Multisymplectic Relative Equilibria, Multiphase Wavetrains, and Coupled NLS Equations

The article begins with a geometric formulation of two-phase wavetrain solutions of coupled nonlinear Schrödinger equations. It is shown that these solutions come in natural four-parameter families, associated with symmetry, and a geometric instability condition can be deduced from the parameter structure that generalizes Roskes' instability criterion. It is then shown that this geometric structure is universal in the sense that it does not depend on the particular equation, only on the structure of the equations. The theory also extends to the case without symmetry, where small divisors may be present, but gives a new formal geometric framework for multiphase wavetrains.     

Classification of Discrete Symmetries of Ordinary Differential Equations

A simple method for determining all discrete point symmetries of a given differential equation has been developed recently. The method uses constant matrices that represent inequivalent automorphisms of the Lie algebra spanned by the Lie point symmetry generators. It may be difficult to obtain these matrices if there are three or more independent generators, because the matrix elements are determined by a large system of algebraic equations. This paper contains a classification of the automorphisms that can occur in the calculation of discrete symmetries of scalar ordinary differential equations, up to equivalence under real point transformations. (The results are also applicable to many partial differential equations.) Where these automorphisms can be realized as point transformations, we list all inequivalent realizations. By using this classification as a look-up table, readers can calculate the discrete point symmetries of a given ordinary differential equation with very little effort.     

Inertial Particle Motion in Recirculating Stokes Flow

Recirculation occurs in many cavity flows. In particular, alveolar flow models have been shown to exhibit recirculation patterns. However, many particles that are inhaled by the lungs do not follow this flow. Instead, they may diffuse into the surrounding flow or possess enough inertia to propel them from fluid particle paths. In this study, we construct a minimal model to observe the behavior of inertial particles caught within a recirculating Stokes flow. We find that, given favorable conditions, inertial particles can be cleared from the cavity or deposited on walls. This depends on the strength of inertia in zero gravity, but can be enhanced when gravity and the orientation of the cavity are taken into account. It is also possible for these effects to balance one another, producing a skewed limit cycle. These combined effects may play a significant part in the retention, deposition, and clearance of aerosols and particulates from alveolar cavities.     

The Stability of Two, Three, and Four Wave Interactions of a Prototype System

The stability of plane wave interactions of coupled nonlinear Schrödinger (CNLS) equations can be analyzed within a bisymplectic framework. This framework is a generalization of the Hamiltonian formulation. The current study considers a family of CNLS equations that are used as a prototype system for studying the combined interaction of unstable and stable component waves in optics. This popular family has a drawback when cast into a bisymplectic framework: the determinant controlling various types of fiber regime is zero. To solve this problem, it is proposed that a limit is taken from a more general CNLS family to the family in question. This method is then bench-marked against known stability results for the simple two plane wave interactions when amplitudes are equal and are found to agree. It is then applied to two wave interactions with unequal amplitudes as well as three and four wave interactions. The latter interactions for this particular system are not spectrally stable. By suggesting a slightly larger family of CNLS equations, it is illustrated that spectral stability can occur. This adapted prototype system may be of use in optics; in particular, to show that long-wave stability is possible given a judicious choice of parameter values.     

Alternating Flow in a Moving Corner

Recent experiments have shown that rapid kinematic mixing occurs in the pulmonary alveoli. Here the Reynolds number is very small, there is recirculation in the alveolar cavity and the alveolar walls move periodically. We have recently shown that non-diffusing particles move chaotically in a two-dimensional model flow with the above features. In parts of the lung, however, there is asynchrony between the wall motion and the ductal flow immediately outside the alveolus. The extent to which this asynchrony affects kinematic mixing in real alveoli is not yet known. The purpose of this paper is to describe the effect of asynchrony on chaotic advection in our two-dimensional model, in order to understand the circumstances in which this becomes significant.