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This paper is devoted to the study of a stochastic epidemiological model which is a variant of the SIR model to which we add an extra factor in the transition rate from susceptible to infected accounting for the inflow of infection due to immigration or environmental sources of infection. This factor yields the formation of new clusters of infections, without having to specify a priori and explicitly their date and place of appearance.We establish an exact deterministic description for such stochastic processes on 1D lattices (finite lines, semi-infinite lines, infinite lines) by showing that the probability of infection at a given point in space and time can be obtained as the solution of a deterministic ODE system on the lattice. Our results allow stochastic initial conditions and arbitrary spatio-temporal heterogeneities on the parameters.We then apply our results to some concrete situations and obtain useful qualitative results and explicit formulae on the macroscopic dynamics and also the local temporal behavior of each individual. In particular, we provide a fine analysis of some aspects of cluster formation through the study of patient-zero problems and the effects of time-varying point sources.Finally, we show that the space-discrete model gives rise to new space-continuous models, which are either ODEs or PDEs, depending on the rescaling regime assumed on the parameters.

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 $\{p_j \}$ and in their individual internal influence rates $\{ q_j \}$. When the network is complete and the heterogeneity is only manifested in $\{p_j \}$ or only in $\{ q_j \}$, it always slows down the diffusion, compared to the corresponding homogeneous network. When, however, consumers are heterogeneous in both $\{p_j\}$ and $\{q_{j}\}$, 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 node to both networks. When the network is not complete, the effect of heterogeneity depends also on its spatial distribution within the network., Comment: 39 pages, 11 figures

One of the most important results in auction theory is that when bidders are symmetric (homogeneous), then under quite general conditions, the seller's expected revenue is independent of the auctio...

A dynamical system is said to be reversible if, given an output, the input can always be recovered in a well-posed manner. Nevertheless, we argue that reversible systems that have a time-reversal symmetry, such as the Nonlinear Schrodinger equation and the ϕ 4 equation can become ”physically irreversible”. By this, we mean that realistically-small experimental errors in measuring the output can lead to dramatic differences between the recovered input and the original one. The loss of reversibility reveals a natural ”arrow of time”, reminiscent of the thermodynamic one, which is the direction in which the radiation is emitted outward. Our results are relevant to imaging and reversal applications in nonlinear optics.

We show theoretically and demonstrate experimentally that collapsing elliptically polarized laser beams experience a nonlinear ellipse rotation that is highly sensitive to small fluctuations in the input power. For arbitrarily small fluctuations in the input power and after a sufficiently large propagation distance, the polarization angle becomes uniformly distributed in $[0,2\ensuremath{\pi}]$ from shot to shot. We term this phenomenon loss of polarization. We perform experiments in fused-silica glass, nitrogen gas, and water and observe a significant increase in the fluctuations of the output polarization angle for elliptically polarized femtosecond pulses as the power is increased beyond the critical power for self-focusing. We also show numerically and confirm experimentally that this effect is more prominent in the anomalous group-velocity dispersion (GVD) regime compared to the normal-GVD regime due to the extended lengths of the filaments for the former. Such effects could play an important role in intense-field light-matter interactions in which elliptically polarized pulses are utilized.

In most models of diffusion of new products, every individual in the social network is a potential adopter. When, however, a fraction α of the individuals cannot adopt the product at any time, the new product percolates (rather than diffuses) in the network, similarly to movement through porous materials. We obtain explicit expressions for the fraction of adopters as a function of time, for complete networks, circular networks, D -dimensional Cartesian networks, small-worlds networks, and scale-free networks. These expressions show that the complex effect of percolation can be captured by two simple aggregate effects: Decreasing the market potential by 1 − α , and reducing the peers effect by ( 1 − α ) k , where k depends on the network type. Hence, percolation of new products is qualitatively similar to diffusion of new products. In particular, there is no threshold value at which a phase transition occurs.

The effect of uncertainties and noise on a quantity of interest (model output) is often better described by its probability density function (PDF) than by its moments. Although density estimation is a common task, the adequacy of approximation methods (surrogate models) for density estimation has not been analyzed before in the uncertainty-quantification (UQ) literature. In this paper, we first show that standard surrogate models (such as generalized polynomial chaos), which are highly accurate for moment estimation, might completely fail to approximate the PDF, even for one-dimensional noise. This is because density estimation requires that the surrogate model accurately approximates the gradient of the quantity of interest, and not just the quantity of interest itself. Hence, we develop a novel spline-based algorithm for density-estimation whose convergence rate in $L^q$ is polynomial in the sampling resolution. This convergence rate is better than that of standard statistical density-estimation methods (such as histograms and kernel density estimators) at dimensions $1 \leq d\leq \frac{5}{2}m$, where $m$ is the spline order. Furthermore, we obtain the convergence rate for density estimation with any surrogate model that approximates the quantity of interest and its gradient in $L^{\infty}$. Finally, we demonstrate our algorithm for problems in nonlinear optics and fluid dynamics.

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., Comment: 29 pages, 16 figures

The inverse equilibrium bidding strategies $\{ v_i(b) \}_{i=1}^n$ in a first-price auction with $n$ asymmetric bidders, where $v_i$ is the value of bidder $i$ and $b$ is the bid, are solutions of a system of $n$ first-order ordinary differential equations, with $2n$ boundary conditions and a free boundary on the right. In this study we show that when the number of bidders is large ($n\gg1$), this problem has a boundary-layer structure with several nonstandard features: (1) The small parameter does not multiply the highest-order derivative. (2) The number of equations goes to infinity as the small parameter goes to zero. (3) The boundary-layer structure is for the derivatives $\{ v_i^\prime(b) \}_{i=1}^n$ but not for $\{ v_i(b) \}_{i=1}^n$. (4) In the boundary-layer region, the solution is the sum of an outer solution in the original variable and an inner solution in the rescaled boundary-layer variable. Using boundary-layer theory, we compute an $O(1/n^3)$ uniform approximation for $\{ v_i(b) \}_{i=1}^n$....

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 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 nonsmooth, however, it could only be solved in a few simple cases. In this study we employ a different methodology that 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. The online appendix is available at https://doi.org/10.1287/opre.2017.1609 .

Traditionally, interactions between laser beams or filaments were considered to be deterministic. We show, however, that in most physical settings, these interactions ultimately become stochastic. Specifically, we show that in the nonlinear propagation of laser beams, the shot-to-shot variation of the nonlinear phase shift increases with distance, and ultimately becomes uniformly distributed in [0, 2π]. 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 outcome of the interaction become impossible. Because the relative phase between the two beams becomes uniformly distributed in [0, 2π], however, the statistics of these stochastic interactions are universal and fully predictable. These statistics can be efficiently computed using a novel universal model for stochastic interactions, even when the noise distribution is unknown.

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 inter-connected (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.

We consider the diffusion of new products in social networks, where consumers who adopt the product can later ``recover'' and stop influencing others to adopt the product. We show that the diffusion is not described by the susceptible-infected-recovered (SIR) model, but rather by a new model, the Bass-SIR model, which combines the Bass model for diffusion of new products with the SIR model for epidemics. The phase transition of consumers from nonadopters to adopters is described by a nonstandard Kolmogorov-Johnson-Mehl-Avrami model, in which clusters growth is limited by adopters' recovery. Therefore, diffusion in the Bass-SIR model only depends on the local structure of the social network, but not on the average distance between consumers. Consequently, unlike the SIR model, a small-worlds structure has a negligible effect on the diffusion. Moreover, unlike the SIR model, there is no threshold value above which the diffusion will peter out. Surprisingly, diffusion on scale-free networks is nearly identical to that on Cartesian ones.

The paper analyzes the revenue of auctions with asymmetric bidders with a large, but finite number of players. We explicitly calculate the seller’s expected revenue in large asymmetric first-price, second-price, and optimal auctions to O(1/n 3) accuracy, where n is the number of players. These calculations show that the revenue differences among these three auction mechanisms scale as ∈2/n 3, where e is the level of asymmetry (heterogeneity) among the distributions of bidders’ valuations. This novel scaling law shows that bidders’ asymmetry already has a negligible effect on revenue ranking of auctions with several (e.g., n = 6) bidders. In contrast, previous results studied only the limiting case n → ∞. We also show that bidders’ asymmetry always reduces the expected revenue in large auctions, but not necessarily in small ones. Finally, we extend the asymptotic O(∈2/n 3) revenue equivalence to a broader class of asymmetric auctions.

Do countries adopt legal norms due to independent, top-down processes, or, do laws spread among nations in a manner which resembles the diffusion of new products and innovations in a social network? We empirically examine this question by employing an influential mathematical model frequently used for analyzing the diffusion of innovations (the Bass model). Our findings indicate that more often than not, the temporal diffusion of legal rules displays a good fit to the Bass model, and suggest that this model can serve as a new methodological tool for studying the diffusion of laws. Applying the Bass model to legal diffusion allows to quantify the influence countries exert on each other in the adoption of specific rules, provides a metric for comparing diffusion across various legal branches, allows to tentatively predict the ongoing spread of specific rules, and sheds light on legal debates in different fields.

Typically, models with a heterogeneous property are considerably harder to analyze than the corresponding homogeneous models, in which the heterogeneous property is replaced by its average value. In this study we show that any outcome of a heterogeneous model that satisfies the two properties of differentiability and symmetry is O (ɛ 2 ) equivalent to the outcome of the corresponding homogeneous model, where ɛ is the level of heterogeneity. We then use this averaging principle to obtain new results in queuing theory, game theory (auctions), and social networks (marketing).

We introduce a new approach for analysis and numerical simulations of asymmetric first-price auctions, which is based on dynamical systems. We apply this approach to asymmetric auctions in which players' valuations are power-law distributed. We utilize a dynamical-systems formulation to provide a proof of the existence and uniqueness of the equilibrium strategies in the cases of two coalitions and of two types of players. In the case of n different players, the singular point of the original system at b = 0 corresponds to a saddle point of the dynamical system with n − 1 admissible directions. This insight enables us to use forward solutions in the analysis and in the numerical simulations, in contrast with previous analytic and numerical studies that used backward solutions. The dynamical-systems approach provides an intuitive explanation for why the standard backward-shooting method for computing the equilibrium strategies is inherently unstable, and enables us to devise a stable forward-shooting method. In particular, in the case of two types of players, this method is extremely simple, as it does not require any shooting.

We study the nonlinear-damping continuation of singular solutions of the critical and supercritical NLS. Our simulations suggest that for generic initial conditions that lead to collapse in the undamped NLS, the solution of the weakly-damped NLS i ψ t ( t , x ) + Δ ψ + | ψ | p − 1 ψ + i δ | ψ | q − 1 ψ = 0 , 0 δ ≪ 1 , is highly asymmetric with respect to the singularity time, and the post-collapse defocusing velocity of the singular core goes to infinity as the damping coefficient δ goes to zero. In the special case of the minimal-power blowup solutions of the critical NLS, the continuation is a minimal-power solution with a higher (but finite) defocusing velocity, whose magnitude increases monotonically with the nonlinear damping exponent q .

The standard method for computing the equilibrium strategies of asymmetric first-price auctions is the backward-shooting method. In this study we show that the backward-shooting method is inherently unstable, and that this instability cannot be eliminated by changing the numerical methodology of the backward solver. Moreover, this instability becomes more severe as the number of players increases. We then present a novel boundary-value method for computing the equilibrium strategies of asymmetric first-price auctions. We demonstrate the robustness and stability of this method for auctions with any number of players, and for players with mixed types of distributions, including distributions with more than one crossing. Finally, we use the boundary-value method to study large auctions with hundreds of players, to compute the asymptotic rate at which large first-price and second-price auctions become revenue equivalent, and to study auctions in which the distributions cannot be ordered according to first-order stochastic dominance.

We present four continuations of the critical nonlinear \schro equation (NLS) beyond the singularity: 1) a sub-threshold power continuation, 2) a shrinking-hole continuation for ring-type solutions, 3) a vanishing nonlinear-damping continuation, and 4) a complex Ginzburg-Landau (CGL) continuation. Using asymptotic analysis, we explicitly calculate the limiting solutions beyond the singularity. These calculations show that for generic initial data that leads to a loglog collapse, the sub-threshold power limit is a Bourgain-Wang solution, both before and after the singularity, and the vanishing nonlinear-damping and CGL limits are a loglog solution before the singularity, and have an infinite-velocity{\rev{expanding core}} after the singularity. Our results suggest that all NLS continuations share the universal feature that after the singularity time $T_c$, the phase of the singular core is only determined up to multiplication by $e^{i\theta}$. As a result, interactions between post-collapse beams (filaments) become chaotic. We also show that when the continuation model leads to a point singularity and preserves the NLS invariance under the transformation $t\rightarrow-t$ and $\psi\rightarrow\psi^\ast$, the singular core of the weak solution is symmetric with respect to $T_c$. Therefore, the sub-threshold power and the{\rev{shrinking}}-hole continuations are symmetric with respect to $T_c$, but continuations which are based on perturbations of the NLS equation are generically asymmetric.

We explicitly calculate the aggregate diffusion dynamics in one-dimensional agent-based models of adoption of new products, without using the mean-field approximation. We then introduce a clusters-dynamics approach, and use it to derive an analytic approximation of the aggregate diffusion dynamics in multidimensional agent-based models. The clusters-dynamics approximation shows that the aggregate diffusion dynamics does not depend on the average distance between individuals, but rather on the expansion rate of clusters of adopters. Therefore, the grid dimension has a large effect on the aggregate adoption dynamics, but a small-world structure and heterogeneity among individuals have only a minor effect. Our results suggest that the one-dimensional model and the Bass model provide a lower bound and an upper bound, respectively, for the aggregate diffusion dynamics in agent-based models with “any” spatial structure.

We prove an asymptotic revenue equivalence among weakly asymmetric auctions with interdependent values, in which bidders have either asymmetric utility functions or asymmetric distributions of signals.

We consider singular solutions of the biharmonic NLS. In the L^2-critical case, the blowup rate is bounded by a quartic-root power law, the solution approaches a self-similar profile, and a finite amount of L^2-norm, which is no less than the critical power, concentrates into the singularity ("strong collapse"). In the L^2-critical and supercritical cases, we use asymptotic analysis and numerical simulations to characterize singular solutions with a peak-type self-similar collapsing core. In the critical case, the blowup rate is slightly faster than a quartic-root, and the self-similar profile is given by the standing-wave ground-state. In the supercritical case, the blowup rate is exactly a quartic-root, and the self-similar profile is a zero-Hamiltonian solution of a nonlinear eigenvalue problem. These findings are verified numerically (up to focusing levels of 10^8) using an adaptive grid method. We also calculate the ground states of the standing-wave equations and the critical power for collapse in two and three dimensions., Comment: 30 pages, 25 figures

We study private-value auctions with n risk-averse bidders, where n is large. We first use asymptotic analysis techniques to calculate explicit approximations oftheequilibriumbidsandoftheseller'srevenueinany k-priceauction(k = 1,2 ,... ). These explicit approximations show that in all large k-price auctions the effect of risk-aversion is O(1/n 2 ) small. Hence, all large k-price auctions with risk-averse bid- ders are O(1/n 2 ) revenue equivalent. The generalization, that all large auctions are O(1/n 2 ) revenue equivalent, is false. Indeed, we show that there exist auction mech- anisms for which the limiting revenue as n −→ ∞ with risk-averse bidders is strictly below the risk-neutral limit. Therefore, these auction mechanisms are not revenue equivalent to large k-price auctions even to leading-order as n −→ ∞.

We present a systematic study of singular vortex solutions of the critical and supercritical two-dimensional nonlinear Schrodinger equation. In particular, we study the critical power for collapse and the asymptotic blowup profile of singular vortices.

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton's method. Our numerical simulations show that Newton's method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities.The one-dimensional results of the current paper create a foundation for the analysis of multi-dimensional problems in the future., Comment: 47 pages, 8 figures

The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.

We present new singular solutions of the nonlinear Schrodinger equation (NLS) i ψ t ( t , r ) + ψ r r + d − 1 r ψ r + | ψ | 2 σ ψ = 0 , 1 d , 2 d ≤ σ ≤ 2 . These solutions collapse with a quasi self-similar ring profile ψ Q , i.e. ψ ∼ ψ Q , where ψ Q = 1 L 1 / σ ( t ) Q ( r − r m ( t ) L ) exp [ i ∫ 0 t d s L 2 ( s ) + i L t 4 L [ α r 2 + ( 1 − α ) ( r − r m ( t ) ) 2 ] ] , L ( t ) is the ring width that vanishes at the singularity, r m ( t ) = r 0 L α ( t ) is the ring radius and α = 2 − σ σ ( d − 1 ) . The blowup rate of these solutions is 1 1 + α for 2 d ≤ σ 2 and 1 d ( 0 α ≤ 1 ), and a square root with a loglog correction (the loglog law) when σ = 2 and 1 d ( α = 0 ). Therefore, the NLS has solutions that collapse with any blowup rate p for 1 / 2 ≤ p 1 . This study extends the results of [G. Fibich, N. Gavish, X. Wang, New singular solutions of the nonlinear Schrodinger equation, Physica D 211 (2005) 193–220] for σ = 1 and d = 2 , and of [P. Raphael, Existence and stability of a solution blowing up on a sphere for a L 2 super critical non linear Schrodinger equation, Duke Math. J. 134 (2) (2006) 199–258] for σ = 2 and d = 2 , to all 2 / d ≤ σ ≤ 2 and 1 d .

In this study we demonstrate how a reference price may affect the degree of price rigidity/ flexibility. For this, we construct a model of reference-price formation, which we use to analyze the effect of asymmetric reference price (cut ‘effects’) on the profitability of price promotions. We derive explicit expressions for the additional profits earned during a promotional period due to consumer perception of a ‘gain’, and for the post-promotion loss of potential profits due to consumer perception of a ‘loss’. We show that when effects of losses on demand are greater than effects of gains (‘loss aversion’), price promotions always lead to a decline in profits. When, however, effects of gains are larger than those of losses, price promotions, as well as reverse price promotions (i.e. price increase) can be profitable. In the latter case we calculate the optimal depth and duration of a price promotion. We also show that reference price can affect price rigidity and flexibility. Copyright # 2007 John Wiley & Sons, Ltd.

We show that chemical kinetics relations can be used to describe processes that involve binding and dissociation reactions that take place on surfaces. From a mathematical perspective, the problem we study is a singular limit of a reaction‐diffusion system in which one of the variables concentrates on a lower‐dimensional set in the limit, while the other continues to diffuse in a fixed domain.

We present new solitarywave solutions of the two-dimensional nonlinear Schrodinger equation on bounded domains (such as rectangles, circles, and annuli). These multipeak necklace solitary waves consist of several identical positive profiles (pearls), such that adjacent pearls have opposite signs. They are stable at low powers, but become unstable at powers well below the critical power for collapse Pcr. This is in contrast with the ground-state (single-pearl) solitary waves on bounded domains, which are stable at any power below Pcr. On annular domains, the ground state solitary waves are radial at low powers, but undergo a symmetry breaking at a threshold power well below Pcr. As in the case of convex bounded domains, necklace solitary waves on the annulus are stable at low powers and become unstable at powers well below Pcr. Unlike on convex bounded domains, however, necklace solitarywaves on the annulus have a second stability regime at powers well above Pcr. For example, when the ratio of the inner to outer radii is 1:2, four-pearl necklaces are stable when their power is between 3.1Pcr and 3.7Pcr. This finding opens the possibility to propagate localized laser beams with substantiallymore power than was possible until now. The instability of necklace solitary waves is excited by perturbations that break the antisymmetry between adjacent pearls, and is manifested by power transfer between pearls. In particular, necklace instability is unrelated to collapse. In order to compute numerically the profile of necklace solitary waves on bounded domains, we introduce a non-spectral variant of Petviashvilis renormalization method., 43 pages

In this chapter we derive the loglog law and the adiabatic laws for solutions of the critical NLS that collapse with the \(\psi _{R^{(0)}}\) profile.

In this chapter we apply the NGO method to strongly nonlinear solutions of the NLS with anomalous dispersion.

In this chapter we study solutions that undergo a quasi self-similar collapse with the \(\psi _{R^{(0)}}\) profile.

In this chapter we study singular shrinking ring solutions of the supercritical subquintic NLS.

In this chapter we present an asymptotic method, the Nonlinear Geometrical Optics (NGO) method, for analyzing the initial self-focusing dynamics of strongly-nonlinear solutions of the NLS