General N‐Dark–Dark Solitons in the Coupled Nonlinear Schrödinger Equations

N‐dark–dark solitons in the integrable coupled NLS equations are derived by the KP‐hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright–bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed‐nonlinearity case, two dark–dark solitons can form a stationary bound state.     

Dynamics of Semi-Discretizations of the Defocusing Nonlinear Schrodinger Equation

It has been demonstrated that the nonlinear Schrödinger(NLS) equation is sensitive to discretizations. In the focusingcase this is due to the homoclinic structure associated withthe NLS equation. In this paper we show that various numericalschemes for the defocusing case are also prone to instabilities,although not as severe as those of the focusing equation. Anintegrable discretization due to Ablowitz and Ladik does notsuffer from the same instabilities. However, it is shown thatit develops a focusing singularity if a threshold conditionis exceeded. Numerical examples illustrating the phenomena pertainingto the defocusing equation are given.     

A coupled finite and boundary spectral element method for linear water-wave propagation problems

A coupled boundary spectral element method (BSEM) and spectral element method (SEM) formulation for the propagation of small-amplitude water waves over variable bathymetries is presented in this work. The wave model is based on the mild-slope equation (MSE), which provides a good approximation of the propagation of water waves over irregular bottom surfaces with slopes up to 1:3. In unbounded domains or infinite regions, space can be divided into two different areas: a central region of interest, where an irregular bathymetry is included, and an exterior infinite region with straight and parallel bathymetric lines. The SEM allows us to model the central region, where any variation of the bathymetry can be considered, while the exterior infinite region is modelled by the BSEM which, combined with the fundamental solution presented by Cerrato et al. [A. Cerrato, J. A. González, L. Rodríguez-Tembleque, Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries, Eng. Anal. Boundary Elem. 62 (2016) 22–34.] can include bathymetries with straight and parallel contour lines. This coupled model combines important advantages of both methods; it benefits from the flexibility of the SEM for the interior region and, at the same time, includes the fulfilment of the Sommerfeld’s radiation condition for the exterior problem, that is provided by the BSEM. The solution approximation inside the elements is constructed by high order Legendre polynomials associated with Legendre–Gauss–Lobatto quadrature points, providing a spectral convergence for both methods. The proposed formulation has been validated in three different benchmark cases with different shapes of the bottom surface. The solutions exhibit the typical p-convergence of spectral methods.     

Co-Existence of Chaos and Stable Periodic Orbits in a Simple Discrete Neural Network

We show that a simple discrete network of two identical neurons can demonstrate chaotic behavior near the origin. This is complementary to the results in Wu and Zhang (Disc. Contin. Dynam. Syst. Series B, 4 (2004), 853-865), where it was shown that the same system can have a large capacity of stable periodic orbits in a region away from the origin.     

On the Nonlinear Development of Görtler Vortices in a Compressible Boundary Layer

The nonlinear development of the Görtler instability in compressible boundary layers on curved walls is considered for vortices of asymptotically large wavenumber. The starting point for our calculations lies in the work of Hall and Lakin (Proc. Roy. Soc. London Ser. A 415:421–444), where the incompressible results were formulated. Without neglecting downstream partial derivatives, the initial development of a vortex from the point where it first starts to grow is calculated. It is shown how the same basic structure that occurs in incompressible flow exists, where the disturbance is confined to a core region bounded above and below by thin shear layers, but that the flow in the core region is of more complicated form than that for incompressible flow.     

On the Quasiconcave Bilevel Programming Problem

Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions     

A reaction-diffusion model for autocatalytic polymerization: II. The initial value problem

An initial-boundary value problem arising from a simple modelfor radical chain polymerization is discussed in detail. Generalproperties of the solution are derived first and it is shownthat a moving interface develops. This separates a region wherethe polymer is sufficiently concentrated for it to be immobilefrom one where it is still free to diffuse. An asymptotic analysisis performed in this latter region, where it is shown that apermanent-form travelling wave (treated in Part I) developsin the long time structure and that this wave travels with itsminimum possible speed. Numerical results for the full initial-boundaryvalue problem are presented which confirm the asymptotic theoryand give results in regions not accessible to this analysis.     

Global Well-Posedness and Scattering for the Defocusing ˙H s-Critical NLS

The authors consider the scattering phenomena of the defocusing H^s-critical NLS. It is shown that if a solution of the defocusing NLS remains bounded in the critical homogeneous Sobolev norm on its maximal interval of existence, then the solution is global and scatters.     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

On the final steps of Newton and higher order methods

The Newton method is one of the most used methods for solving nonlinear system of equations when the Jacobian matrix is nonsingular. The method converges to a solution with Q-order two for initial points sufficiently close to the solution. The method of Halley and the method of Chebyshev are among methods that have local and cubic rate of convergence. Combining these methods with a backtracking and curvilinear strategy for unconstrained optimization problems these methods have been shown to be globally convergent. The backtracking forces a strict decrease of the function of the unconstrained optimization problem. It is shown that no damping of the step in the backtracking routine is needed close to a strict local minimizer and the global method behaves as a local method. The local behavior for the unconstrained optimization problem is investigated by considering problems with two unknowns and it is shown that there are no significant differences in the region where the global method turn into a local method for second and third order methods. Further, the final steps to reach a predefined tolerance are investigated. It is shown that the region where the higher order methods terminate in one or two iteration is significantly larger than the corresponding region for Newton’s method.     

Long-time decay of the solutions of the Davey—Stewartson II equations

Summary Using the method of inverse scattering, the sup-norms of the solutions of the Davey—Stewartson II equations are shown to decay in the order of 1/¦t¦ as ¦t¦ goes to infinity. In the focusing case this result is obtained for small initial data, whereas in the defocusing case it is obtained for general initial data.     

Comments on the non-stationary chaos

Non-stationary chaos is a universal phenomenon in non-hyperbolic dynamical systems. Basic problems regarding the non-stationarity are discussed from ergodic-theoretical viewpoints. By use of a simple system, it is shown that “the law of large number” as well as “the law of small number” break down in the non-stationary regime. The non-stationarity in dynamical systems proposes a crucial problem underlying in the transitional region between chance and necessity, where non-observable processes behind reality interplay with observable ones. The incompleteness of statistical ensembles is discussed from the Karamata's theory. Finally, the significance of the stationary/non-stationary interface is emphasized in relation to the universality of 1/f fluctuations.     

Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows

We demonstrate a numerical approach for solving the one-dimensional non-linear weakly dispersive Serre equations. By introducing a new conserved quantity the Serre equations can be written in conservation law form, where the velocity is recovered from the conserved quantities at each time step by solving an auxiliary elliptic equation. Numerical techniques for solving equations in conservative law form can then be applied to solve the Serre equations. We demonstrate how this is achieved. The system of conservation equations are solved using the finite volume method and the associated elliptic equation for the velocity is solved using a finite difference method. This robust approach allows us to accurately solve problems with steep gradients in the flow, such as those generated by discontinuities in the initial conditions.The method is shown to be accurate, simple to implement and stable for a range of problems including flows with steep gradients and variable bathymetry.     

A well-balanced van Leer-type numerical scheme for shallow water equations with variable topography

A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves.     

Residual currents and corridor of flow in the Rio de la Plata

The Rio de la Plata is a large and shallow water body that discharges onto the Atlantic Ocean. The main driving forces for the river flow are the bathymetry, tides, the outflow from the Paraná and Uruguay rivers and the winds. A numerical model covering the entire river was set up with the objective of increasing our understanding of the hydrographical features and morphological dynamics in the Estuary. The simulations revealed a counter-clockwise residual circulation in the Samborombón Bay and an eastward net flow near the Uruguayan coast. The residual flow is forced by both the tides and the bathymetry. The residence time for the entire river ranges from 40 to 80 days. However, residence times above 120 days was found in the Samborombón Bay. Three corridors of flow have been identified.     

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space , then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the defocusing CSP equation from the single‐component extended Kadomtsev‐Petviashvili (KP) hierarchy by the reduction method. As a by‐product, the N‐dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.     

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Rectangular matrix solutions of the defocusing nonlinear Schrödinger equation (dNLS) are studied in quarter-plane and semi-strip. Evolution of the corresponding Weyl–Titchmarsh (Weyl) function is described in terms of the initial Weyl function and boundary conditions. In the next step, the initial Weyl function is recovered (for the quarter-plane case) from the long-time asymptotics of the wave function considered at the boundary. Thus, it is shown that the evolution of the Weyl function is uniquely defined by the boundary conditions. Moreover, a procedure to recover solutions of dNLS (uniquely defined by the boundary conditions) is given. In a somewhat different way, the same boundary value problem is also dealt with in a semi-strip (for the case of a quasi-analytic initial condition).     

Estimates on enstrophy, palinstrophy, and invariant measures for 2-D turbulence

We construct semi-integral curves which bound the projection of the global attractor of the 2-D Navier-Stokes equations in the plane spanned by enstrophy and palinstrophy. Of particular interest are certain regions of the plane where palinstrophy dominates enstrophy. Previous work shows that if solutions on the global attractor spend a significant amount of time in such a region, then there is a cascade of enstrophy to smaller length scales, one of the main features of 2-D turbulence theory. The semi-integral curves divide the plane into regions having limited ranges for the direction of the flow. This allows us to estimate the average time it would take for an intermittent solution to burst into a region of large palinstrophy. We also derive a sharp, universal upper bound on the average palinstrophy and show that it is achieved only for forces that admit statistical steady states where the nonlinear term is zero.     

Efficient Simulation of Markov Chains Using Segmentation

A methodology is proposed that is suitable for efficient simulation of continuous-time Markov chains that are nearly-completely decomposable. For such Markov chains the effort to adequately explore the state space via Crude Monte Carlo (CMC) simulation can be extremely large. The purpose of this paper is to provide a fast alternative to the standard CMC algorithm, which we call Aggregate Monte Carlo (AMC). The idea of the AMC algorithm is to reduce the jumping back and forth of the Markov chain in small subregions of the state space. We accomplish this by aggregating such problem regions into single states. We discuss two methods to identify collections of states where the Markov chain may become ‘trapped’: the stochastic watershed segmentation from image analysis, and a graph-theoretic decomposition method. As a motivating application, we consider the problem of estimating the charge carrier mobility of disordered organic semiconductors, which contain low-energy regions in which the charge carrier can quickly become stuck. It is shown that the AMC estimator for the charge carrier mobility reduces computational costs by several orders of magnitude compared to the CMC estimator.     

Optimal Myopic Allocation of a Product with Fixed Lifetime

Allocation policies of a perishable product from a regional centre to n locations in the region are analysed. Optimal myopic rules are derived for two general classes of policies: rotation policies, where unused product that is not outdated is returned to the centre; and retention policies, where returns to the centre are not possible. Costs are charged for every unit short or outdated. It is shown that the optimal myopic rule minimizes both shortage and outdate costs for one period, it is simple to implement in a realistic environment and is independent of the unit costs. Analytic solutions are given for several demand distributions. Finally, an example of application in blood management is presented.