Any-order propagation of the nonlinear Schroedinger equation

We derive an exact propagation scheme for nonlinear Schroedinger equations. This scheme is entirely analogous to the propagation of linear Schroedinger equations. We accomplish this by defining a special operator whose algebraic properties ensure the correct propagation. As applications, we provide a simple proof of a recent conjecture regarding higher-order integrators for the Gross-Pitaevskii equation, extend it to multi-component equations, and to a new class of integrators.Comment: 10 pages, no figures, submitted to Phys. Rev.

Engaging stakeholders in research to address water-energy-food (WEF) nexus challenges

The water–energy–food (WEF) nexus has become a popular, and potentially powerful, frame through which to analyse interactions and interdependencies between these three systems. Though the case for transdisciplinary research in this space has been made, the extent of stakeholder engagement in research remains limited with stakeholders most commonly incorporated in research as end-users. Yet, stakeholders interact with nexus issues in a variety of ways, consequently there is much that collaboration might offer to develop nexus research and enhance its application. This paper outlines four aspects of nexus research and considers the value and potential challenges for transdisciplinary research in each. We focus on assessing and visualising nexus systems; understanding governance and capacity building; the importance of scale; and the implications of future change. The paper then proceeds to describe a novel mixed-method study that deeply integrates stakeholder knowledge with insights from multiple disciplines. We argue that mixed-method research designs—in this case orientated around a number of cases studies—are best suited to understanding and addressing real-world nexus challenges, with their inevitable complex, non-linear system characteristics. Moreover, integrating multiple forms of knowledge in the manner described in this paper enables research to assess the potential for, and processes of, scaling-up innovations in the nexus space, to contribute insights to policy and decision making

Interaction of Nonlinear Schr\"odinger Solitons with an External Potential

Employing a particularly suitable higher order symplectic integration algorithm, we integrate the 1-$d$ nonlinear Schr\"odinger equation numerically for solitons moving in external potentials. In particular, we study the scattering off an interface separating two regions of constant potential. We find that the soliton can break up into two solitons, eventually accompanied by radiation of non-solitary waves. Reflection coefficients and inelasticities are computed as functions of the height of the potential step and of its steepness.Comment: 14 pages, uuencoded PS-file including 10 figure

High-frequency homogenization for periodic media

This article is available open access through the publisher’s website at the link below. Copyright @ 2010 The Royal Society.An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.NSERC (Canada) and the EPSRC

Momentum and energy preserving integrators for nonholonomic dynamics

In this paper, we propose a geometric integrator for nonholonomic mechanical systems. It can be applied to discrete Lagrangian systems specified through a discrete Lagrangian defined on QxQ, where Q is the configuration manifold, and a (generally nonintegrable) distribution in TQ. In the proposed method, a discretization of the constraints is not required. We show that the method preserves the discrete nonholonomic momentum map, and also that the nonholonomic constraints are preserved in average. We study in particular the case where Q has a Lie group structure and the discrete Lagrangian and/or nonholonomic constraints have various invariance properties, and show that the method is also energy-preserving in some important cases.Comment: 18 pages, 6 figures; v2: example and figures added, minor correction to example 2; v3: added section on nonholonomic Stoermer-Verlet metho

Many-body symbolic dynamics of a classical oscillator chain

We study a certain type of the celebrated Fermi-Pasta-Ulam particle chain, namely the inverted FPU model, where the inter-particle potential has a form of a quartic double well. Numerical evidence is given in support of a simple symbolic description of dynamics (in the regime of sufficiently high potential barrier between the wells) in terms of an (approximate) Markov process. The corresponding transition matrix is formally identical to a ferromagnetic Heisenberg quantum spin-1/2 chain with long range coupling, whose diagonalization yields accurate estimates for a class of time correlation functions of the model.Comment: 22 pages in LaTeX with 14 figures; submitted to Nonlinearity ; corrected page offset proble

Explicit Lie-Poisson integration and the Euler equations

We give a wide class of Lie-Poisson systems for which explicit, Lie-Poisson integrators, preserving all Casimirs, can be constructed. The integrators are extremely simple. Examples are the rigid body, a moment truncation, and a new, fast algorithm for the sine-bracket truncation of the 2D Euler equations.Comment: 7 pages, compile with AMSTEX; 2 figures available from autho

Current in open quantum systems

We show that a dissipative current component is present in the dynamics generated by a Liouville-master equation, in addition to the usual component associated with Hamiltonian evolution. The dissipative component originates from coarse graining in time, implicit in a master equation, and needs to be included to preserve current continuity. We derive an explicit expression for the dissipative current in the context of the Markov approximation. Finally, we illustrate our approach with a simple numerical example, in which a quantum particle is coupled to a harmonic phonon bath and dissipation is described by the Pauli master equation.Comment: To appear in Phys. Rev. Let

Model selection in High-Dimensions: A Quadratic-risk based approach

In this article we propose a general class of risk measures which can be used for data based evaluation of parametric models. The loss function is defined as generalized quadratic distance between the true density and the proposed model. These distances are characterized by a simple quadratic form structure that is adaptable through the choice of a nonnegative definite kernel and a bandwidth parameter. Using asymptotic results for the quadratic distances we build a quick-to-compute approximation for the risk function. Its derivation is analogous to the Akaike Information Criterion (AIC), but unlike AIC, the quadratic risk is a global comparison tool. The method does not require resampling, a great advantage when point estimators are expensive to compute. The method is illustrated using the problem of selecting the number of components in a mixture model, where it is shown that, by using an appropriate kernel, the method is computationally straightforward in arbitrarily high data dimensions. In this same context it is shown that the method has some clear advantages over AIC and BIC.Comment: Updated with reviewer suggestion