On the convergence of difference schemes for one nonlocal boundary-value problem

We consider a nonlocal boundary-value problem for the Poisson equation in a rectangular domain. Dirichlet conditions are posed on a pair of adjacent sides of a rectangle, and integral constraints are given instead of standard boundary conditions on the other pair. The corresponding difference scheme is constructed and investigated; an a priori estimate of the solution is obtained with the help of energy inequality method. Discretization error estimate that is compatible with the smoothness of the solution sought is obtained.     

Quantum Hall bilayer digital amplifier

A quantum Hall double layer system at total Landau level filling factor ν=1, seeded by a modulated ac-voltage input signal of frequency inside the forbidden band gap, is demonstrated to work as a digital amplifier of ultra weak signals. The theory employs to the sine-Gordon model and it is confirmed by the first experimental observations of nonlinear bistable transmission in a pendula chain.     

Nonlinear band gap transmission in optical waveguide arrays

The effect of nonlinear transmission in coupled optical waveguide arrays is theoretically investigated and a realistic experimental setup is suggested. The beam is injected in a single boundary waveguide, linear refractive index of which (n(0)) is larger than refractive indexes (n) of other identical waveguides in the array. Particularly, the effect holds if omega(n(0)-n)/c>2Q, where Q is a linear coupling constant between array waveguides, omega is a carrier wave frequency, and c is a light velocity. Numerical experiments show that the energy transfers from the boundary waveguide to the waveguide array above a certain threshold intensity of the injected beam. This effect is due to the creation and the propagation of gap solitons in full analogy with a similar phenomenon in sine-Gordon lattice [Phys. Rev. Lett. 89, 134102 (2002)]].     

Nonadiabatic Landau-Zener tunneling in waveguide arrays with a step in the refractive index

Landau-Zener tunneling is discussed in connection with optical waveguide arrays. Light injected in a specific band of the Bloch spectrum in the propagation constant can be transmitted to another band, changing its physical properties. This can be achieved using two coupled waveguide arrays with different refractive indices. The step in the refractive index causes wave "acceleration" and thus induces strongly nonadiabatic Landau-Zener tunneling. Theoretically, the analysis is performed by considering a Schr?dinger equation in a periodic potential with a step. The region of physical parameters where this phenomenon can occur is analytically determined and a realistic experimental setup is suggested. Its application could allow the realization of light filters.     

The Effect of Number of Arms on the Aggregation Behavior of Thermoresponsive Poly(N-isopropylacrylamide) Star Polymers

The thermoresponsive nature of aqueous solutions of poly(N-isopropylacrylamide) (PNIPAAM) star polymers containing 2, 3, 4, and 6 arms has been investigated by turbidity, dynamic light scattering, rheology, and rheo-SALS. Simulations of the thermosensitive nature of the single star polymers have also been conducted. Some of the samples form aggregates even at temperatures significantly below the lower critical solution temperature (LCST) of PNIPAAM. Increasing concentration and number of arms promotes associations at low temperatures. When the temperature is raised, there is a competition between size increase due to enhanced aggregation and a size reduction caused by contraction. Monte Carlo simulations show that the single stars contract with increasing temperature, and that this contraction is more pronounced when the number of arms is increased. Some samples exhibit a minimum in the turbidity data after the initial increase at the cloud point. The combined rheology and rheo-SALS data suggest that this is due to a fragmentation of the aggregates followed by re-aggregation at even higher temperatures. Although the 6-arm star polymer aggregates more than the other stars at low temperatures, the more compact structure renders it less prone to aggregation at temperatures above the cloud point.     

Chipot’s method for a one-dimensional Kirchhoff static equation

The paper deals with a method of solution of a nonlinear integro-differential static string equation. The accuracy of the method is discussed. The numerical examples are given.     

The anti-FPU problem

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.     

Effects of addition of anionic and cationic surfactants to poly(N-isopropylacrylamide) microgels with and without acrylic acid groups

The interaction of the anionic surfactant sodium dodecyl sulfate (SDS) and the cationic surfactant hexadecyl trimethyl ammonium bromide with poly(N-isopropylacrylamide) (PNIPAAM) microgels with and without poly(acrylic acid) (PAA) was investigated by means of dynamic light scattering (DLS), zeta potential, and turbidimetry measurements. The DLS results show that the PNIPAAM microgels with PAA will contract when an anionic or cationic surfactant is added to the suspension, while the PNIPAAM microgels without PAA expand in the presence of an ionic surfactant. A collapse of the PNIPAAM microgels is observed when the temperature is increased. From the zeta potential measurements, it is observed that the charge density of PNIPAAM microgels in the presence of an ionic surfactant is significantly affected by temperature and the attachment of the negatively charged PAA groups. The turbidity measurements clearly indicate that the interaction between PNIPAAM and SDS is more pronounced than that of the cationic surfactant.     

Nonlinear dynamics in double square-well potentials

The first example of the coexistence of Josephson oscillations with a self-trapping regime is found in the context of the coherent nonlinear dynamics in a double square-well potential. We prove the simultaneous existence of symmetric, antisymmetric, and asymmetric stationary solutions of the associated Gross-Pitaevskii equation, which explains this macroscopic bistability. We illustrate and confirm the effect with numerical simulations. This property allows suggesting experiments with Bose-Einstein condensates in engineered optical lattices or with weakly coupled optical waveguide arrays. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 292–303, August, 2007.     

Coexistence of Josephson oscillations and a novel self-trapping regime in optical waveguide arrays

Considering the coherent nonlinear dynamics between two weakly linked optical waveguide arrays, we find the first example of coexistence of Josephson oscillations with a novel self-trapping regime. This macroscopic bistability is explained by proving analytically the simultaneous existence of symmetric, antisymmetric, and asymmetric stationary solutions of the associated nonlinear Schr?dinger equation. The effect is illustrated and confirmed by numerical simulations. This property allows us to conceive an optical switch based on the variation of the refractive index of the linking central waveguide.