Christopher J. Howls

We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.

We study a discrete variant of the Airy equation, formulated as an advance-delay equation, to reveal that discretization induces the higher-order Stokes phenomenon, which is not present in the continuous Airy function and is typically only encountered in solutions to third-order or higher linear homogeneous, or nonlinear, differential equations. Using steepest descent and direct series methods, we derive asymptotic solutions and the Stokes structure. Our analysis shows that discretization produces a more intricate Stokes structure, containing higher-order Stokes phenomena and infinite accumulations of Stokes and anti-Stokes curves. The latter feature is a strictly nonlinear effect in continuous differential equations. We show that this unusual behavior can be generated in a discrete equation from a linear discretization. Numerical simulations confirm the predictions, and a direct comparison with the continuous Airy equation explains how the discretization alters the Stokes structure.

Waves propagating near an event horizon display interesting features including logarithmic phase singularities and caustics. We consider an acoustic horizon in a flowing Bose-Einstein condensate where the elementary excitations obey the Bogoliubov dispersion relation. In the hamiltonian ray theory the solutions undergo a broken pitchfork bifurcation near the horizon and one might therefore expect the associated wave structure to be given by a Pearcey function, this being the universal wave function that dresses catastrophes with two control parameters. However, the wave function is in fact an Airy-type function supplemented by a logarithmic phase term, a novel type of wave catastrophe. Similar wave functions arise in aeroacoustic flows from jet engines and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from previous authors in that we analyze the behaviour of the integral representation of the wave function using exponential coordinates. This allows for a different treatment of the branches that gives rise to an analysis based purely on saddlepoint expansions, which resolve the multiple real and complex waves that interact at the horizon and its companion caustic. We find that the horizon is a physical manifestation of a Stokes surface, marking the place where a wave is born, and that the horizon and the caustic do not in general coincide: the finite spatial region between them delineates a broadened horizon.

Waves propagating near an event horizon display interesting features including logarithmic phase singularities and caustics. We consider an acoustic horizon in a flowing Bose–Einstein condensate where the elementary excitations obey the Bogoliubov dispersion relation. In the Hamiltonian ray theory the solutions undergo a broken pitchfork bifurcation near the horizon and one might therefore expect the associated wave structure to be given by a Pearcey function, this being the universal wave function that dresses catastrophes with two control parameters. However, the wave function is in fact an Airy-type function supplemented by a logarithmic phase term, a novel type of wave catastrophe. Similar wave functions arise in aeroacoustic flows from jet engines, path integrals in radio astronomy, and also gravitational horizons if dispersion which violates Lorentz symmetry in the UV is included. The approach we take differs from most previous authors in that we analyze the behavior of the integral representation of the wave function using exponential coordinates. This allows for a different treatment of the branch cuts and gives rise to an analysis based purely on saddlepoint expansions. We are thereby able to resolve the multiple real and complex waves that interact at the horizon and its companion caustic. We find that the horizon is a physical manifestation of a Stokes surface, marking the place where a wave is born, and that the horizon and the caustic do not in general coincide: the finite spatial region between them delineates a broadened horizon.

Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to ‘resum’ series to obtain more efficient approximations, and have been successfully widely applied in the study of continuous linear and nonlinear, single and multidimensional problems. In particular, a method known as transasymptotic resummation can be used to describe continuous behaviour occurring on multiple scales without the need for asymptotic matching. Here we apply transasymptotic resummation to discrete systems and show that it may be used to naturally and efficiently describe discrete delayed bifurcations, or ‘canards’, in singularly-perturbed variants of the logistic map which contain delayed period-doubling bifurcations. We use transasymptotic resummation to approximate the solutions, and describe the behaviour of the solution across the bifurcations. This approach has two significant advantages: it may be applied in systematic fashion even across multiple bifurcations, and the exponential multipliers encode information about the bifurcations that are used to explain effects seen in the solution behaviour.

In this paper we develop a novel ray solver for the time-harmonic linearized Euler equations used to predict high-frequency flow–acoustic interaction effects from point sources in subsonic mean jet flows. The solver incorporates solutions to three generic ray problems found in free-space flows: the multiplicity of rays at a receiver point, propagation of complex rays and unphysical divergences at caustics. We show that these respective problems can be overcome by an appropriate boundary value reformulation of the nonlinear ray equations, a bifurcation-theory-inspired complex continuation, and an appeal to the uniform functions of catastrophe theory. The effectiveness of the solver is demonstrated for sources embedded in isothermal parallel and spreading jets, with the fields generated containing a wide variety of caustic structures. Solutions are presented across a large range of receiver angles in the far field, both downstream, where evanescent complex rays generate the cone of silence, and upstream, where multiple real rays are organized about a newly observed cusp caustic. The stability of the caustics is verified for both jets by their persistence under parametric changes of the flow and source. We show the continuation of these caustics as surfaces into the near field is complicated due to a dense caustic network, featuring a chain of locally hyperbolic umbilic caustics, generated by the tangency of rays as they are channelled upstream within the jet.

The downstream propagation of high-frequency acoustic waves from a point source in a subsonic jet obeying Lilley's equation is well known to be organized around the so-called ‘cone of silence’, a fold catastrophe across which the amplitude may be modelled uniformly using Airy functions. Here we show that acoustic waves not only unexpectedly propagate upstream, but also are organized at constant distance from the point source around a cusp catastrophe with amplitude modelled locally by the Pearcey function. Furthermore, the cone of silence is revealed to be a cross-section of a swallowtail catastrophe. One consequence of these discoveries is that the peak acoustic field upstream is not only structurally stable but also at a similar level to the known downstream field. The fine structure of the upstream cusp is blurred out by distributions of symmetric acoustic sources, but peak upstream acoustic beaming persists when asymmetries are introduced, from either arrays of discrete point sources or perturbed continuum ring source distributions. These results may pose interesting questions for future novel jet-aircraft engine designs where asymmetric source distributions arise.