Editorial for the Special Issue on Functional Data Analysis and Related Topics

This Special Issue of the Journal of Multivariate Analysis (JMVA), which comprises a survey and 19 research papers, takes its roots in the Fourth International Workshop on Functional and Operatorial Statistics held in A Coruña, Spain, June 15–17, 2017, and dedicated to the memory of Peter Hall. However, this issue extends far beyond the scope of IWFOS-2017 and includes contributions from several of the world’s leading research groups in functional data analysis (FDA). All papers were peer-reviewed according to the journal’s high academic standards

Collapse and stable self-trapping for Bose-Einstein condensates with 1/r^b type attractive interatomic interaction potential

We consider dynamics of Bose-Einstein condensates with long-range attractive interaction proportional to $1/r^b$ and arbitrary angular dependence. It is shown exactly that collapse of Bose-Einstein condensate without contact interactions is possible only for $b\ge 2$. Case $b=2$ is critical and requires number of particles to exceed critical value to allow collapse. Critical collapse in that case is strong one trapping into collapsing region a finite number of particles. Case $b>2$ is supercritical with expected weak collapse which traps rapidly decreasing number of particles during approach to collapse. For $b<2$ singularity at $r=0$ is not strong enough to allow collapse but attractive $1/r^b$ interaction admits stable self-trapping even in absence of external trapping potential

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

Collapse in the nonlocal nonlinear Schr\"odinger equation

We discuss spatial dynamics and collapse scenarios of localized waves governed by the nonlinear Schr\"{o}dinger equation with nonlocal nonlinearity. Firstly, we prove that for arbitrary nonsingular attractive nonlocal nonlinear interaction in arbitrary dimension collapse does not occur. Then we study in detail the effect of singular nonlocal kernels in arbitrary dimension using both, Lyapunoff's method and virial identities. We find that for for a one-dimensional case, i.e. for $n=1$, collapse cannot happen for nonlocal nonlinearity. On the other hand, for spatial dimension $n\geq2$ and singular kernel $\sim 1/r^\alpha$, no collapse takes place if $\alpha<2$, whereas collapse is possible if $\alpha\ge2$. Self-similar solutions allow us to find an expression for the critical distance (or time) at which collapse should occur in the particular case of $\sim 1/r^2$ kernels. Moreover, different evolution scenarios for the three dimensional physically relevant case of Bose Einstein condensate are studied numerically for both, the ground state and a higher order toroidal state with and without an additional local repulsive nonlinear interaction. In particular, we show that presence of an additional local repulsive term can prevent collapse in those cases

Strong Collapse Turbulence in Quintic Nonlinear Schr\"odinger Equation

We consider the quintic one dimensional nonlinear Schr\"odinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time forming forced turbulence. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. In statistical steady state of the developed turbulence the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly-Gaussian while the large amplitude tail of probability density function (PDF) is strongly non-Gaussian with power-like behavior. The small amplitude nearly-Gaussian fluctuations seed formation of large collapse events. The universal spatio-temporal form of these events together with the PDF for their maximum amplitudes define the power-like tail of PDF for large amplitude fluctuations, i.e., the intermittency of strong turbulence.Comment: 14 pages, 17 figure

Noise suppression and enhanced focusability in plasma Raman amplifier with multi-frequency pump

Laser pulse compression/amplification through Raman backscattering in plasmas can be facilitated by using multi-frequency pump laser beams. The efficiency of amplification is increased by suppressing the Raman instability of thermal fluctuations and seed precursors. Also the focusability of the amplified radiation is enhanced due to the suppression of large-scale longitudinal speckles in the pump wave structure

Ultrashort filaments of light in weakly-ionized, optically-transparent media

Modern laser sources nowadays deliver ultrashort light pulses reaching few cycles in duration, high energies beyond the Joule level and peak powers exceeding several terawatt (TW). When such pulses propagate through optically-transparent media, they first self-focus in space and grow in intensity, until they generate a tenuous plasma by photo-ionization. For free electron densities and beam intensities below their breakdown limits, these pulses evolve as self-guided objects, resulting from successive equilibria between the Kerr focusing process, the chromatic dispersion of the medium, and the defocusing action of the electron plasma. Discovered one decade ago, this self-channeling mechanism reveals a new physics, widely extending the frontiers of nonlinear optics. Implications include long-distance propagation of TW beams in the atmosphere, supercontinuum emission, pulse shortening as well as high-order harmonic generation. This review presents the landmarks of the 10-odd-year progress in this field. Particular emphasis is laid to the theoretical modeling of the propagation equations, whose physical ingredients are discussed from numerical simulations. Differences between femtosecond pulses propagating in gaseous or condensed materials are underlined. Attention is also paid to the multifilamentation instability of broad, powerful beams, breaking up the energy distribution into small-scale cells along the optical path. The robustness of the resulting filaments in adverse weathers, their large conical emission exploited for multipollutant remote sensing, nonlinear spectroscopy, and the possibility to guide electric discharges in air are finally addressed on the basis of experimental results.Comment: 50 pages, 38 figure