-averse bidders, where n is a large number. We first u... more We study private-value auctions with n risk-averse bidders, where n is a large number. We first use asymptotic techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . . ), and use these explicit approximations to show that all large k-price auctions with risk-averse bidders are O(1/n) revenue equivalent. We then prove that there exist auction mechanisms for which the limiting revenue as n −→ ∞ in the case of risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even in the limit as n −→ ∞. Finally, we formulate a general condition under which the limiting revenue with risk-averse bidders is equal to the risk-neutral limit. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, [email protected] Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gu...
In mathematical models with uncertainties and noise, the calculation of a deterministic "qua... more In mathematical models with uncertainties and noise, the calculation of a deterministic "quantity of interest" (model output) is often replaced by the calculation of its moments (mean, standard deviation, etc.) and probability density function. Standard methods for these tasks are either statistical (Monte-Carlo, kernel density estimators, etc.) or spectral approximations (e.g., generalized polynomial chaos). In this paper we present a novel spline-based algorithm for these tasks. Our method offers significant advantages over the existing methods for density estimation, including a guaranteed convergence rate which is at least cubic in the number of samples. Furthermore, although spectral methods can approximate moments with exponential accuracy, the spline-based approximation is often more accurate when the sample size is small. We also show how to approximate the moments and density of non-smooth quantities of interest, which is often prohibitive in spectral methods. Fin...
We show that all laser beams gradually lose their initial phase information in nonlinear propagat... more We show that all laser beams gradually lose their initial phase information in nonlinear propagation. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in-or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the interaction outcome become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of the interaction outcome are universal, and can be efficiently computed using a polynomial-chaos approach, even when the distributions of the noise sources are unknown.
We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who ... more We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who adopt the product can later "recover" and stop influencing their peers to adopt the product. To gain insight into the effect of the social network structure on the diffusion, we focus on two extreme cases. In the "most-connected" configuration where all consumers are interconnected (complete network), averaging over all consumers leads to an aggregate model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. In the "least-connected" configuration where consumers are arranged on a circle and each consumer can only be influenced by his left neighbor (one-sided 1D network), averaging over all consumers leads to a different aggregate model which is linear, and can be solved explicitly. We conjecture that for any other network, the diffusion is bounded from below and from above by that on a one-sided 1D network and on a complete network, respectively. When consumers are arranged on a circle and each consumer can be influenced by his left and right neighbors (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of non-recovering adopters, where the diffusion on one-sided and on two-sided 1D networks is identical. We also propose a nonlinear model for recoveries, and show that consumers' heterogeneity has a negligible effect on the aggregate diffusion.
ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filament... more ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filamentation distance of intense laser pulses in the atmosphere can be extended and controlled with a simple double‐lens setup. Using this method we were able to achieve a 20‐fold delay of the filamentation distance of non‐chirped 120 fs pulses propagating in air, from 16 m to 330 m. At 330 m, the collapsing pulse is sufficiently powerful to create plasma filaments. We also show that the scatter of the filaments at 330 m can be significantly reduced by tilting the second lens. We derive a simple formula for the filamentation distance, and confirm its agreement with the experimental results. We also observe that delaying the onset of filamentation increases the filament length. To the best of our knowledge, this is the longest distance reported in the literature at which plasma filaments were created and controlled. Finally, we show that the peak power at the onset of collapse is significantly higher with the double‐lens setup, compared with the standard negative chirping approach.
Two random traffic streams are competing for the service time of a single server (multiplexer). T... more Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling ...
Service providers, such as cell phone carriers, often offer three-part tariff plans that consist ... more Service providers, such as cell phone carriers, often offer three-part tariff plans that consist of three levers: A fixed fee, an allowance of free units, and a price per each unit above the allowance. In previous studies the optimal three-part tariff contract was characterized using the standard first-order conditions approach. Because this optimization problem is non-smooth, however, it could only be solved in a few simple cases. In this study we employ a different methodology which is based on obtaining a global bound for the firm profit, and then showing that this bound is attained by the optimal plan. This approach allows us to explicitly calculate the optimal three-part tariff plan under quite general conditions, where consumers are rational, they have a general utility function, they experience psychological costs when they exceed the number of free units, they have deterministic or stochastic consumption rates, they are homogeneous or heterogeneous, and the firm costs are fixed or depend on the usage level.
In this Letter we i n troduce a systematic perturbation method for analyzing the e ect of small p... more In this Letter we i n troduce a systematic perturbation method for analyzing the e ect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schr odinger equation PNLS to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpli ed, depending on whether PNLS is power conserving or not. An importantand somewhat surprising result is that various small defocusingperturbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations.
We show that small negative fourth-order dispersion can arrest spatiotemporal collapse of ultrash... more We show that small negative fourth-order dispersion can arrest spatiotemporal collapse of ultrashort pulses with anomalous dispersion in a planar waveguide with pure Kerr nonlinearity, resulting in ͑2 1 1͒D optical bullets. Similarly to solitons, these bullets undergo elastic collisions. Since these bullets can self-trap from noisy Gaussian input beams and propagate without any power losses, this result may be used to realize experimentally stable, nondissipative optical bullets.
An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schrödinger ... more An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schrödinger equation that is valid from the early stages of self-focusing until the blowup point. The adiabatic law leads to an analytical formula for the location of the blowup point and can be used to estimate the effects of various small perturbations on self-focusing. The results of the analysis are confirmed by numerical simulations.
The standard explanation for multiple filamentation of laser pulses is that it is caused by noise... more The standard explanation for multiple filamentation of laser pulses is that it is caused by noise in the input beam. We propose an alternative explanation that is based on deterministic vectorial (polarization) effects. We present numerical simulations in support of the vectorial-effects explanation and suggest a simple experiment for deciding whether multiple filamentation is due to vectorial effects.
To study the effect of boundaries on diffusion of new products, we introduce two novel analytic t... more To study the effect of boundaries on diffusion of new products, we introduce two novel analytic tools: The indifference principle, which enables us to explicitly compute the aggregate diffusion on various networks, and the dominance principle, which enables us to rank the diffusion on different networks. Using these principles, we prove our main result that on a finite line, one-sided diffusion (i.e., when each consumer can only be influenced by her left neighbor) is strictly slower than two-sided diffusion (i.e., when each consumer can be influenced by her left and right neighbor). This is different from the periodic case of diffusion on a circle, where one-sided and two-sided diffusion are identical. We observe numerically similar results in higher dimensions.
Preprint Series of Department of Mathematics, Hokkaido University, 2007
We study analytically and numerically the stability of the standing waves for a nonlinear Schrödi... more We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in H 1 rad (R) and unstable in H 1 (R) under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blow-up in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.
We consider nonlinear bound states of the nonlinear Schrödinger equation i∂ z φ(z, x) = −∂ 2 x φ ... more We consider nonlinear bound states of the nonlinear Schrödinger equation i∂ z φ(z, x) = −∂ 2 x φ − (1 + m(N x))|φ| p−1 φ, in the presence of a nonlinear periodic microstructure m(N x). This equation models the propagation of laser beams in a medium whose nonlinear refractive index is modulated in the transverse direction, and also arises in the study of Bose-Einstein Condensation (BEC) in a medium with a spatially dependent scattering length. In the nonlinear optics context, N = r beam /r ms denotes the ratio of beam width to microstructure characteristic scale. We study the profiles of the nonlinear bound states using a multiple scale (homogenization) expansion for N ≫ 1 (wide beams), a perturbation analysis for N ≪ 1 (narrow beams) and numerical simulations for N = O(1). In the subcritical case p < 5, beams centered at local maxima of the microstructure are stable. Furthermore, beams centered at local minima of the microstructure are unstable to general (asymmetric) perturbations but stable relative to symmetric perturbations. In the critical case p = 5, a nonlinear microstructure can only stabilize narrow beams centered at a local maximum of the microstructure, provided that the microstructure also satisfies a certain local condition. Even in this case, the stability region is very small so that small (O(10 −2)) perturbations can destabilize the beam. Therefore, such beams are "mathematically" stable but "physically" unstable.
The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation ... more The formation of singularities of self-focusing solutions of the nonlinear Schr odinger equation NLS in critical dimension is characterized by a delicate balance between the focusing nonlinearity and di raction Laplacian, and is thus very sensitive to small perturbations. In this paper we i n troduce a systematic perturbation theory for analyzing the e ect of additional small terms on self focusing, in which the perturbed critical NLS is reduced to a simpler system of modulation equations that do not depend on the spatial variables transverse to the beam axis. The modulation equations can be further simpli ed, depending on whether the perturbed NLS is power conserving or not. We review previous applications of modulation theory and present several new ones that include: Dispersive saturating nonlinearities, self-focusing with Debye relaxation, the Davey Stewartson equations, self-focusing in optical ber arrays and the e ect of randomness. An important and somewhat surprising result is that various small defocusing perturbations lead to a generic form of the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations. In the special case of the unperturbed critical NLS, modulation theory leads to a new adiabatic law for the rate of blowup which is accurate from the early stages of self-focusing and remains valid up to the singularity point. This adiabatic law preserves the lens transformation property of critical NLS and it leads to an analytic formula for the location of the singularity as a function of the initial pulse power, radial distribution and focusing angle. The asymptotic limit of this law agrees with the known loglog blowup behavior. However, the loglog behavior is reached only after huge ampli cations of the initial amplitude, at which point the physical basis of NLS is in doubt. We also include in this paper a new condition for blowup of solutions in critical NLS and an improved version of the Dawes-Marburger formula for the blowup location of Gaussian pulses.
Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this qu... more Does a new product spread faster among heterogeneous or homogeneous consumers? We analyze this question using the stochastic discrete Bass model in which consumers may differ in their individual external influence rates [Formula: see text] and in their individual internal influence rates [Formula: see text]. When the network is complete and the heterogeneity is only manifested in [Formula: see text] or only in [Formula: see text], it always slows down the diffusion, compared with the corresponding homogeneous network. When, however, consumers are heterogeneous in both [Formula: see text] and [Formula: see text], heterogeneity slows down the diffusion in some cases but accelerates it in others. Moreover, the dominance between the heterogeneous and homogeneous adoption levels is global in time in some cases but changes with time in others. Perhaps surprisingly, global dominance between two networks is not always preserved under “additive transformations”, such as adding an identical n...
We study private-value auctions with n risk-averse bidders, where n is a large number. We first u... more We study private-value auctions with n risk-averse bidders, where n is a large number. We first use asymptotic techniques to calculate explicit approximations of the equilibrium bids and of the seller’s revenue in any k-price auction (k = 1, 2, . . . ), and use these explicit approximations to show that all large k-price auctions with risk-averse bidders are O(1/n) revenue equivalent. We then prove that there exist auction mechanisms for which the limiting revenue as n −→ ∞ in the case of risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even in the limit as n −→ ∞. Finally, we formulate a general condition under which the limiting revenue with risk-averse bidders is equal to the risk-neutral limit. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel, [email protected] Department of Industrial Engineering and Management, Faculty of Engineering Sciences, Ben-Gu...
In mathematical models with uncertainties and noise, the calculation of a deterministic "qua... more In mathematical models with uncertainties and noise, the calculation of a deterministic "quantity of interest" (model output) is often replaced by the calculation of its moments (mean, standard deviation, etc.) and probability density function. Standard methods for these tasks are either statistical (Monte-Carlo, kernel density estimators, etc.) or spectral approximations (e.g., generalized polynomial chaos). In this paper we present a novel spline-based algorithm for these tasks. Our method offers significant advantages over the existing methods for density estimation, including a guaranteed convergence rate which is at least cubic in the number of samples. Furthermore, although spectral methods can approximate moments with exponential accuracy, the spline-based approximation is often more accurate when the sample size is small. We also show how to approximate the moments and density of non-smooth quantities of interest, which is often prohibitive in spectral methods. Fin...
We show that all laser beams gradually lose their initial phase information in nonlinear propagat... more We show that all laser beams gradually lose their initial phase information in nonlinear propagation. Therefore, if two beams travel a sufficiently long distance before interacting, it is not possible to predict whether they would intersect in-or out-of-phase. Hence, if the underlying propagation model is non-integrable, deterministic predictions and control of the interaction outcome become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of the interaction outcome are universal, and can be efficiently computed using a polynomial-chaos approach, even when the distributions of the noise sources are unknown.
We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who ... more We consider the diffusion of new products in the discrete Bass-SIR model, in which consumers who adopt the product can later "recover" and stop influencing their peers to adopt the product. To gain insight into the effect of the social network structure on the diffusion, we focus on two extreme cases. In the "most-connected" configuration where all consumers are interconnected (complete network), averaging over all consumers leads to an aggregate model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. In the "least-connected" configuration where consumers are arranged on a circle and each consumer can only be influenced by his left neighbor (one-sided 1D network), averaging over all consumers leads to a different aggregate model which is linear, and can be solved explicitly. We conjecture that for any other network, the diffusion is bounded from below and from above by that on a one-sided 1D network and on a complete network, respectively. When consumers are arranged on a circle and each consumer can be influenced by his left and right neighbors (two-sided 1D network), the diffusion is strictly faster than on a one-sided 1D network. This is different from the case of non-recovering adopters, where the diffusion on one-sided and on two-sided 1D networks is identical. We also propose a nonlinear model for recoveries, and show that consumers' heterogeneity has a negligible effect on the aggregate diffusion.
ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filament... more ABSTRACT We show experimentally for ultrashort laser pulses propagating in air, that the filamentation distance of intense laser pulses in the atmosphere can be extended and controlled with a simple double‐lens setup. Using this method we were able to achieve a 20‐fold delay of the filamentation distance of non‐chirped 120 fs pulses propagating in air, from 16 m to 330 m. At 330 m, the collapsing pulse is sufficiently powerful to create plasma filaments. We also show that the scatter of the filaments at 330 m can be significantly reduced by tilting the second lens. We derive a simple formula for the filamentation distance, and confirm its agreement with the experimental results. We also observe that delaying the onset of filamentation increases the filament length. To the best of our knowledge, this is the longest distance reported in the literature at which plasma filaments were created and controlled. Finally, we show that the peak power at the onset of collapse is significantly higher with the double‐lens setup, compared with the standard negative chirping approach.
Two random traffic streams are competing for the service time of a single server (multiplexer). T... more Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling ...
Service providers, such as cell phone carriers, often offer three-part tariff plans that consist ... more Service providers, such as cell phone carriers, often offer three-part tariff plans that consist of three levers: A fixed fee, an allowance of free units, and a price per each unit above the allowance. In previous studies the optimal three-part tariff contract was characterized using the standard first-order conditions approach. Because this optimization problem is non-smooth, however, it could only be solved in a few simple cases. In this study we employ a different methodology which is based on obtaining a global bound for the firm profit, and then showing that this bound is attained by the optimal plan. This approach allows us to explicitly calculate the optimal three-part tariff plan under quite general conditions, where consumers are rational, they have a general utility function, they experience psychological costs when they exceed the number of free units, they have deterministic or stochastic consumption rates, they are homogeneous or heterogeneous, and the firm costs are fixed or depend on the usage level.
In this Letter we i n troduce a systematic perturbation method for analyzing the e ect of small p... more In this Letter we i n troduce a systematic perturbation method for analyzing the e ect of small perturbations on critical self focusing by reducing the perturbed critical nonlinear Schr odinger equation PNLS to a simpler system of modulation equations that do not depend on the transverse variables. The modulation equations can be further simpli ed, depending on whether PNLS is power conserving or not. An importantand somewhat surprising result is that various small defocusingperturbations lead to a universal form for the modulation equations, whose solutions have slowly decaying focusing-defocusing oscillations.
We show that small negative fourth-order dispersion can arrest spatiotemporal collapse of ultrash... more We show that small negative fourth-order dispersion can arrest spatiotemporal collapse of ultrashort pulses with anomalous dispersion in a planar waveguide with pure Kerr nonlinearity, resulting in ͑2 1 1͒D optical bullets. Similarly to solitons, these bullets undergo elastic collisions. Since these bullets can self-trap from noisy Gaussian input beams and propagate without any power losses, this result may be used to realize experimentally stable, nondissipative optical bullets.
An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schrödinger ... more An adiabatic approach is used to derive a new law for self-focusing in the nonlinear Schrödinger equation that is valid from the early stages of self-focusing until the blowup point. The adiabatic law leads to an analytical formula for the location of the blowup point and can be used to estimate the effects of various small perturbations on self-focusing. The results of the analysis are confirmed by numerical simulations.
The standard explanation for multiple filamentation of laser pulses is that it is caused by noise... more The standard explanation for multiple filamentation of laser pulses is that it is caused by noise in the input beam. We propose an alternative explanation that is based on deterministic vectorial (polarization) effects. We present numerical simulations in support of the vectorial-effects explanation and suggest a simple experiment for deciding whether multiple filamentation is due to vectorial effects.
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Papers by Gadi Fibich