Dynamics of soliton-like solutions for slowly varying, generalized gKdV equations: refraction vs. reflection

In this work we continue the description of soliton-like solutions of some slowly varying, subcritical gKdV equations. In this opportunity we describe, almost completely, the allowed behaviors: either the soliton is refracted, or it is reflected by the potential, depending on its initial energy. This last result describes a new type of soliton-like solution for gKdV equations, also present in the NLS case. Moreover, we prove that the solution is not pure at infinity, unlike the standard gKdV soliton.Comment: 51 pages, submitte

Solitons in cavity-QED arrays containing interacting qubits

We reveal the existence of polariton soliton solutions in the array of weakly coupled optical cavities, each containing an ensemble of interacting qubits. An effective complex Ginzburg-Landau equation is derived in the continuum limit taking into account the effects of cavity field dissipation and qubit dephasing. We have shown that an enhancement of the induced nonlinearity can be achieved by two order of the magnitude with a negative interaction strength which implies a large negative qubit-field detuning as well. Bright solitons are found to be supported under perturbations only in the upper (optical) branch of polaritons, for which the corresponding group velocity is controlled by tuning the interacting strength. With the help of perturbation theory for solitons, we also demonstrate that the group velocity of these polariton solitons is suppressed by the diffusion process

Spatial Solitons in Media with Delayed-Response Optical Nonlinearities

Near-soliton scanning light-beam propagation in media with both delayed-response Kerr-type and thermal nonlinearities is analyzed. The delayed-response part of the Kerr nonlinearity is shown to be competitive as compared to the thermal nonlinearity, and relevant contributions to a distortion of the soliton form and phase can be mutually compensated. This quasi-soliton beam propagation regime keeps properties of the incli- ned self-trapped channel.Comment: 7 pages, to be published in Europhys. Let

Spatial Optical Solitons due to Multistep Cascading

We introduce a novel class of parametric optical solitons supported simultaneously by two second-order nonlinear cascading processes, second-harmonic generation and sum-frequency mixing. We obtain, analytically and numerically, the solutions for three-wave spatial solitons and show that the presence of an additional cascading mechanism can change dramatically the properties and stability of two-wave quadratic solitary waves.Comment: 6 pages, 4 figure

New Instantiations of the CRYPTO 2017 Masking Schemes

At CRYPTO 2017, Belaïd et al. presented two new private multiplication algorithms over finite fields, to be used in secure masking schemes. To date, these algorithms have the lowest known complexity in terms of bilinear multiplication and random masks respectively, both being linear in the number of shares $d+1$. Yet, a practical drawback of both algorithms is that their safe instantiation relies on finding matrices satisfying certain conditions. In their work, Belaïd et al. only address these up to $d=2$ and 3 for the first and second algorithm respectively, limiting so far the practical usefulness of their constructions. In this paper, we use in turn an algebraic, heuristic, and experimental approach to find many more safe instances of Belaïd et al.\u27s algorithms. This results in explicit instantiations up to order $d = 6$ over large fields, and up to $d = 4$ over practically relevant fields such as $\mathbb{F}_{2^8}$

Weakly collisional Landau damping and three-dimensional Bernstein-Greene-Kruskal modes: New results on old problems

Landau damping and Bernstein-Greene-Kruskal (BGK) modes are among the most fundamental concepts in plasma physics. While the former describes the surprising damping of linear plasma waves in a collisionless plasma, the latter describes exact undamped nonlinear solutions of the Vlasov equation. There does exist a relationship between the two: Landau damping can be described as the phase-mixing of undamped eigenmodes, the so-called Case-Van Kampen modes, which can be viewed as BGK modes in the linear limit. While these concepts have been around for a long time, unexpected new results are still being discovered. For Landau damping, we show that the textbook picture of phase-mixing is altered profoundly in the presence of collision. In particular, the continuous spectrum of Case-Van Kampen modes is eliminated and replaced by a discrete spectrum, even in the limit of zero collision. Furthermore, we show that these discrete eigenmodes form a complete set of solutions. Landau-damped solutions are then recovered as true eigenmodes (which they are not in the collisionless theory). For BGK modes, our interest is motivated by recent discoveries of electrostatic solitary waves in magnetospheric plasmas. While one-dimensional BGK theory is quite mature, there appear to be no exact three-dimensional solutions in the literature (except for the limiting case when the magnetic field is sufficiently strong so that one can apply the guiding-center approximation). We show, in fact, that two- and three-dimensional solutions that depend only on energy do not exist. However, if solutions depend on both energy and angular momentum, we can construct exact three-dimensional solutions for the unmagnetized case, and two-dimensional solutions for the case with a finite magnetic field. The latter are shown to be exact, fully electromagnetic solutions of the steady-state Vlasov-Poisson-Amp\`ere system

Soliton propagation through a disordered system:statistics of the transmission delay

We have studied the soliton propagation through a segment containing random pointlike scatterers. In the limit of small concentration of scatterers when the mean distance between the scatterers is larger than the soliton width, a method has been developed for obtaining the statistical characteristics of the soliton transmission through the segment. The method is applicable for any classical particle traversing through a disordered segment with the given velocity transformation after each act of scattering. In the case of weak scattering and relatively short disordered segment the transmission time delay of a fast soliton is mostly determined by the shifts of the soliton center after each act of scattering. For sufficiently long segments the main contribution to the delay is due to the shifts of the amplitude and velocity of a fast soliton after each scatterer. Corresponding crossover lengths for both cases of light and heavy solitons have been obtained. We have also calculated the exact probability density function of the soliton transmission time delay for a sufficiently long segment. In the case of weak identical scatterers the latter is a universal function which depends on a sole parameter—the mean number of scatterers in a segment

Shock waves in the dissipative Toda lattice

We consider the propagation of a shock wave (SW) in the damped Toda lattice. The SW is a moving boundary between two semi-infinite lattice domains with different densities. A steadily moving SW may exist if the damping in the lattice is represented by an ``inner'' friction, which is a discrete analog of the second viscosity in hydrodynamics. The problem can be considered analytically in the continuum approximation, and the analysis produces an explicit relation between the SW's velocity and the densities of the two phases. Numerical simulations of the lattice equations of motion demonstrate that a stable SW establishes if the initial velocity is directed towards the less dense phase; in the opposite case, the wave gradually spreads out. The numerically found equilibrium velocity of the SW turns out to be in a very good agreement with the analytical formula even in a strongly discrete case. If the initial velocity is essentially different from the one determined by the densities (but has the correct sign), the velocity does not significantly alter, but instead the SW adjusts itself to the given velocity by sending another SW in the opposite direction.Comment: 10 pages in LaTeX, 5 figures available upon regues

Nonlinear local parallel acceleration of electrons through Landau trapping by oblique whistler mode waves in the outer radiation belt

International audienceSimultaneous observations of electron velocity distributions and chorus waves by the Van Allen Probe B are analyzed to identify long-lasting (more than 6 h) signatures of electron Landau resonant interactions with oblique chorus waves in the outer radiation belt. Such Landau resonant interactions result in the trapping of ˜1-10 keV electrons and their acceleration up to 100-300 keV. This kind of process becomes important for oblique whistler mode waves having a significant electric field component along the background magnetic field. In the inhomogeneous geomagnetic field, such resonant interactions then lead to the formation of a plateau in the parallel (with respect to the geomagnetic field) velocity distribution due to trapping of electrons into the wave effective potential. We demonstrate that the electron energy corresponding to the observed plateau remains in very good agreement with the energy required for Landau resonant interaction with the simultaneously measured oblique chorus waves over 6 h and a wide range of L shells (from 4 to 6) in the outer belt. The efficient parallel acceleration modifies electron pitch angle distributions at energies ˜50-200 keV, allowing us to distinguish the energized population. The observed energy range and the density of accelerated electrons are in reasonable agreement with test particle numerical simulations

Singularites in the Bousseneq equation and in the generalized KdV equation

In this paper, two kinds of the exact singular solutions are obtained by the improved homogeneous balance (HB) method and a nonlinear transformation. The two exact solutions show that special singular wave patterns exists in the classical model of some nonlinear wave problems