Existence of multi-solitary waves with logarithmic relative distances for the NLS equation

We construct 2-solitary wave solutions with logarithmic distance to the nonlinear Schrödinger equation,

$\mathrm{i}{?}_{t}u+\mathrm{\Delta }u+|u{|}^{p?1}u=0,t\in \mathbb{R},x\in {\mathbb{R}}^d,$

in mass-subcritical cases $1<p<1+\frac{4}{d}$ and mass-supercritical cases $1+\frac{4}{d}<p<\frac{d+2}{d?2}$, i.e. solutions $u(t)$ satisfying

${6u(t)?{\mathrm{e}}^{\mathrm{i}\gamma (t)}\sum _{k=1}^{2}Q(??x_k(t))6}_{H^1}\to 0$

and

$|x_1(t)?x_2(t)|～2\log ?t,\text{as}t\to +\infty ,$

where Q is the ground state. The logarithmic distance is related to strong interactions between solitary waves.In the integrable case ($d=1$ and $p=3$), the existence of such solutions is known by inverse scattering (E. Olmedilla, Multiple pole solutions of the nonlinear Schrödinger equation, Physica D 25 (1987) 330–346; T. Zakharov, A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. JETP 34 (1972) 62–69). The mass-critical case $p=1+\frac{4}{d}$ exhibits a specific behavior related to blow-up, previously studied in Y. Martel, P. Raphaël (Strongly interacting blow up bubbles for the mass critical NLS, Ann. Sci. Éc. Norm. Supér. 51 (2018) 701–737).     

On some Hasse principles for homogeneous spaces of algebraic groups over global fields of positive characteristic

We extend to global function fields some Hasse principles for homogeneous spaces of connected linear algebraic groups proved earlier by several authors in the case of number fields. We also give some applications.     

Involutionn-aire et ses applications a l'étud des hyperquadriques dans les espaces euclidiens et non-euclidiens an dimensions

Sans résumé Présenté par G. Hajós     

Fullerene C60-perylene-3,4:9,10-bis(dicarboximide) light-harvesting dyads: spacer-length and bay-substituent effects on intramolecular singlet and triplet energy transfer

Novel covalent fullerene C(60)-perylene-3,4:9,10-bis(dicarboximide) (C(60)-PDI) dyads (1-4) were synthesized and characterized. Their electrochemical and photophysical properties were investigated. Electrochemical studies show that the reduction potential of PDI can be tuned relative to C(60) by molecular engineering through altering the substituents on the PDI bay region. It was demonstrated using steady-state and time-resolved spectroscopy that a quantitative, photoinduced energy transfer takes place from the PDI moiety, acting as a light-harvesting antenna, to the C(60) unit, playing the role of energy acceptor. The bay-substitution (tetrachloro [1 and 2] or tetra-tert-butylphenoxy [3 and 4]) of the PDI antenna and the linkage length (C(2) [1 and 3] or C(5) [2 and 4]) to the C(60) acceptor are important parameters in the kinetics of energy transfer. Femtosecond transient absorption spectroscopy indicates singlet-singlet energy-transfer times (from the PDI to the C(60) unit) of 0.4 and 5 ps (1), 4.5 and 27 ps (2), 0.8 and 12 ps (3), and 7 and 50 ps (4), these values being ascribed to two different conformers for each C(60)-PDI system. Subsequent triplet-triplet energy-transfer times (from the C(60) unit to the PDI) are slower and in the order of 0.8 ns (1), 6.2 ns (2), 2.7 ns (3), and 9 ns (4). Nanosecond transient absorption spectroscopy of final PDI triplet states show a marked influence of the bay substitution (tetrachloro- or tetra-tert-butylphenoxy), and triplet-state lifetimes (10-20 micros) and the PDI triplet quantum yields (0.75-0.52) were estimated. The spectroscopy showed no substantial solvent effect upon comparing toluene (non-polar) to benzonitrile (polar), indicating that no electron transfer is occurring in these systems.     

Oxygen-17 nuclear magnetic resonance: The effects of remote unsaturation on 17o-chemical shifts in polycyclic ethers

Natural-abundance ¹⁷O-NMR spectra of 7-oxanorbornane exo-3-oxatricyclo [3.2.1.0^2.4]octane and their unsaturated derivatives (endo cyclic and exocyclic double bonds) have been measured. Linear correlation laws were observed for $\text{δ}{}_{\mathrm{o}}\text{δ}{}_{\mathrm{c}}$ of these ethers/corresponding hydrocarbons. The “cyclization shifts” for δ_o in ethers were not correlated by the “cyclization shifts” for δ_c of the corresponding hydrocarbons.     

Modifications de la position C-8 de purines nucleosides par action de reactifs electrophiles sur leurs derives silyles et lithies

The alkylation of 8-lithiopurine trimethylsilyl nucleosides at carbon 8 can be carried out conveniently by reaction with various electrophilic reagents.     

Modification de la position 8 des purines nucleosides et de l'adenosine monophosphate cyclique-3′,5′. Reactions de couplage catalytique des organo-magnesiens avec les bromo-8 purines ribosides et bromo-8 adenosine monophosphate cyclique-3′,5′ silyles en presence de dichloro-bis-triphenylphosphine palladium

The palladium catalysed condensation of Grignard reagents with silyl derivatives of 8-bromopurine nucleosides and 8-bromoadenosine-3′,5′ cyclic monophosphate is a convenient method for the preparation of free 8-alkylpurine nucleosides and 8-alkyladenosine-3′,5′ cyclic monophosphate.