Propagation and interaction of solitons for nonintegrable equations

We describe an approach to the construction of multi-soliton asymptotic solutions for nonintegrable equations. The general idea is realized in the case of N waves, N = 1, 2, 3, and for the KdV-type equation with nonlinearity u ⁴. A brief review of asymptotic methods as well as results of numerical simulation are included.     

Ground State of <Emphasis Type="Italic">N</Emphasis> Coupled Nonlinear Schrödinger Equations in <Emphasis Type="Italic">R</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>,<Emphasis Type="Italic">n</Emphasis>≤3

We establish some general theorems for the existence and nonexistence of ground state solutions of steady-state N coupled nonlinear Schrödinger equations. The sign of coupling constants β _ij ’s is crucial for the existence of ground state solutions. When all β _ij ’s are positive and the matrix Σ is positively definite, there exists a ground state solution which is radially symmetric. However, if all β _ij ’s are negative, or one of β _ij ’s is negative and the matrix Σ is positively definite, there is no ground state solution. Furthermore, we find a bound state solution which is non-radially symmetric when N=3.     

Phase Diagram of a Generalized ABC Model on the Interval

We study the equilibrium phase diagram of a generalized ABC model on an interval of the one-dimensional lattice: each site i=1,…,N is occupied by a particle of type α=A,B,C, with the average density of each particle species N _α /N=r _α fixed. These particles interact via a mean field nonreflection-symmetric pair interaction. The interaction need not be invariant under cyclic permutation of the particle species as in the standard ABC model studied earlier. We prove in some cases and conjecture in others that the scaled infinite system N→∞, i/N→x∈[0,1] has a unique density profile ρ _α (x) except for some special values of the r _α for which the system undergoes a second order phase transition from a uniform to a nonuniform periodic profile at a critical temperature \(T_{c}=3\sqrt{r_{A} r_{B} r_{C}}/2\pi\).     

Electromagnetic-field amplification in finite one-dimensional photonic crystals

The electromagnetic-field distribution in a finite one-dimensional photonic crystal is studied using the numerical solution of Maxwell’s equations by the transfer-matrix method. The dependence of the transmission coefficient T on the period d (or the wavelength λ) has the characteristic form with M–1 (M is the number of periods in the structure) maxima with T = 1 in the allowed band of an infinite crystal and zero values in the forbidden band. The field-modulus distribution E(x) in the structure for parameters that correspond to the transmission maxima closest to the boundaries of forbidden bands has maxima at the center of the structure; the value at the maximum considerably exceeds the incident-field strength. For the number of periods M ~ 50, more than an order of magnitude increase in the field amplification is observed. The numerical results are interpreted with an analytic theory constructed by representing the solution in the form of a linear combination of counterpropagating Floquet modes in a periodic structure.     

Justification of the Lattice Equation for a Nonlinear Elliptic Problem with a Periodic Potential

We justify the use of the lattice equation (the discrete nonlinear Schrödinger equation) for the tight-binding approximation of stationary localized solutions in the context of a continuous nonlinear elliptic problem with a periodic potential. We rely on properties of the Floquet band-gap spectrum and the Fourier–Bloch decomposition for a linear Schrödinger operator with a periodic potential. Solutions of the nonlinear elliptic problem are represented in terms of Wannier functions and the problem is reduced, using elliptic theory, to a set of nonlinear algebraic equations solvable with the Implicit Function Theorem. Our analysis is developed for a class of piecewise-constant periodic potentials with disjoint spectral bands, which reduce, in a singular limit, to a periodic sequence of infinite walls of a non-zero width. The discrete nonlinear Schrödinger equation is applied to classify localized solutions of the Gross–Pitaevskii equation with a periodic potential.     

Entanglement measure for pure <Emphasis Type="Italic">M</Emphasis> ⊗ <Emphasis Type="Italic">N</Emphasis> bipartite quantum states

We propose an entanglement measure for pure M ? N bipartite quantum states. We obtain the measure by generalizing the equivalent measure for a 2 ? 2 system, via a 2 ? 3 system, to the general bipartite case. The measure emphasizes the role Bell states have, both for forming the measure and for experimentally measuring the entanglement. The form of the measure is similar to the generalized concurrence. In the case of 2 ? 3 systems, we prove that our measure, which is directly measurable, equals the concurrence. It is also shown that, in order to measure the entanglement, it is sufficient to measure the projections of the state onto a maximum of M(M ? 1)N(N ? 1)/2 Bell states.     

Universal Low-energy Behavior in a Quantum Lorentz Gas with Gross-Pitaevskii Potentials

We consider a quantum particle interacting with N obstacles, whose positions are independently chosen according to a given probability density, through a two-body potential of the form N²V (Nx) (Gross-Pitaevskii potential). We show convergence of the N dependent one-particle Hamiltonian to a limiting Hamiltonian where the quantum particle experiences an effective potential depending only on the scattering length of the unscaled potential and the density of the obstacles. In this sense our Lorentz gas model exhibits a universal behavior for N large. Moreover we explicitely characterize the fluctuations around the limit operator. Our model can be considered as a simplified model for scattering of slow neutrons from condensed matter.     

Quark microscopic picture for vector and pseudoscalar mesons in the nucleon and their quasielastic knockout by high-energy electrons

A technique for projecting a multiquark wave function in the microscopic model of a ³P⁰ scalar fluctuation onto the virtual-decay channels N → N + ρ and N → N + π is formulated (at a more general level for the latter than previously). The amplitude for the electromagnetic transition ρ + γ _T ^* → π in electron-induced quasielastic rho-meson knockout followed by rho-meson conversion to a pion is considered. Theoretical results obtained in this way are contrasted against available experimental data, and reasonable agreement is found for cross-section values. This confirms a universal character of the ³P₀ model. The precision of relevant experiments is as yet insufficient for comparing the momentum distribution of the rho meson from the channel N → N + ρ with its theoretical counterpart.     

Effect of intrinsic defects on the electronic structure of BN nanotubes

The effect of intrinsic defects on the electronic structure of boron-nitrogen nanotubes (5, 5) and (9, 0) is investigated by the method of linearized associated cylindrical waves. Nanotubes with extended defects of substitution N _B of a boron atom by a nitrogen atom and, vice versa, nitrogen by boron BN with an impurity concentration of 1.5 to 5% are considered. It is shown that the presence of such defects significantly affects the band structure of boron-nitrogen nanotubes. A defect band D^π(B, N) is formed in the bandgap, which sharply reduces the width of the gap. The presence of impurities also affects the valence band: the widths of s, sp, and p_π bands change and the gap between s and sp bands is partially filled. These effects may be detected experimentally by, e.g., optical and photoelectron spectroscopy.     

Analytical study of charged boson stars with large scalar self-couplings

We give good approximate analytic solutions for spherical charged boson stars in the large scalar-self-coupling limit in general relativity. We show that if the charge e and mass m of the scalar field nearly satisfy the critical relation \(e^2\approx Gm^2\) (where G is the Newton constant), our analytic expressions for stable solutions agree well with the numerical solutions.     

Weyl’s Type Estimates on the Eigenvalues of Critical Schrödinger Operators

We consider on a bounded domain \(\Omega \subset {\mathbb{R}}^N\) , the Schrödinger operator ? Δ ? V supplemented with Dirichlet boundary solutions. The potential V is either the critical inverse square potential V(x) = (N ? 2)²/4|x|^?2 or the critical borderline potential V(x) =  (1/4)dist(x, ?Ω)^?2. We present explicit asymptotic estimates on the eigenvalues of the critical Schrödinger operator in each case, based on recent results on improved Hardy–Sobolev type inequalities.     

Observational Constraints on the Chaplygin Gas with Inverse Power Law Potential in Braneworld Inflation

We study Chaplygin gas as a candidate for inflation in the context of braneworld inflationary model. We investigate this model in the framework of the Randall–Sundrum type II, considering a original and generalized Chaplygin gas. We use inverse power law potential to examine the behavior of some inflationary spectrum parameters such as the spectral index n_s, the ratio r and the running of the scalar spectral index dn_s/dlnk, our results are in agreement with recent observational data for a particular choice of e-folding number N and parameters space of the model.     

The trinification model <Emphasis Type="Italic">SU</Emphasis>(3)<Superscript>3</Superscript> from orbifolds for fuzzy spheres

In this review, we consider an N = 4 supersymmetric SU(3N) gauge theory defined on the Minkowski spacetime. Then we apply an orbifold projection leading to an N = 1 supersymmetric SU(N)³ model, with a truncated particle spectrum. Then, we present the dynamical generation of (twisted) fuzzy spheres as vacuum solutions of the projected field theory, breaking the SU(N)³ spontaneously to a chiral effective theory with unbroken gauge group the trinification group, SU(3)³.     

Raman Scattering in Lonsdaleite

We present the results of our study of a polycrystalline diamond–lonsdaleite powder from the Popigai crater (Siberia) using UV micro-Raman scattering and synchrotron X-ray diffraction. By subtracting two experimental Raman spectra of diamond–lonsdaleite samples with a close ratio of the diamond and lonsdaleite fractions, when the maximum contribution to the difference spectrum is related to the difference of the lonsdaleite contributions and not to the change in the diamond band width, we have managed to obtain the spectrum of “pure” lonsdaleite. Its deconvolution has allowed us to identify all three Raman-active vibrational modes E_2g, A_1g, and E_1g whose positions agree well with the results of ab initio calculations.     

Neutral Particle Motion around a Schwarzschild Black Hole in Modified Gravity

We investigate circular motion of neutral test particles on equatorial plane near a black hole in scalar-tensor-vector gravity. We consider three cases (i) α < G/G_N (ii) α = G/G_N and (iii) α > G/G_N to find the regions where motion can exist. The corresponding effective potential, energy, angular momentum and center of mass energy are evaluated. Further, we define four different cases for α > G/G_N and identify stable and unstable regions of circular orbits. It is found that circular orbits having zero angular momentum exist at r = αGNM due to repulsive gravity effects. We conclude that the structure of stable regions for α < G/G_N as well as α = G/G_N case is completely different from that of α > G/G_N.     

Massive hyper-Kähler sigma models and BPS domain walls

With the non-Abelian hyper-Kähler quotient by U(M) and SU(M) gauge groups, we give the massive hyper-Kähler sigma models that are not toric in the N=1 superfield formalism. The U(M) quotient gives N!/[M!(N-M)!] (N is the number of flavors) discrete vacua that may allow various types of domain walls, whereas the SU(M) quotient gives no discrete vacua. We derive a BPS domain-wall solution in the case of N = 2 and M = 1 in the U(M) quotient model.     

Effect of incommensurate potential on the resonant tunneling through Majorana bound states on the topological superconductor chains

We study the transport through the Kitaev chain with incommensurate potentials coupled to two normal leads by the numerical operator method. We find a quantized linear conductance of e ² / h, which is independent to the disorder strength and the gate voltage in a wide range, signaling the Majorana bound states. While the incommensurate potential suppresses the current at finite voltage bias, and then narrows the linear response regime of the I-V curve which exhibits two plateaus corresponding to the superconducting gap and the band edge, respectively. The linear conductance abruptly drops to zero as the disorder strength reaches the critical value 2g _s + 2Δ with Δ the p-wave pairing amplitude and g _s the hopping between neighbor sites, corresponding to the transition from the topological superconducting phase to the Anderson localized phase. Changing the gate voltage also causes an abrupt drop of the linear conductance by driving the chain into the topologically trivial superconducting phase, whose I-V curve exhibits an exponential shape.     

Rigidity of the Laughlin Liquid

We consider general N-particle wave functions that have the form of a product of the Laughlin state with filling factor \(1/\ell \) and an analytic function of the N variables. This is the most general form of a wave function that can arise through a perturbation of the Laughlin state by external potentials or impurities, while staying in the lowest Landau level and maintaining the strong correlations of the original state. We show that the perturbation can only shift or lower the 1-particle density but nowhere increase it above a maximum value. Consequences of this bound for the response of the Laughlin state to external fields are discussed.     

On the Mean Field and Classical Limits of Quantum Mechanics

The main result in this paper is a new inequality bearing on solutions of the N-body linear Schrödinger equation and of the mean field Hartree equation. This inequality implies that the mean field limit of the quantum mechanics of N identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation. This result applies to the case of C^1,1 interaction potentials. The quantity measuring the approximation of the N-body quantum dynamics by its mean field limit is analogous to the Monge–Kantorovich (or Wasserstein) distance with exponent 2. The inequality satisfied by this quantity is reminiscent of the work of Dobrushin on the mean field limit in classical mechanics [Func. Anal. Appl. 13, 115–123, (1979)]. Our approach to this problem is based on a direct analysis of the N-particle Liouville equation, and avoids using techniques based on the BBGKY hierarchy or on second quantization.