Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation

We consider the propagation of wave packets for the nonlinear Schrödinger equation, in the semi-classical limit. We establish the existence of a critical size for the initial data, in terms of the Planck constant: if the initial data are too small, the nonlinearity is negligible up to the Ehrenfest time. If the initial data have the critical size, then at leading order the wave function propagates like a coherent state whose envelope is given by a nonlinear equation, up to a time of the same order as the Ehrenfest time. We also prove a nonlinear superposition principle for these nonlinear wave packets.     

Dispersive blow-up for nonlinear Schrödinger equations revisited

The possibility of finite-time, dispersive blow-up for nonlinear equations of Schrödinger type is revisited. This mathematical phenomena is one of the conceivable explanations for oceanic and optical rogue waves. In dimension one, the fact that dispersive blow up does occur for nonlinear Schrödinger equations already appears in [9]. In the present work, the existing results are extended in several ways. In one direction, the theory is broadened to include the Davey–Stewartson and Gross–Pitaevskii equations. In another, dispersive blow up is shown to obtain for nonlinear Schrödinger equations in spatial dimensions larger than one and for more general power-law nonlinearities. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel's formula is obtained.     

Tautomeric polymorphism in salicylideneamine derivatives: an X-ray diffraction and solid-state NMR study

The crystal structure of the N-(3-hydroxysalicylidene)-4-methoxyaniline has been studied by single-crystal X-ray diffraction and solid-state NMR spectroscopy. This is the first example of a Schiff base derived from 3-hydroxysalicylaldehyde which displays in the asymmetric unit, four distinct molecules linked together in the crystal lattice by two types of intermolecular O–HO hydrogen bonds and formed by two independent tetramers. The ¹³C CPMAS NMR study corroborates the above results; the presence of different tautomeric equilibria in the same crystal structure is demonstrated and a qualitative estimation of the equilibrium mixture composition is given.     

Synthesis,characterization, and photoresponsive behavior of new azobenzene‐containing polyethers

Two new polyethers, bearing azobenzene moiety in the side chain, were synthesized in excellent yields by means of anionic polymerization of 4‐glycidyloxyazobenzene and 4‐cyano‐4′‐glycidyloxyazobenzene (leading to azo‐P1 and azo‐P2 polymers, respectively) with the system polyiminophosphazene base t‐Bu‐P₄/3,5‐di‐tert‐butylphenol as initiator. The polymers were characterized with respect to their molecular weights, structure, and calorimetric features. The polyether bearing cyanoazobenzene group in the side chain was found to exhibit nematic phase up to 200 °C. E–Z isomerization of both polymers in tetrahydrofuran solution, after irradiating with UV light at 364 nm for 15 min, was investigated by means of UV–visible absorption spectroscopy. In the case of glycidylic monomers as well as the resulting polymers, E–Z isomerization was also investigated by means of ¹H NMR, by direct irradiation in the NMR probe in deuterated 1,1,2,2‐tetrachloroethane solution. By this technique, in the case of 4‐cyano‐4′‐glycidyloxyazobenzene, it was found that irradiation led to a photostationary state corresponding to an amount of Z isomer equal to 25%. For azo‐P1 polymer, Z–E or “reverse” isomerization was investigated at 60, 70, or 80 °C directly in the NMR probe; as expected, the process followed a first‐order rate law. The kinetic constants as well as the activation energy for the process were determined in this temperature range. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 5426–5436, 2009     

Implications of large-cardinal principles in homotopical localization

The existence of arbitrary cohomological localizations on the homotopy category of spaces has remained unproved since Bousfield settled the same problem for homology theories in the decade of 1970. This is related with another open question, namely whether or not every homotopy idempotent functor on spaces is an f-localization for some map f. We prove that both questions have an affirmative answer assuming the validity of a suitable large-cardinal axiom from set theory (Vopěnka's principle). We also show that it is impossible to prove that all homotopy idempotent functors are f-localizations using the ordinary ZFC axioms of set theory (Zermelo-Fraenkel axioms with the axiom of choice), since a counterexample can be displayed under the assumption that all cardinals are nonmeasurable, which is consistent with ZFC.     

Observation of B Meson decays to b1pi and b1K

We present the results of searches for decays of B mesons to final states with a b1 meson and a charged pion or kaon. The data, collected with the BABAR detector at the Stanford Linear Accelerator Center, represent 382x10(6) BB[over ] pairs produced in e+e- annihilation. The results for the branching fractions are, in units of 10(-6), B(B+-->b1(0)pi+)=6.7+/-1.7+/-1.0, B(B+-->b1(0)K+)=9.1+/-1.7+/-1.0, B(B0-->b1(-/+)pi(+/-))=10.9+/-1.2+/-0.9, and B(B0-->b1(-)K+)=7.4+/-1.0+/-1.0, with the assumption that B(b1-->omega pi)=1. We also measure charge and flavor asymmetries A(ch)(B+-->b1(0)pi+)=0.05+/-0.16+/-0.02, Ach(B+-->b1(0)K+)=-0.46+/-0.20+/-0.02, A(ch)(B0-->b1(-/+)pi(+/-))=-0.05+/-0.10+/-0.02, C(B0-->b1(-/+)pi(+/-))=-0.22+/-0.23+/-0.05, DeltaC(B0-->b1(-/+)pi(+/-))=-1.04+/-0.23+/-0.08, and A(ch)(B0-->b1(-)K+)=-0.07+/-0.12+/-0.02. The first error quoted is statistical, and the second systematic.     

Amplitude analysis of the B+/--->phiK*(892)+/- decay

We perform an amplitude analysis of B+/--->phi(1020)K*(892)+/- decay with a sample of about 384 x 10(6) BB[over ] pairs recorded with the BABAR detector. Overall, twelve parameters are measured, including the fractions of longitudinal fL and parity-odd transverse f perpendicular amplitudes, branching fraction, strong phases, and six parameters sensitive to CP violation. We use the dependence on the Kpi invariant mass of the interference between the JP=1(-) and 0+ Kpi components to resolve the discrete ambiguity in the determination of the strong and weak phases. Our measurements of fL=0.49+/-0.05+/-0.03, f perpendicular=0.21+/-0.05+/-0.02, and the strong phases point to the presence of a substantial helicity-plus amplitude from a presently unknown source.     

On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential

We consider the nonlinear Schrödinger equation with a harmonic potential in the presence of two combined energy-subcritical power nonlinearities. We assume that the larger power is defocusing, and the smaller power is focusing. Such a framework includes physical models, and ensures that finite energy solutions are global in time. We address the questions of the existence and the orbital stability of the set of standing waves. Given the mathematical features of the equation (external potential and inhomogeneous nonlinearity), the set of parameters for which standing waves exist in unclear. In the two-dimensional case, we adapt the method of fundamental frequency solutions, introduced by the second author in the higher-dimensional case without potential. This makes it possible to describe accurately the set of fundamental frequency standing waves and ground states, and to prove its orbital stability.     

Characterization of transversal homothetic solutions in the n-body problem

Homothetic solutions of the n-body problem can be seen as heteroclinic orbits when the dynamical variables are changed via the McGehee blow-up and the time is suitably scaled. Transversality of the invariant asymptotic manifolds which contain the heteroclinic orbits is related to some structural stability. We fully characterize the cases in which such transversality is obtained for the n-body problem in any dimension.