The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian G 2 (C 5) in choosing the numerical integrator, and the role of the Hodge star operator for relating 2 (C 5) and 3 (C 5) and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity.

Hamiltonian evolution equations which are equivariant with respect to the action of a Lie group are models for physical phenomena such as oceanographic flows, optical fibres and atmospheric flows, and such systems often have a wide variety of solitary-wave or front solutions. In this paper, we present a new symplectic framework for analysing the spectral problem associated with the linearization about such solitary waves and fronts. At the heart of the analysis is a multi-symplectic formulation of Hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions, with a third symplectic structure -with two-form denoted by -associated with a coordinate frame moving at the speed of the wave. This leads to a geometric decomposition and symplectification of the Evans function formulation for the linearization about solitary waves and fronts.We introduce the concept of the symplectic Evans matrix, a matrix consisting of restricted -symplectic forms. By applying Hodge duality to the exterior algebra formulation of the Evans function, we find that the zeros of the Evans function correspond to zeros of the determinant of the symplectic Evans matrix. Based on this formulation, we prove several new properties of the Evans function. Restricting the spectral parameter λ to the real axis, we obtain rigorous results on the derivatives of the Evans function near the origin, based solely on the abstract geometry of the equations, and results for the large |λ| behaviour which use primarily the symplectic structure, but also extend to the non-symplectic case.The Lie group symmetry affects the Evans function by generating zero eigenvalues of large multiplicity in the so-called systems at infinity. We present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation.By combining all these results, a new rigorous sufficient condition for instability of solitary waves and fronts is obtained. 2 Thomas J. Bridges & Gianne DerksThe theory applies to a large class of solitary waves and fronts including waves which are bi-asymptotic to a nonconstant manifold of states as |x| tends to infinity. To illustrate the theory, it is applied to three examples: a Boussinesq model from oceanography, a class of nonlinear Schrödinger equations from optics and a nonlinear Klein-Gordon equation from atmospheric dynamics.

Sleep changes across the lifespan, with a delay in sleep timing and a reduction in slow wave sleep seen in adolescence, followed by further reductions in slow wave sleep but a gradual drift to earlier timing during healthy ageing. The mechanisms underlying changes in sleep timing are unclear: are they primarily related to changes in circadian processes, or to a reduction in the neural activity dependent build up of homeostatic sleep pressure during wake, or both? We review existing studies of age-related changes to sleep and explore how mathematical models can explain observed changes. Model simulations show that typical changes in sleep timing and duration, from adolesence to old age, can be understood in two ways: either as a consequence of a simultaneous reduction in the amplitude of the circadian wake-propensity rhythm and the neural activity dependent build-up of homeostatic sleep pressure during wake; or as a consequence of reduced homeostatic sleep pressure alone. A reduction in the homeostatic pressure also explains greater vulnerability of sleep to disruption and reduced daytime sleep-propensity in healthy ageing. This review highlights the important role of sleep homeostasis in sleep timing. It shows that the same phenotypic response may have multiple underlying causes, and identifies aspects of sleep to target to correct delayed sleep in adolescents and advanced sleep in later life.

Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.

The linear stability of solitary-wave or front solutions of Hamiltonian evolutionary equations, which are equivariant with respect to a Lie group, is studied. The organizing centre for the analysis is a multisymplectic formulation of Hamiltonian partial differential equations (PDEs) where a distinct symplectic operator is assigned for time and space. This separation of symplectic structures leads to new characterizations of the following components of the analysis. The states at infinity are characterized as manifolds of relative equilibria associated with the spatial symplectic structure. The momentum of the connecting orbit, or shape of the solitary wave, considered as a heteroclinic orbit in a phase-space representation, is given a new characterization as a one-form on the tangent space to the heteroclinic manifold and this one-form is a restriction of the temporal symplectic structure. For the linear stability analysis, a new symplectic characterization of the Evans function and its derivatives are obtained, leading to an abstract geometric proof of instability for a large class of solitary-wave states of equivariant Hamiltonian evolutionary PDEs. The theory sheds new light on several well-known models: the gKdV equation, a Boussinesq system and a nonlinear wave equation. The generalization to solitary waves associated with multidimensional heteroclinic manifolds and the implications for solitary waves or fronts which are biasymptotic to invariant manifolds such as periodic states are also discussed.

Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction-drift-diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman-Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

Streamer ionization fronts are pulled fronts propagating into a linearly unstable state; the spatial decay of the initial condition of a planar front selects dynamically one specific long time attractor out of a continuous family. A stability analysis for perturbations in the transverse direction has to take these features into account. In this paper we show how to apply the Evans function in a weighted space for this stability analysis. Zeros of the Evans function indicate the intersection of the stable and unstable manifolds; they are used to determine the eigenvalues. Within this Evans function framework, a numerical dynamical systems method for the calculation of the dispersion relation as an eigenvalue problem is defined and dispersion curves for different values of the electron diffusion constant and of the electric field ahead of the front are derived. Numerical solutions of the initial value problem confirm the eigenvalue calculations. The numerical work is complemented with an analysis of the Evans function leading to analytical expressions for the dispersion relation in the limit of small and large wave numbers. The paper concludes with a fit formula for intermediate wave numbers. This empirical fit supports the conjecture that the smallest unstable wave length of the Laplacian instability is proportional to the diffusion length that characterizes the leading edge of the pulled ionization front.

The linear stability problem for solitary wave states of the Kawahara-or fifthorder KdV-type-equation and its generalizations is considered. A new formulation of the stability problem in terms of the symplectic Evans matrix is presented. The formulation is based on a multisymplectification of the Kawahara equation, and leads to a new characterization of the basic solitary wave, including changes in the state at infinity represented by embedding the solitary wave in a multiparameter family. The theory is used to give a rigorous geometric sufficient condition for instability. The theory is abstract and applies to a wide range of solitary wave states. For example, the theory is applied to the families of solitary waves found by Kichenassamy-Olver and Levandosky.