pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual

pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path for elliptic systems of PDEs over bounded 2D domains, based on the Matlab pdetoolbox. The new features include a more efficient use of FEM, easier switching between different single parameter continuations, genuine multi-parameter continuation (e.g., fold continuation), more efficient implementation of nonlinear boundary conditions, cylinder and torus geometries (i.e., periodic boundary conditions), and a general interface for adding auxiliary equations like mass conservation or phase equations for continuation of traveling waves. The package (library, demos, manuals) can be downloaded at www.staff.uni-oldenburg.de/hannes.uecker/pde2pat

Correspondence between theory and practice of a Beerkan infiltration experiment

The Beerkan infiltration experiment is carried out by inserting the ring a short depth into the soil and establishing a positive head of water on the infiltration surface for at least a part of the run. Nevertheless, the data are analyzed by assuming a fully unconfined infiltration process (ring insertion depth, d = 0 cm) and a null ponded depth of water (H = 0 cm). The influence of ring insertion and ponded water on an infiltration process of 2 h sampled every minute was tested in this numerical investigation. Five soils varying from sand to silt loam, three ring radii (5–15 cm), and the Beerkan-specific range of values for both d and H (between 0 and 1 cm) were considered. The differences between the theoretical (d = H = 0 cm) and the practical (d = H = 1 cm) setups varied from −10.4 to +8.6% for the mean infiltration rate and from −10.2 to +8.3% for the final cumulative infiltration. These differences were small, and they decreased in absolute value by considering a soil-dependent ring radius. In particular, nearly negligible differences were detected using a small ring in coarse-textured soils and a large ring in fine-textured soils. In the coarser soils, inserting the ring and establishing a ponded depth of water did not alter the estimated coefficients of the two-parameter infiltration model appreciably with the cumulative linearization method, because these coefficients differed between the theoretical and practical setups by no more than 9.2%. In fine soils, linearization could not be possible regardless of the considered setup, or it was the use of d = H = 1 cm instead of d = H = 0 cm that impeded a convincing linearization of the data. In conclusion, the good correspondence, in many circumstances, between the theoretical and the practical Beerkan infiltration experiment reinforced the interest in this simple experiment as a practical means to collect infiltration data in the field

Surface Gap Soliton Ground States for the Nonlinear Schr\"{o}dinger Equation

We consider the nonlinear Schr\"{o}dinger equation $(-\Delta +V(x))u = \Gamma(x) |u|^{p-1}u$, $x\in \R^n$ with $V(x) = V_1(x) \chi_{\{x_1>0\}}(x)+V_2(x) \chi_{\{x_1<0\}}(x)$ and $\Gamma(x) = \Gamma_1(x) \chi_{\{x_1>0\}}(x)+\Gamma_2(x) \chi_{\{x_1<0\}}(x)$ and with $V_1, V_2, \Gamma_1, \Gamma_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min \sigma(-\Delta +V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1, \Gamma\equiv \Gamma_1$ and $V\equiv V_2, \Gamma\equiv \Gamma_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\tfrac{d^2}{dx^2} +V_1(x)$ and $-\tfrac{d^2}{dx^2} +V_2(x)$.Comment: definition of ground and bound states added, assumption (H2) weakened (sign changing nonlinearity is now allowed); 33 pages, 4 figure

Vortex families near a spectral edge in the Gross-Pitaevskii equation with a two-dimensional periodic potential

We examine numerically vortex families near band edges of the Bloch wave spectrum in the Gross--Pitaevskii equation with a two-dimensional periodic potential and in the discrete nonlinear Schroedinger equation. We show that besides vortex families that terminate at a small distance from the band edges via fold bifurcations there exist vortex families that are continued all way to the band edges.Comment: 12 pages, 8 figure

Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier--Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross--Pitaevskii equation and the coupled-mode equations are obtained for a finite-time interval.Comment: 32 pages, 16 figure

Temperature dependence of binary and ternary recombination of H3+ ions with electron

We study binary and the recently discovered process of ternary He-assisted recombination of H3+ ions with electrons in a low temperature afterglow plasma. The experiments are carried out over a broad range of pressures and temperatures of an afterglow plasma in a helium buffer gas. Binary and He-assisted ternary recombination are observed and the corresponding recombination rate coefficients are extracted for temperatures from 77 K to 330 K. We describe the observed ternary recombination as a two-step mechanism: First, a rotationally-excited long-lived neutral molecule H3* is formed in electron-H3+ collisions. Second, the H3* molecule collides with a helium atom that leads to the formation of a very long-lived Rydberg state with high orbital momentum. We present calculations of the lifetimes of H3* and of the ternary recombination rate coefficients for para and ortho-H3+. The calculations show a large difference between the ternary recombination rate coefficients of ortho- and para-H3+ at temperatures below 300 K. The measured binary and ternary rate coefficients are in reasonable agreement with the calculated values.Comment: 15 page

Investigating biological activity spectrum for novel styrylquinazoline analogues

In this study, series of ring-substituted 2-styrylquinazolin-4(3H)-one and 4-chloro-2-styrylquinazoline derivatives were prepared. The syntheses of the discussed compounds are presented. The compounds were analyzed by RP-HPLC to determine lipophilicity. They were tested for their inhibitory activity on photosynthetic electron transport (PET) in spinach (Spinacia oleracea L.) chloroplasts. Primary in vitro screening of the synthesized compounds was also performed against four mycobacterial strains and against eight fungal strains. Several compounds showed biological activity comparable with or higher than that of the standard isoniazid. It was found that the electronic properties of the R substituent, and not the total lipophilicity of the compound, were decisive for the photosynthesis-inhibiting activity of tested compounds

Continuation for thin film hydrodynamics and related scalar problems

This chapter illustrates how to apply continuation techniques in the analysis of a particular class of nonlinear kinetic equations that describe the time evolution through transport equations for a single scalar field like a densities or interface profiles of various types. We first systematically introduce these equations as gradient dynamics combining mass-conserving and nonmass-conserving fluxes followed by a discussion of nonvariational amendmends and a brief introduction to their analysis by numerical continuation. The approach is first applied to a number of common examples of variational equations, namely, Allen-Cahn- and Cahn-Hilliard-type equations including certain thin-film equations for partially wetting liquids on homogeneous and heterogeneous substrates as well as Swift-Hohenberg and Phase-Field-Crystal equations. Second we consider nonvariational examples as the Kuramoto-Sivashinsky equation, convective Allen-Cahn and Cahn-Hilliard equations and thin-film equations describing stationary sliding drops and a transversal front instability in a dip-coating. Through the different examples we illustrate how to employ the numerical tools provided by the packages auto07p and pde2path to determine steady, stationary and time-periodic solutions in one and two dimensions and the resulting bifurcation diagrams. The incorporation of boundary conditions and integral side conditions is also discussed as well as problem-specific implementation issues

A comparison of the neuroprotective efficacy of newly developed oximes (K117, K127) and currently available oxime (obidoxime) in tabun-poisoned rats

The potency of newly developed bispyridinium compounds (K117, K127) to reduce tabun-induced acute neurotoxic signs and symptoms was compared with currently available oxime (obidoxime) using functional observational battery. The neuroprotective effects of atropine alone and atropine combined with one of three bispyridinium oximes (K117, K127, obidoxime) on rats poisoned with tabun at a sublethal dose (180 μg/kg i.m.; 80% of LD50 value) were studied. Tabun-induced neurotoxicity was monitored using a functional observational battery and automatic measurement of motor activity at 24 h following tabun challenge. The results indicated that all tested oximes combined with atropine enabled tabun-poisoned rats to survive 24 h following tabun challenge while one tabun-poisoned rats died within 24 h after tabun poisoning when the rats were treated with atropine alone. Newly developed oxime K127 combined with atropine was the most effective in decreasing tabun-induced neurotoxicity in the case of sublethal poisonings among all oximes tested. Nevertheless, the differences of neuroprotective efficacy between K127 and obidoxime are not sufficient to replace obidoxime by K127 for the treatment of acute tabun poisonings