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.     

Collapse of optical vortices

We theoretically and experimentally investigate the self-focusing of optical vortices in Kerr media. We observe collapse to a distinct self-similar profile, which becomes unstable to azimuthal perturbations. We analyze the azimuthal modulational instability for ring-shaped vortices and predict the number of azimuthal maxima solely as a function of power and topological charge. In our experiments, the observed multiple-filamentation patterns are in excellent agreement with our theoretical analysis.     

In-situ UV-visible study of Pd nanocluster formation in solution

The formation of Pd nanoclusters in solution is studied. This system has two types of light-absorbing species: Pd ions which absorb light via electronic transitions and Pd clusters and aggregates which absorb light via valence-conduction transitions and also scatter light due to their nanometric dimensions. Here we monitor these dynamic changes using UV-visible spectroscopy. The reduction and clustering concentration profiles are extracted from the raw data using a combination of net analyte signal (NAS) and principal component analysis (PCA) methods. PdCl2, Pd(OAc)2 and Pd(NO3)2 are used as Pd2+ precursors and various tetra-n-octylammonium carboxylates are applied as reducing and stabilising agents. This in situ approach enables the quantification of both the reduction of the Pd2+ ions and the growth of the Pd clusters. Kinetic models that account for ion reduction, cluster growth and aggregation are presented and the influence of the counteranions and the reducing agents on these processes is discussed.     

感应耦合非热等离子体催化转化二氧化碳

射频感应耦合等离子体能够在室温条件下选择活化并分解二氧化碳,本文通过调控等离子体放电条件(气体流量、放电频率等)获得了较高的一氧化碳产率. 研究发现网状金属催化剂在二氧化碳等离子体中能促进氧原子的复合反应,从而有效抑制其与目标产物一氧化碳的逆反应,提升反应整体效率. 本文为实时转化过剩的可再生电能(来自太阳能、风能、潮汐能等)为高附加值一氧化碳中的化学能提供了一个可行的方案.     

High-order numerical solution of the nonlinear Helmholtz equation with axial symmetry

The nonlinear Helmholtz (NLH) equation models the propagation of intense laser beams in a Kerr medium. The NLH takes into account the effects of nonparaxiality and backward scattering that are neglected in the more common nonlinear Schrödinger model. In [G. Fibich, S. Tsynkov, High-order two-way artificial boundary conditions for nonlinear wave propagation with backscattering, J. Comput. Phys., 171 (2001) 632–677] and [G. Fibich, S. Tsynkov, Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions, J. Comput. Phys., 210 (2005) 183–224], a novel high-order numerical method for solving the NLH was introduced and implemented in the case of a two-dimensional Cartesian geometry. The NLH was solved iteratively, using the separation of variables and a special nonlocal two-way artificial boundary condition applied to the resulting decoupled linear systems. In the current paper, we propose a major improvement to the previous method. Instead of using LU decomposition after the separation of variables, we employ an efficient summation rule that evaluates convolution with the discrete Green's function. We also extend the method to a three-dimensional setting with cylindrical symmetry, under both Dirichlet and Sommerfeld-type transverse boundary conditions.     

Symposium 9: Analytical problems and biological significance of cytokines in body fluids

Analytical and Bioanalytical Chemistry -     

Size direction games over the real line. I

Two related infinite two person games of perfect information are defined and studied.     

Conversion of CO2 by non-Thermal Inductively-Coupled Plasma Catalysis

CO\begin{document}$ _2 $\end{document} decomposition is a very strongly endothermic reaction where very high temperatures are required to thermally dissociate CO\begin{document}$ _2 $\end{document}. Radio frequency inductively-coupled plasma enables to selectively activate and dissociate CO\begin{document}$ _2 $\end{document} at room temperature. Tuning the flow rate and the frequency of the radio frequency inductively-coupled plasma gives high yields of CO under mild conditions. Finally the discovery of a plasma catalytic effect has been demonstrated for CO\begin{document}$ _2 $\end{document} dissociation that shows a significant increase of the CO yield by metallic meshes. The metallic meshes become catalysts under exposure to plasma to activate the recombination reaction of atomic O to yield O\begin{document}$ _2 $\end{document}, thereby reducing the reaction to convert CO back to CO\begin{document}$ _2 $\end{document}. Inductively-coupled hybrid plasma catalysis allows access to study and to utilize high CO\begin{document}$ _2 $\end{document} conversion in a non-thermal plasma regime. This advance offers opportunities to investigate the possibility to use radio frequency inductively-coupled plasma to store superfluous renewable electricity into high-valuable CO in time where the price of renewable electricity is plunging.