Nondispersive solutions to the L 2-critical Half-Wave Equation

We consider the focusing L ²-critical half-wave equation in one space dimension, $$i \partial_t u = D u - |u|^2 u$$ , where D denotes the first-order fractional derivative. Standard arguments show that there is a critical threshold ${M_{*} > 0}$ such that all H ^1/2 solutions with ${\|u\|_{L^2} < M_*}$ extend globally in time, while solutions with ${\|u\|_{L^2} \geq M_*}$ may develop singularities in finite time. In this paper, we first prove the existence of a family of traveling waves with subcritical arbitrarily small mass. We then give a second example of nondispersive dynamics and show the existence of finite-time blowup solutions with minimal mass ${\|u_0\|_{L^2} = M_*}$ . More precisely, we construct a family of minimal mass blowup solutions that are parametrized by the energy E ₀ > 0 and the linear momentum ${P_0 \in \mathbb{R}}$ . In particular, our main result (and its proof) can be seen as a model scenario of minimal mass blowup for L ²-critical nonlinear PDEs with nonlocal dispersion.     

Around rationality of cycles

We prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.     

Dewetting of thin polymer films near the glass transition

Dewetting of ultrathin polymer films near the glass transition exhibits unexpected front morphologies [G. Reiter, Phys. Rev. Lett. 87, 186101 (2001)]. Here, we present the first theoretical attempt to understand these features, focusing on the shear-thinning behavior of these films. We analyze the profile of the dewetting film, and characterize the time evolution of the dry region radius, R(d)(t), and of the rim height, h(m)(t). After a transient time, depending on the initial thickness, h(m)(t) grows like square root[t] while R(d)(t) increases like exp(square root[t]). Different regimes of growth are expected, depending on the initial film thickness and experimental time range.     

Theoretical and experimental investigation of the structural and spectroscopic properties of coumarin 343 fluoroionophores

A joint theoretical-experimental investigation has been carried out to unravel the details of the complexation of cations by fluoroionophores based on coumarin 343 and to interpret the modifications in the ligand and also in the coumarin structural, electronic, magnetic, and vibrational properties. It is confirmed that C343-dea (1) complexes the cations by both the lactone and the amide oxygen atoms whereas for C343-crown (2) and C343-dibenzocrown (3), the cations are complexed by the oxygen atoms of the lactone as well as those of the crown ligand. These complexations induce geometric modifications, which are delocalized over the coumarin backbone and are related to electronic reorganizations that modify the spectroscopic signatures. This paper analyzes these signatures and shows how they are related as well as how they can be used to monitor the complexation process. Upon complexation, the UV-visible absorption spectra display a bathochromic shift of the most intense electronic transition; this shift is generally larger for the most flexible compound 1 as well as when complexing divalent cations. NMR spectra bear many signatures of the complexation, of which the most remarkable ones are the large shielding of C(1) and the large deshieldings of C(9) and C(16). Additional makers of complexation are highlighted in the IR vibrational spectra, in particular the bands associated with the lactone and amide CO vibrations, which are downshifted when the corresponding CO is involved in the complexation mode and, otherwise, upshifted. A high degree of consistency characterizes the different geometrical, electronic, magnetic, and vibrational signatures, which substantiates the assignment of the modes of complexation in 1-3. In addition, the agreement between the experimental data and the theoretical values is rather satisfactory, in that it at least enables us to interpret the spectral signatures.     

Hydrogenation properties of KSi and NaSi Zintl phases

The recently reported KSi-KSiH(3) system can store 4.3 wt% of hydrogen reversibly with slow kinetics of several hours for complete absorption at 373 K and complete desorption at 473 K. From the kinetics measured at different temperatures, the Arrhenius plots give activation energies (E(a)) of 56.0 ± 5.7 kJ mol(-1) and 121 ± 17 kJ mol(-1) for the absorption and desorption processes, respectively. Ball-milling with 10 wt% of carbon strongly improves the kinetics of the system, i.e. specifically the initial rate of absorption becomes about one order of magnitude faster than that of pristine KSi. However, this fast absorption causes a disproportionation into KH and K(8)Si(46), instead of forming the KSiH(3) hydride from a slow absorption. This disproportionation, due to the formation of stable KH, leads to a total loss of reversibility. In a similar situation, when the pristine Zintl NaSi phase absorbs hydrogen, it likewise disproportionates into NaH and Na(8)Si(46), indicating a very poorly reversible reaction.