Repulsive Case Research Articles

Minimal energy solutions for repulsive nonlinear Schrödinger systems

In this paper we establish existence and nonexistence results concerning fully nontrivial minimal energy solutions of the nonlinear Schrödinger system−Δu+u=|u|2q−2u+b|u|q−2u|v|qin Rn,−Δv+ω2v=|v|2q−2v+b|u|q|v|q−2vin Rn. We consider the repulsive case b<0 and assume that the exponent q satisfies 1<q<nn−2 in case n≥3 and 1<q<∞ in case n=1 or n=2. For space dimensions n≥2 and arbitrary b<0 we prove the existence of fully nontrivial nonnegative solutions which converge to a solution of some optimal partition problem as b→−∞. In case n=1 we prove that minimal energy solutions exist provided the coupling parameter b has small absolute value whereas fully nontrivial solutions do not exist if 1<q≤2 and b has large absolute value.

Spectroscopy for a few atoms harmonically trapped in one dimension

Spectroscopic labels for a few particles with spin that are harmonically trapped in one dimension with effectively zero-range interactions are provided by quantum numbers that characterize the symmetries of the Hamiltonian: permutations of identical particles, parity inversion, and the separability of the center-of-mass. The exact solutions for the noninteracting and infinitely repulsive cases are reduced with respect to these symmetries. This reduction explains how states of single-component and multicomponent fermions and bosons transform under adiabatic evolution from noninteracting to strong hard-core repulsion. These spectroscopic methods also clarify previous analytic and numerical results for intermediate values of interaction strength. Several examples, including adiabatic mapping for two-component fermionic states in the cases $N=3--5$, are provided.

Multiple mixed states of nodal solutions for nonlinear Schrödinger systems

In this paper we develop a general critical point theory to deal with existence and locations of multiple critical points produced by minimax methods in relation to multiple invariant sets of the associated gradient flow. The motivation is to study non-trivial nodal solutions with each component sign-changing for a class of nonlinear Schrodinger systems which arise from Bose–Einstein condensates theory. Our general method allows us to obtain infinitely many mixed states of nodal solutions for the repulsive case.

Space Moments of the Vlasov--Poisson System: Propagation and Regularity

We consider weak solutions to the three dimensional Vlasov--Poisson system of equations, in either the attractive or the repulsive case, which are generated by data with finite space moments of order 2. We obtain a propagation result for any space moment of order $>2$. Under some suitable assumptions, a uniform bound on the macroscopic density is derived, leading to a uniqueness statement. Both results follow from the Hölder regularity satisfied by the corresponding DiPerna--Lions flow.

A New Stochastic Approach for the Simulation of Agglomeration between Colloidal Particles

This paper presents a stochastic approach for the simulation of particle agglomeration, which is addressed as a two-step process: first, particles are transported by the flow toward each other (collision step) and, second, short-ranged particle-particle interactions lead either to the formation of an agglomerate or prevent it (adhesion step). Particle collisions are treated in the framework of Lagrangian approaches where the motions of a large number of particles are explicitly tracked. The key idea to detect collisions is to account for the whole continuous relative trajectory of particle pairs within each time step and not only the initial and final relative distances between two possible colliding partners at the beginning and at the end of the time steps. The present paper is thus the continuation of a previous work (Mohaupt M., Minier, J.-P., Tanière, A. A new approach for the detection of particle interactions for large-inertia and colloidal particles in a turbulent flow, Int. J. Multiphase Flow, 2011, 37, 746-755) and is devoted to an extension of the approach to the treatment of particle agglomeration. For that purpose, the attachment step is modeled using the DLVO theory (Derjaguin and Landau, Verwey and Overbeek) which describes particle-particle interactions as the sum of van der Waals and electrostatic forces. The attachment step is coupled with the collision step using a common energy balance approach, where particles are assumed to agglomerate only if their relative kinetic energy is high enough to overcome the maximum repulsive interaction energy between particles. Numerical results obtained with this model are shown to compare well with available experimental data on agglomeration. These promising results assert the applicability of the present modeling approach over a whole range of particle sizes (even nanoscopic) and solution conditions (both attractive and repulsive cases).

Stationary solutions for the1+1nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates in small potentials

Stationary solutions for the 1+1 cubic nonlinear Schrödinger equation modeling repulsive Bose-Einstein condensates (BEC) in a small potential are obtained through a form of nonlinear perturbation. In particular, for sufficiently small potentials, we determine the perturbation theory of stationary solutions, by use of an expansion in Jacobi elliptic functions. This idea was explored before in order to obtain exact solutions [Bronski, Carr, Deconinck, and Kutz, Phys. Rev. Lett. 86, 1402 (2001)], where the potential itself was fixed to be a Jacobi elliptic function, thereby reducing the nonlinear ODE into an algebraic equation, (which could be easily solved). However, in the present paper, we outline the perturbation method for completely general potentials, assuming only that such potentials are locally small. We do not need to assume that the nonlinearity is small, as we perform a sort of nonlinear perturbation by allowing the zeroth-order perturbation term to be governed by a nonlinear equation. This allows us to consider even poorly behaved potentials, so long as they are bounded locally. We demonstrate the effectiveness of this approach by considering a number of specific potentials: for the simplest potentials, and we recover results from the literature, while for more complicated potentials, our results are new. Dark soliton solutions are constructed explicitly for some cases, and we obtain the known one-soliton tanh-type solution in the simplest setting for the repulsive BEC. Note that we limit our results to the repulsive case; similar results can be obtained for the attractive BEC case.

Combinatorics of Generalized Bethe Equations

A generalization of the Bethe ansatz equations is studied, where a scalar two-particle S-matrix has several zeroes and poles in the complex plane, as opposed to the ordinary single pole/zero case. For the repulsive case (no complex roots), the main result is the enumeration of all distinct solutions to the Bethe equations in terms of the Fuss-Catalan numbers. Two new combinatorial interpretations of the Fuss-Catalan and related numbers are obtained. On the one hand, they count regular orbits of the permutation group in certain factor modules over Z^M, and on the other hand, they count integer points in certain M-dimensional polytopes.

Existence and Nonlinear Stability of Stationary States for the Semi-Relativistic Schrödinger-Poisson System

We study the stationary states of the semi-relativistic Schrödinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of (Abou Salem et al. in Dyn Partial Differ Equ 9(2):121–132, 2012) for the semi-relativistic Schrödinger-Poisson system to spaces with higher regularity.

Two-Step Glass Transition Induced by Attractive Interactions in Quasi-Two-Dimensional Suspensions of Ellipsoidal Particles

We study experimentally the glass transition dynamics in quasi-two-dimensional suspensions of colloidal ellipsoids, aspect ratio α=2.1, with repulsive as well as attractive interactions. For the purely repulsive case, we find that the orientational and translational glass transitions occur at the same area fraction. Strikingly, for intermediate depletion attraction strengths, we find that the orientational glass transition precedes the translational one. By quantifying structure and dynamics, we show that quasi-long-range ordering is promoted at these attraction strengths, which subsequently results in a two-step glass transition. Most interestingly, within experimental certainty, we observe reentrant glass dynamics only in the translational degrees of freedom.

Collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs in one-dimensional harmonic confinement

We investigate finite number effects in collisions between two states of an initially well known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4. In the non-interacting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left/right number uncertainty as our primary measure of entanglement. Additionally we study the time-evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create qualitatively different behavior to repulsive interactions, due to the presence of bound states (quantum solitons) and explain the processes behind this.

Adherence of Model Molecules to Silica Surfaces: First Principle Calculations

Abstract The adherence of “model molecules” methylene blue and eosine Y (“positive” and “negatively” charged respectively) to crystal SiO2 surfaces is studied from first principle calculations at the DFT level. Adsorption energies are calculated which follow the experimental threads obtained elsewhere (Rivera et al., 2013). We study the quantum nature of the electronic charge transfer between the surface and the molecules, showing the localized and delocalized patterns associated to the repulsive and attractive case respectively.

Domain wall interactions at a cross-shaped vertex

The interaction of two domain walls (DWs) at a cross-shaped vertex fabricated from two ferromagnetic nanowires has been experimentally investigated. Both magnetostatically repulsive and attractive interactions have been probed. It is found that in the repulsive case, a passing DW may directly induce the depinning of another that is already pinned at a vertex. This effect can be qualitatively described by considering only simple, magnetostatic-charge-based arguments. In the attractive case, however, asymmetric pinning is found, with complete suppression of depinning possible. This observed effect is contrary to simple charge-based arguments and highlights the need for full micromagnetic characterization of the DW interactions in more complex systems.

Emergent magnetic monopoles in frustrated magnetic systems

Emergent magnetic monopoles in frustrated magnetic systems

Single-particle excitation spectrum in 1D ultracold fermionic optical lattices

We investigate the properties of fermions trapped in a one-dimensional (1D) optical lattice by using the density-matrix renormalization (DMRG) group method. Owing to a harmonic confinement potential inherent in real experiments, system becomes inhomogeneous structure. Under certain conditions for weak-repulsive interaction, we find that the single-particle excitation spectrum structure changes to band branching and discrete bound-state states as increasing the trap strength. Additionally, we consider the case of strong-repulsive interacting regimes which local density profile show coexist with central Mott-plateau phase with surrounded by metallic regions. As increasing the trap strength, we find the breakdown of 1D Tomonaga-Luttinger (TL) liquid which is alternative to an effective doping into the Mott phase. We will present the various kinds of striking spectra, show the comparison of repulsive interaction case with attractive interaction one.

Segregated and Synchronized Vector Solutions for Nonlinear Schrödinger Systems

We consider the following nonlinear Schrodinger system in $${\mathbb{R}^3}$$ $$\left\{\begin{array}{ll}-\Delta u + P(|x|)u = \mu u^{2}u + \beta v^2u,\quad x \in \mathbb{R}^3,\\-\Delta v + Q(|x|)v = \nu v^{2}v + \beta u^2v,\quad x \in \mathbb{R}^3,\end{array}\right.$$ where P(r) and Q(r) are positive radial potentials, $${\mu > 0, \nu > 0}$$ and $${\beta \in \mathbb{R}}$$ is a coupling constant. This type of system arises, in particular, in models in Bose–Einstein condensates theory. We examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type, and in the attractive case we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Depending upon the system being repulsive or attractive, our results exhibit distinct characteristic features of vector solutions.

Effect of grafting on nanoparticle segregation in polymer/nanoparticle blends near a substrate

Nanoparticles in polymer films have shown the tendency to migrate to the substrate due to an entropic-based attractive depletion interaction between the particles and the substrate. It is also known that polymer-grafted nanoparticles show better dispersion in a polymer matrix. Here, molecular dynamics simulations are employed to study the effect of grafting on the nanoparticle segregation to the substrate. The nanoparticles were modeled as spheres and the polymers as bead-spring chains. The polymers of the grafts and the matrix are identical in nature. For a purely repulsive system, the nanoparticle density near the surface was found to decrease as the length of grafted chains and the number of grafts increased and in the bulk, the nanoparticles are well-dispersed. Whereas, in case of attractive systems with interparticle interactions on the order of thermal energy, the nanoparticles segregated to the substrate even more strongly, essentially forming clusters on the wall and in the bulk. However, due to the presence of grafted chains on the nanoparticles, the clusters formed in the bulk are structurally anisotropic. The effect of grafts on nanoparticle segregation to the surface was found to be qualitatively similar to the purely repulsive case.

Energy Spectrum of Two-Particle Cruciform Quantum Wires

We analyze the energy spectrum of a heterostructure with two particles separated in two crossed wires with parabolic confinement along their axes. We show that the energy spectrum in the limit of extremely strong confinement does not depend on the type of the interaction, and it is similar to the spectrum of a harmonic oscillator. On the contrary, when the confinement is very weak, the two-particle energy spectra for the cases of attraction and repulsion are very different. We present curves for the dependence of the energy levels on the lengths of the wires.

On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system

The paper concerns multiplicity of vector solutions for nonlinear Schrödinger systems, in particular of semi-positive solutions. New variational techniques are developed to study the existence of this type of solutions. Asymptotic behaviors are examined in various parameter regimes including both attractive and repulsive cases.

Dipole-dipole interaction in a photonic crystal nanocavity

Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190 China(Dated: December 30, 2011)Dipole-dipole interaction between two two-level ‘atoms’ in photonic crystal nanocavity is investi-gated based on finite-difference time domain algorithm. This method includes both real and virtualphoton effects and can be applied for dipoles with different transition frequencies in both weak andstrong coupling regimes. Numerical validations have been made for dipoles in vacuum and in anideal planar microcavity. For dipoles located in photonic crystal nanocavity, it is found that thecooperative decay parameters and the dipole-dipole interaction potential strongly depend on the fol-lowing four factors: the atomic position, the atomic transition frequency, the resonance frequency,and the cavity quality factor. Properly arranging the positions of the two atoms, we can acquireequal value of the cooperative decay parameters and the local coupling strength. Large cooperativedecay parameters can be achieved when transition frequency is equal to the resonance frequency.For transition frequency varying in a domain of the cavity linewidth around the resonance frequency,dipole-dipole interaction potential changes continuously from attractive to repulsive case. Largervalue and sharper change of cooperative parameters and dipole-dipole interaction can be obtainedfor higher quality factor. Our results provide some manipulative approaches for dipole-dipole in-teraction with potential application in various fields such as quantum computation and quantuminformation processing based on solid state nanocavity and quantum dot system.

Mathematical predominance of Dirichlet condition for the one-dimensional Coulomb potential

We restrict a quantum particle under a Coulombian potential (i.e., the Schrödinger operator with inverse of the distance potential) to three-dimensional tubes along the x axis and diameter ɛ, and study the confining limit ɛ → 0. In the repulsive case we prove a strong resolvent convergence to a one-dimensional limit operator, which presents Dirichlet boundary condition at the origin. Due to the possibility of the falling of the particle in the center of force, in the attractive case we need to regularize the potential and also prove a norm resolvent convergence to the Dirichlet operator at the origin. Thus, it is argued that, among the infinitely many self-adjoint realizations of the corresponding problem in one dimension, the Dirichlet boundary condition at the origin is the reasonable one-dimensional limit.