Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

Abstract

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schrödinger equation (see work of Zakharov, Glassey, and Ogawa–Tsutsumi) to show that certain classes of negative energy solutions must blowup in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.     

A bicapped trigonal–prismatic lead complex: catena‐poly[[[aqua­bis(1,10‐phenanthroline‐κ2N,N′)lead(II)]‐μ‐squarato‐κO1:κ2O2,O3] dihydrate]

The asymmetric unit of the title compound, {[Pb(C₄O₄)(C₁₂H₈N₂)₂(H₂O)]·2H₂O}_n, contains one squarate dianion, two phenanthroline (phen) ligands and one aqua ligand all coordinated to Pb, and two solvent water mol­ecules. The eight‐coordinate Pb metal ion displays a distorted bicapped trigonal–prismatic coordination environment, defined by three squarate O atoms, four N atoms from two chelating phen ligands and one O atom from the coordinated water mol­ecule. The crystal structure contains chains of squarate‐1,2,3‐bridged Pb^II ions running in the [010] direction. These polymeric chains are linked to one another via offset face‐to‐face π–π inter­actions between the phen ligands, which lead to a two‐dimensional network extending along the (001) plane. The crystal structure is also stabilized by O—H⋯O inter­molecular hydrogen‐bond inter­actions, forming a three‐dimensional network.     

(2,9‐Dimethyl‐1,10‐phenanthroline‐κ2N,N′)(pyridine‐2,6‐dicarboxylato‐κ3O2,N,O6)copper(II) trihydrate

The copper(II) centre in the mononuclear title complex, [Cu(C₇H₃NO₄)(C₁₄H₁₂N₂)]·3H₂O, is surrounded by one bidentate 2,9‐dimethyl‐1,10‐phenanthroline (dmphen) ligand and one tridentate pyridine‐2,6‐dicarboxylate ligand, and exhibits a distorted square‐pyramidal geometry. The crystal packing involves both hydrogen‐bonding and π–π inter­actions. The solvent water mol­ecules link monomers to one another through hydrogen‐bonding inter­actions, forming ladder‐like chains in the bc plane. Face‐to‐face and slipped π–π inter­actions also occur between dmphen rings of neighboring mol­ecules and are responsible for inter­chain packing.     

Bis(acesulfamato‐κ2N3,O4)bis­(2‐amino­pyrimidine‐κN1)copper(II)

In the crystal structure of the title compound, bis­(2‐amino­pyrimidine‐κN¹)bis­[6‐meth­yl‐1,2,3‐oxathia­zin‐4(3H)‐one 2,2‐dioxide(1−)‐κ²N³,O⁴]copper(II), [Cu(C₄H₄NO₄S)₂(C₄H₅N₃)₂], the first mixed‐ligand complex of acesulfame, the Cu^II centre resides on a centre of symmetry and has an octa­hedral geometry that is distorted both by the presence of four‐membered chelate rings and by the Jahn–Teller effect. The equatorial plane is formed by the N atoms of two amino­pyrimidine (ampym) ligands and by the weakly basic carbonyl O atoms of the acesulfamate ligands, while the more basic deprotonated N atoms of these ligands are in the elongated axial positions with a strong misdirected valence. The crystal is stabilized by pyrimidine ring stacking and by inter­molecular hydrogen bonding involving the NH₂ moiety of the ampym ligand and the carbon­yl O atom of the acesulfamate moiety.     

An efficient method for synthesis of organophosphorus compounds in aqueous media

A convenient and facile one-pot synthesis of stable phosphorus ylides and 1,4-diionic organophosphorus compounds is reported by the reaction of triphenylphosphine,dialkylacetylenedicarboxylates and N-H,C-H or S-H acids in the presence of polyethyleneglycol (PEG),β-cyclodextrin(β-CD),glycerine(Gly) or ethyleneglycol(EG) in water.This methodology is of interest due to the use of water as a solvent,thus minimizing the cost operational hazards,and environmental pollution.     

Synthesis and electrochemistry of tetrakis(7-coumarinthio-4-methyl)-phthalocyanines,and preparation of their cinnamic acid and sodium cinnamate derivatives

Novel water soluble free-base, Zn(II) and Co(II) metallo phthalocyanines with four cinnamic acid moieties were prepared from the corresponding tetrakis(7-coumarinthio-4-methyl)-phthalocyanine by the lactone ring opening reaction. The new compounds were purified and characterized by elemental analysis, ¹H NMR, Maldi-TOF, FT-IR and UV–Vis spectral data. Cyclic and differential pulse voltammetry and in situ spectroelectrochemistry of the 2,9,16,23-tetrakis(7-coumarinthio-4-methyl) substituted free-base 1a, Zn(II) 1b and Co(II) 1c phthalocyanines, employed as the starting compounds have been studied. This allowed us not only to identify metal- and phthalocyanine ring-based redox processes of the complexes, but also the effect of aggregation on these processes.     

Structural, spectroscopic characterization and EPR studies of tetramethylammonium-hydrogen fumarate-fumaric acid complex

The tetramethylammonium-hydrogen fumarate-fumaric acid (hereafter, [TMAHF-FA]) complex was synthesized and characterized by spectroscopic (IR), structural (XRD) and electron paramagnetic resonance (EPR) techniques. In the crystal structure, tetramethylammonium cation lies on a mirror plane of the space group P2₁/m. Crystal structure analysis reveals that there is a large degree of proton sharing between the fumaric acid and hydrogen fumarate, with the H atom lying almost symmetrically between the donor and acceptor sites, as evidenced by the long O-H and short H?O distances [1.19(3) Å, 1.26(3) Å], respectively. γ-Irradiation damage centers in [TMAHF-FA] single crystal have been investigated by EPR at room temperature. The spectra indicated the existence of radical. The EPR spectra recorded in the three mutually perpendicular planes have shown two magnetically distinct paramagnetic sites in monoclinic lattice. The principal values of g and hyperfine constants for both sites were calculated. IR spectrum was resolved and transitions were assigned based on the molecular structure.     

Optical solitons and modulation instability analysis of the (1+1)-dimensional coupled nonlinear Schrödinger equation

This study successfully reveals the dark, singular solitons, periodic wave and singular periodic wave solutions of the (1+1)-dimensional coupled nonlinear Schrödinger equation by using the extended rational sine-cosine and rational sinh-cosh methods. The modulation instability analysis of the governing model is presented. By using the suitable values of the parameters involved, the 2-, 3-dimensional and the contour graphs of some of the reported solutions are plotted.