Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations

We consider the focusing nonlinear Schr\"odinger equations $i\partial_t u+\Delta u +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities p_c with $p_c=1+\frac{4}{N}$ and 0<\e\ll 1. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up tim

Ring-type singular solutions of the biharmonic nonlinear Schrodinger equation

We present new singular solutions of the biharmonic nonlinear Schrodinger equation in dimension d and nonlinearity exponent 2\sigma+1. These solutions collapse with the quasi self-similar ring profile, with ring width L(t) that vanishes at singularity, and radius proportional to L^\alpha, where \alpha=(4-\sigma)/(\sigma(d-1)). The blowup rate of these solutions is 1/(3+\alpha) for 4/d\le\sigma<4, and slightly faster than 1/4 for \sigma=4. These solutions are analogous to the ring-type solutions of the nonlinear Schrodinger equation.Comment: 21 pages, 13 figures, research articl

Continuations of the nonlinear Schr\"odinger equation beyond the singularity

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

Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere

In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from $\mathbb{R}^2$ into $S^2$.Comment: 14 pages, 5 figures, final version to be published in Nonlinearit

Interaction between habitat limitation and dispersal limitation is modulated by species life history and external conditions: a stochastic matrix model approach

Traditionally, species absence in a community is ascribed either to dispersal limitation (i.e., the inability of propagules of a species to reach a site) or to habitat limitation (abiotic or biotic conditions of a site prevent species from forming a viable population); sowing experiments can then distinguish between these two mechanisms. In our view, the situation is even more complicated. To demonstrate the complexity of the problem, we designed and applied simulations based on an extension of matrix models covering effects of propagule pressure and habitat limitation, and reflecting various characteristics of a species and of a habitat. These included life history, fecundity, seed bank viability of a species, habitat carrying capacity and disturbances. All the investigated factors affected proportion of occupied habitats. Whereas they can, to a large extent, compensate for each other, simultaneous decrease of habitat suitability and propagule input can be detrimental to the survival of a population. Our model demonstrated that in many cases, the absence of a species in a community is of stochastic nature, and result of interaction of species life history and various external conditions, and thus cannot be simply attributed to a single cause. The model results are supported with examples of case studies. The results also explain some well-known ecological phenomena, as decrease of niche breadth from the center to the margins of area of distribution. Finally, the model also suggests some caveats in interpretation of the results of sowing experiments. | Supporting Information Supporting Information </supplementary-material

Generalized Neighbor-Interaction Models Induced by Nonlinear Lattices

It is shown that the tight-binding approximation of the nonlinear Schr\"odinger equation with a periodic linear potential and periodic in space nonlinearity coefficient gives rise to a number of nonlinear lattices with complex, both linear and nonlinear, neighbor interactions. The obtained lattices present non-standard possibilities, among which we mention a quasi-linear regime, where the pulse dynamics obeys essentially the linear Schr{\"o}dinger equation. We analyze the properties of such models both in connection with their modulational stability, as well as in regard to the existence and stability of their localized solitary wave solutions

Azimuthally polarized spatial dark solitons: exact solutions of Maxwell's equations in a Kerr medium

Spatial Kerr solitons, typically associated with the standard paraxial nonlinear Schroedinger equation, are shown to exist to all nonparaxial orders, as exact solutions of Maxwell's equations in the presence of vectorial Kerr effect. More precisely, we prove the existence of azimuthally polarized, spatial, dark soliton solutions of Maxwell's equations, while exact linearly polarized (2+1)-D solitons do not exist. Our ab initio approach predicts the existence of dark solitons up to an upper value of the maximum field amplitude, corresponding to a minimum soliton width of about one fourth of the wavelength.Comment: 4 pages, 4 figure

Finite-Band-width Effects on the Transition Temperature and NMR Relaxation Rate of Impure Superconductors

We study the thermodynamic properties of impure superconductors by explicitly taking into consideration the finiteness of electronic bandwidths within the phonon-mediated Eliashberg formalism. For a finite electronic bandwidth, the superconducting transition temperature, $T_c$, is suppressed by nonmagnetic impurity scatterings. This is a consequence of a reduction in the effective electron-phonon coupling, $\lambda_{eff}$. The reduced $\lambda_{eff}$ is reflected in the observation that the coherence peak in $1/(T_1 T)$, where $T_1$ is the nuclear spin-lattice relaxation time and $T$ is the temperature, is enhanced by impurity scatterings for a finite bandwidth. Calculations are presented for $T_c$ and $1/(T_1 T)$ as bandwidths and impurity scattering rates are varied. Implications for doped C$_{60}$ superconductors are discussed in connection with $T_c$ and $1/T_1$ measurements.Comment: 10 pages. REVTeX. 5 postscript figures. Scheduled to be published in Physical Review B, March 1. The previous submission is revised and two figures are adde

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to an amplitude instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength

Statistics and Characteristics of Spatio-Temporally Rare Intense Events in Complex Ginzburg-Landau Models

We study the statistics and characteristics of rare intense events in two types of two dimensional Complex Ginzburg-Landau (CGL) equation based models. Our numerical simulations show finite amplitude collapse-like solutions which approach the infinite amplitude solutions of the nonlinear Schr\"{o}dinger (NLS) equation in an appropriate parameter regime. We also determine the probability distribution function (PDF) of the amplitude of the CGL solutions, which is found to be approximately described by a stretched exponential distribution, $P(|A|) \approx e^{-|A|^\eta}$, where $\eta < 1$. This non-Gaussian PDF is explained by the nonlinear characteristics of individual bursts combined with the statistics of bursts. Our results suggest a general picture in which an incoherent background of weakly interacting waves, occasionally, `by chance', initiates intense, coherent, self-reinforcing, highly nonlinear events.Comment: 7 pages, 9 figure