Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.     

The quality coefficient as performance assessment parameter of straight line calibration curves in relationship with the number of calibration points

Knowledge of the response function (y?=?f(x)) is essential in the validation of quantitative analysis methods as it describes the mathematical relationship between measurable responses and the concentrations or quantities of the analyte in the sample within a suitable range. The most common response function used is a straight line obtained by ordinary least squares (OLS) regression. Suitability of calibration lines obtained by OLS regression might be verified by calculation of a quality coefficient (QC_mean). Mathematical modelling performed previously showed that with respect to critical limit values for g, which controls the symmetry of the prediction interval of the abscissa value obtained from the confidence intervals around the OLS calibration curve, a corresponding quality coefficient value exists as a quality performance parameter which is related to the spread of the abscissa values around their mean. In this paper, new mathematical models are developed to demonstrate to which extend also the number n of calibration points (x _i ,y _i ) defines the required value for the quality coefficient (QC_mean) for different values of g. From these models, it could be established that the attribution of a critical limit value to QC_mean as a performance parameter for straight line calibration cannot be arbitrary chosen but has to rely on the mathematical model relating QC_mean, the g-value, the number n of calibration points and the spread of the x _i -values around their mean. Practical measures for analysts are provided which tend to lower the g-value of straight calibration lines beneath critical values and enable to improve the quality of the calibration line applied for analysis, as demonstrated in an elaborated example.     

Measurement of the neutron radius of 208Pb through parity violation in electron scattering

We report the first measurement of the parity-violating asymmetry A(PV) in the elastic scattering of polarized electrons from 208Pb. A(PV) is sensitive to the radius of the neutron distribution (R(n)). The result A(PV)=0.656±0.060(stat)±0.014(syst) ppm corresponds to a difference between the radii of the neutron and proton distributions R(n)-R(p)=0.33(-0.18)(+0.16) fm and provides the first electroweak observation of the neutron skin which is expected in a heavy, neutron-rich nucleus.     

Influence of ion properties on the equilibrium and transport properties of electrolyte solutions

Strong electrolytes are described in the framework of the primitive model in which the solvent is regarded as a dielectric continuum, using the mean spherical approximation. The analytical solution of the equilibrium and transport properties is dependent on the ions' diameters and valencies. For hydrated or nonspherical ions, an effective diameter must be fitted. A sensitivity study of the osmotic coefficient and the transport coefficients is performed on theoretical 1-1, 2-1, and 3-1 electrolytes, up to a total ion concentration of 2 mol/L.     

Classification tree models for the prediction of blood-brain barrier passage of drugs

The use of classification trees for modeling and predicting the passage of molecules through the blood-brain barrier was evaluated. The models were built and evaluated using a data set of 147 molecules extracted from the literature. In the first step, single classification trees were built and evaluated for their predictive abilities. In the second step, attempts were made to improve the predictive abilities using a set of 150 classification trees in a boosting approach. Two boosting algorithms, discrete and real adaptive boosting, were used and compared. High-predictive classification trees were obtained for the data set used, and the models could be improved with boosting. In the context of this research, discrete adaptive boosting gives slightly better results than real adaptive boosting.     

The orbital stability of the cnoidal waves of the Korteweg-de Vries equation

The cnoidal wave solution of the integrable Korteweg-de Vries equation is the most basic of its periodic solutions. Following earlier work where the linear stability of these solutions was established, we prove in this Letter that cnoidal waves are (nonlinearly) orbitally stable with respect to so-called subharmonic perturbations: perturbations that are periodic with period any integer multiple of the cnoidal-wave period. Our method of proof combines the construction of an appropriate Lyapunov function with the seminal results of Grillakis, Shatah and Strauss (1987, 1990) [17] and [18]. The integrability of the Korteweg-de Vries equation is used in that we need the presence of at least one extra conserved quantity in addition to those expected from the Lie point symmetries of the equation.     

Interface Problems for Dispersive Equations

The interface problem for the linear Schrödinger equations in one‐dimensional piecewise homogeneous domains is examined by providing an explicit solution in each domain. The location of the interfaces is known and the continuity of the wave function and a jump in their derivative at the interface are the only conditions imposed. The problem of two semi‐infinite domains and that of two finite‐sized domains are examined in detail. The problem and the method considered here extend that of an earlier paper by Deconinck et al. (2014) [1]. The dispersive nature of the problem presents additional difficulties that are addressed here.     

Dynamics and Stability of Bose-Einstein Condensates: The Nonlinear Schrödinger Equation with Periodic Potential

Summary. The cubic nonlinear Schr?dinger equation with a lattice potential is used to model a periodic dilute-gas Bose-Einstein condensate. Both two- and three-dimensional condensates are considered, for atomic species with either repulsive or attractive interactions. A family of exact solutions and corresponding potential is presented in terms of elliptic functions. The dynamical stability of these exact solutions is examined using both analytical and numerical methods. For condensates with repulsive atomic interactions, all stable, trivial-phase solutions are off-set from the zero level. For condensates with attractive atomic interactions, no stable solutions are found, in contrast to the one-dimensional case [8].     

High-order upwind residual distribution schemes on isoparametric curved elements

Residual distribution schemes on curved geometries are discussed in the context of higher order spatial discretization for hyperbolic conservation laws. The discrete solution, defined by a Finite Element space based on triangular Lagrangian P_k elements, is globally continuous. A natural sub-triangulation of these elements allows to reuse the simple distribution schemes previously developed for linear P₁ triangles. The paper introduces curved elements with piecewise quadratic and cubic approximation of the boundaries of the domain, using standard sub- or isoparametric transformation. Numerical results for the Euler equations confirm the predicted order of accuracy, showing the importance of a higher order approximation of the geometry.