Exponential decay of solutions to nonlinear elliptic equations with potentials

Exponential decay estimates are obtained for complex-valued solutions to nonlinear elliptic equations in \mathbbRⁿ ,\mathbb{R}^{n} , where the linear term is given by Schr?dinger operators H =  − Δ  +  V with nonnegative potentials V and the nonlinear term is given by a single power with subcritical Sobolev exponent in the attractive case. We describe specific rates of decay in terms of V, some of which are shown to be optimal. Moreover, our estimates provide a unified understanding of two distinct cases in the available literature, namely, the vanishing potential case V = 0 and the harmonic potential case V(x) = |x|².     

Modulation of fluorescence through coassembly of molecules in organic nanostructures

This paper describes the fluorescence of bimolecular coassemblies that form one-dimensional nanostructures. One molecule is a fluorescent peptide amphiphile containing its branched stilbene chromophore covalently linked to the hydrophilic end of the amphiphile, and the second molecule is a shorter, nonfluorescent peptide amphiphile of complementary charge. Using circular dichroism we observe that mixing both molecules results in coassemblies that exhibit a beta-sheet signature in the peptide region indicative of these types of nanostructures. The nature of the coassembly is dependent on the molar ratio of each component, and the changing CD spectra suggest the formation of domains along the length of the nanofibers with decreasing concentrations of the fluorescent component. In coassemblies with dilute concentrations of the fluorophore, we observe an increase in fluorescence intensity and quantum yield, as well as chiral transfer to the achiral segment of the fluorescent peptide amphiphile. The coassemblies studied containing a fluorescent component at a low molar ratio exhibit fluorescence resonance energy transfer to fluorescent acceptors in solution. When the nonfluorescent peptide amphiphile component is designed to bind the important bioactive polysaccharide heparin, a selective transfer of energy is observed between fluorescein-tagged heparin and the coassemblies in both dilute solution and in macroscopic gels.     

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A major difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves. This is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing-wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the nonradial repulsive case, all bound states are unstable, and the instability is manifested by a lateral drift away from the defect, sometimes in combination with a finite-width instability or a blowup instability.     

On a decay property of solutions to the Haraux-Weissler equation

We give a sufficient condition that non-radial H¹-solutions to the Haraux-Weissler equation should belong to the weighted Sobolev space , where ρ is the weight function exp(|x|²/4). Our result provides, in some sense, a connection between the solutions obtained by ODE method and those by variational approach in the space .     

Molecular design of polymer coatings capable of photo-triggered stress relaxation via dynamic covalent bond exchange

Polymer coatings are frequently used to modify surface properties of inorganic substrates. However, the disparity in physical properties between polymer film and substrate often leads to residual stress development, which can be deleterious to the overall performance of coated materials. This work reports the molecular design of polymer films that dissipate stress upon irradiation with ultraviolet (UV) light. These polymers are synthesized by post-polymerization modification of the reactive polymer, poly(2-vinyl-4,4-dimethyl azlactone), to introduce dynamic crosslinks capable of light-initiated addition transfer fragmentation chemistry. Using a custom-built optical cantilever, contrasting film stress responses are observed between films containing dynamic bonds and analogous control films after UV light irradiation, which indicate successful stress relaxation. Further experiments demonstrate the complete relaxation of residual stress in dynamic films after an extended exposure, thereby generating a “stress-free” film. Films fabricated using this approach can be easily tailored to incorporate additional moieties to introduce desired surface properties for future application in a wide array of coatings.     

Pendular-state spectroscopy of the S(1)-S(0) electronic transition of 9-cyanoanthracene

Fluorescence excitation spectra of the S(1)-S(0) origin band of 9-cyanoanthracene have been observed under a uniform electric field up to 200 kV/cm to explore pendular-state spectrum of an asymmetric-top molecule close to the strong field limit. The observed spectra exhibit distinct evolution of the band contour as a function of the applied electric field, which are much different from each other for different excitation configurations. An approximate method suitable for spectrum simulations of large asymmetric-top molecules in a pendular condition is developed for the analysis of the experimental results. The comparison of the observed and simulated spectra shows that the spectra are well ascribed in terms of the pendular-state selection rules, which have recently been derived from theoretical consideration of the pendular-limit representation of energy levels and spectra.     

Stability of Standing Waves for Nonlinear Schrödinger Equations with Inhomogeneous Nonlinearities

The effect of inhomogeneity of nonlinear medium is discussed concerning the stability of standing waves e^{i ω t}ϕ_ω(x) for a nonlinear Schr?dinger equation with an inhomogeneous nonlinearity V (x)|u|^{p − 1}u, where V (x) is proportional to the electron density. Here, ω > 0 and ϕ_ω(x) is a ground state of the stationary problem. When V (x) behaves like |x|^−b at infinity, where 0 < b < 2, we show that e^{i ω t}ϕ_ω(x) is stable for p < 1 + (4 − 2b)/n and sufficiently small ω > 0. The main point of this paper is to analyze the linearized operator at standing wave solution for the case of V (x) = |x|^−b. Then, this analysis yields a stability result for the case of more general, inhomogeneous V (x) by a certain perturbation method. Communicated by Bernard Helffer submitted 14/07/04, accepted 28/02/05     

Photo-induced aromatic assembly of benzocycloalka[1,2-b] furan and spiro[furan-2(3H),1′-benzocycloalkane] derivatives

The photo-induced benzannulation of benzocycloalka[1,2-b]furans produced hydrohelicene-type compounds in good yields. A similar photoreaction of the spiro[furan-2(3H),1′-benzocycloalkane]s afforded dihydrophenalene derivatives in moderate yields. Benzo[kl]xanthene was also formed by a similar photoreaction of spiro[furan-2(3H),9′-xanthene]. The reaction pathway was elucidated by the ¹H NMR spectrum of the reaction mixture and the alternative synthesis of the intermediate.