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.     

Theory of the vortex matter transformations in high-Tc superconductor YBCO

Flux line lattice in type II superconductors undergoes a transition into a "disordered" phase such as vortex liquid or vortex glass, due to thermal fluctuations and random quenched disorder. We quantitatively describe the competition between the thermal fluctuations and the disorder using the Ginzburg-Landau approach. The following T-H phase diagram of YBCO emerges. There are just two distinct thermodynamical phases, the homogeneous and the crystalline one, separated by a single first order transition line. The line, however, makes a wiggle near the experimentally claimed critical point at 12 T. The "critical point" is reinterpreted as a (noncritical) Kauzmann point in which the latent heat vanishes and the line is parallel to the T axis. The magnetization, the entropy, and the specific heat discontinuities at melting compare well with experiments.     

Evolution of localization in frequency for modulated light pulses in a recirculating fiber loop

We present an experimental demonstration of the evolution of localization in frequency of light pulses that are repeatedly kicked by phase modulation and then propagated along equally spaced lengths of fiber with weak dispersion. The experiment was performed with a long fiber recirculating loop that allows us to follow the pulse's spectral changes after each cycle.     

Hydrodynamic singularities and clustering in a freely cooling inelastic gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite-time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.     

Experimental study of the stochastic nature of the pulsation self-starting process in passive mode locking

We present an experimental study of the probabilistic nature of pulsation self-starting in passively mode-locked lasers. It is a Poissonian process that results from a noise-activated switching barrier. The switching rate from cw operation to pulsation when the laser pump level is turned on has an exponential dependence that is inversely proportional to the square of the laser power.     

Radiation-induced dechlorination of carbon tetrachloride. II. The kinetics of liquid-phase reactions of trichloromethyl radicals in n-hexane solutions

The kinetics of the gamma-radiation induced free-radical chain reaction in solutions of carbon tetrachloride in n-hexane (RH) was investigated in the temperature range of 297.5–373°K. Trichloromethyl radicals were produced by the reaction of radiolytically generated hexyl radicals with carbon tetrachloride. The kinetics of the following reactions were investigated: The following rate expression was obtained: where θ = 2.303 RT in kcal/mole. The error limits are the standard deviation from the least mean square Arrhenius plots. It is concluded that reaction (4) is diffusion controlled in the liquid phase, and that the activation energies in the gas and liquid phases of reaction (3) are equal.     

Analysis of an elasto-plastic thick walled sphere loaded by internal and external pressure

The non-linear behaviour of a thick walled sphere under internal and external pressure is considered. The material is assumed to be incrementally elasto-plastic in the classical sense. No restriction on the magnitude of the deformation is imposed. For internal pressure a solution is obtained in terms of closed integrals. It is shown that the three dimensional solution reduces exactly to the membrane solution when the thickness of the shell becomes very small.Numerical examples are given for some practical materials used in the aerospace industry. It is shown that at a certain critical pressure instability of the second kind (inflation) is obtained.     

Rings and boxes in dissipative environments

We study a particle on a ring in presence of a dissipative Caldeira-Leggett environment and derive its response to a dc field. We find, through a 2-loop renormalization group analysis, that a large dissipation parameter η flows to a fixed point η(R)=η(c)=?/2π. We also reexamine the mapping of this problem to that of the Coulomb box and show that the relaxation resistance, of recent interest, is quantized for large η. For finite η>η(c) we find that a certain average of the relaxation resistance is quantized. We propose a box experiment to measure a quantized noise.     

Representations of product systems over semigroups and dilations of commuting CP maps

We prove that every pair of commuting CP maps on a von Neumann algebra M can be dilated to a commuting pair of endomorphisms (on a larger von Neumann algebra). To achieve this, we first prove that every completely contractive representation of a product system of C^∗-correspondences over the semigroup N² can be dilated to an isometric (or Toeplitz) representation.