The use of Borel-summation in the establishment of non-existence of certain travelling-wave solutions of the Kuramoto-Sivashinsky equation

In Grimshaw and Hooper (Physica D 50, 231–238 (1991)) an asymptotic analysis was developed to establish non-existence of a certain class of travelling-wave solutions of the Kuramoto-Sivashinsky equation due to the presence of exponentially small but non-zero terms. Here we show that Borel-summation leads to the main result in a simpler and more robust manner.     

Vibrational characteristics of Balinese gamelan metallophones

A study of the eight metallophone pairs from a Balinese gamelan semara dana has been conducted. Acoustical recordings of metallophone bars being struck were used to examine ratios of overtone frequencies to the fundamental. Results showed large variability in the number and ratios of overtones present. Scanning laser Doppler vibrometry measurements made on several bars also revealed great variability in mode shapes present. The distribution of prominent overtones and their modal shapes do not appear to match those of Western metallophones. Notably, the overall gamelan metallophone characteristics are quite dissimilar to the glockenspiel, which disagrees with previous studies.     

Steady gap solitons in a coupled Korteweg-de Vries system: A dynamical systems approach

In this article we search for steady gap soliton solutions of a set of coupled Korteweg-de Vries equations. These solutions arise whenever there is a narrow gap in the linear spectrum. For small amplitudes we use a dynamical systems approach combined with a normal form analysis to find a canonical equation set. When truncated at the cubic order in amplitude we can exhibit explicit solutions which agree with those found earlier by an asymptotic analysis. We argue that two of these solutions persist under perturbation by the higher-order terms.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

Markov chain models for delinquency: Transition matrix estimation and forecasting

A Markov chain is a natural probability model for accounts receivable. For example, accounts that are ‘current’ this month have a probability of moving next month into ‘current’, ‘delinquent’ or ‘paid‐off’ states. If the transition matrix of the Markov chain were known, forecasts could be formed for future months for each state. This paper applies a Markov chain model to subprime loans that appear neither homogeneous nor stationary. Innovative estimation methods for the transition matrix are proposed. Bayes and empirical Bayes estimators are derived where the population is divided into segments or subpopulations whose transition matrices differ in some, but not all entries. Loan‐level models for key transition matrix entries can be constructed where loan‐level covariates capture the non‐stationarity of the transition matrix. Prediction is illustrated on a $7 billion portfolio of subprime fixed first mortgages and the forecasts show good agreement with actual balances in the delinquency states. Copyright © 2010 John Wiley & Sons, Ltd.     

Transient Linear Growth and Nonlinear Effects

We study a simple model consisting of two coupled nonlinear ordinary differential equations to display the interplay between transient linear growth and nonlinear effects. Specifically, we seek to examine any possible links between the initial conditions that yield maximum transient linear growth and those initial conditions that first lead to attraction to a stable critical point other than the origin. Our results show that any such link is only apparent when the nonlinearity is small.     

Approximate Analytical and Numerical Solutions of the Stationary Ostrovsky Equation

Approximate stationary solutions of the Ostrovsky equation describing long weakly nonlinear waves in a rotating liquid are constructed. These solutions may be regarded as a periodic sequence of arcs of parabolas containing Korteweg-de Vries solitons at the junctures. Results of numerical computations of the dynamics of the approximate solutions obtained from the nonstationary Ostrovsky equation are presented. It is found that, in the presence of negative dispersion, the shape of a stationary wave is well predicted by the approximate theory, whereas the calculated wave velocity differs slightly from the theoretical value. The stationary solutions in media with positive dispersion are evidently unstable (at least for sufficiently strong rotation), and numerical computations demonstrate a complicated picture of nonstationary destruction.