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Tony Lyons

Publications and source records attributed to Tony Lyons.

16 recordsLinked to original sources
arXiv2023

Relational space-time and de Broglie waves

Relative motion of particles is examined in the context of relational space-time. It is shown that de Broglie waves may be derived as a representation of the coordinate maps between the rest-frames of these particles. Energy and momentum are not absolute characteristics of these particles, they are understood as parameters of the coordinate maps between their rest-frames. It is also demonstrated the position of a particle is not an absolute, it is contingent on the frame of reference used to observe the particle.

physics.gen-ph↗
arXiv2022

On the intrinsic wave-particle duality of coordinate maps

The relative motion of material, point-like observers is analysed in terms of coordinate maps between the respective rest-frame of each observer. Under the assumption these maps are $C^{2}$-regular, conservation laws are deduced, which in turn are found to yield wave equations familiar as scalar and spinor fields. Wave-particle dualism is then understood as a property of these coordinate maps as opposed to an inherent property of the point-like observers themselves. Additionally it is shown that the relative acceleration of these observers can only be realised in this context by the introduction of additional potential fields. It is shown that if each observer is to be associated with an inertial frame of reference, then the potential field is necessarily a massless gauge field.

physics.gen-ph↗
arXiv2022

Particle paths in equatorial flows

We investigate particle trajectories in equatorial flows with geophysical corrections caused by the earth's rotation. Particle trajectories in the flows are constructed using pairs of analytic functions defined over the labelling space used in the Lagrangian formalism. Several classes of flow are investigated, and the physical regime in which each is valid is determined using the pressure distribution function of the flow, while the vorticity distribution of each flow is also calculated and found to be effected by earth's rotation.

physics.flu-dyn↗
arXiv2022

Variable eddy viscosities in the atmospheric boundary layer from ageostrophic wind-speed profiles

We generate explicit height-dependent eddy viscosity coefficients in the Ekman layer from convex wind speed profiles. The solutions we obtain are parameterized in terms of the relative deflection angle between the wind directions at the top and bottom of the flow, as well as the geostrophic wind speed and a velocity scale we interpret as the transfer rate of horizontal momentum in the vertical direction. The solutions may be used to infer the thickness of the Ekman layer for a variety of deflection angles different from deflection angle of the classic Ekman spiral.

physics.flu-dyn↗
arXiv2022

Geophysical internal equatorial waves of extreme form

The existence of internal geophysical waves of extreme form is confirmed and an explicit solution presented. The flow is confined to a layer lying above an eastward current while the mean horizontal flow of the solutions is westward, thus incorporating flow reversal in the fluid.

physics.flu-dyn↗
arXiv2019

Camassa-Holm cuspons, solitons and their interactions via the dressing method

A dressing method is applied to a matrix Lax pair for the Camassa-Holm equation, thereby allowing for the construction of several global solutions of the system. In particular solutions of system of soliton and cuspon type are constructed explicitly. The interactions between soliton and cuspon solutions of the system are investigated. The geometric aspects of the Camassa-Holm equation ar re-examined in terms of quantities which can be explicitly constructed via the inverse scattering method.

nlin.SI↗
arXiv2017

The dynamic pressure in deep-water extreme Stokes waves

In this paper we consider the dynamic pressure in a deep-water extreme Stokes wave. While the presence of stagnation points introduces a number of mathematical complications, maximum principles are applied to analyse the dynamic pressure in the fluid body by means of an excision process. It is shown that the dynamic pressure attains its maximum value beneath the wave crest and its minimum beneath the wave trough, while it decreases in moving away from the crest line along any streamline.

physics.flu-dyn↗
arXiv2017

Solitons of shallow-water models from energy-dependent spectral problems

The current work investigates the soliton solutions of the Kaup-Boussinesq equation using the Inverse Scattering Transform method. We outline the construction of the Riemann-Hilbert problem for a pair energy-dependent spectral problems for the system, which we then use to construct the solution of this hydrodynamic system.

math-ph↗
arXiv2016

The pressure distribution in extreme Stokes waves

In this paper we prove that the pressure beneath an extreme Stokes wave over finite depth is strictly increasing with depth. Additionally it is shown that the pressure decreases in moving between a crest-line and trough-line, while it is stationary with respect to the horizontal coordinate along these lines themselves.

math.AP↗
arXiv2015

The pressure in a deep-water Stokes wave of greatest height

In this paper we investigate the qualitative behaviour of the pressure function beneath an extreme Stokes wave over infinite depth. The presence of a stagnation point at the wave-crest of an extreme Stokes wave introduces a number of mathematical difficulties resulting in the irregularity of the free surface profile. It will be proven that the pressure decreases in the horizontal direction between a crest-line and subsequent trough-line, except along these lines themselves where the pressure is stationary with respect to the horizontal coordinate. In addition we will prove that the pressure strictly increases with depth throughout the fluid body.

math.AP↗
arXiv2014

Two-component equations modelling water waves with constant vorticity

In this paper we derive a two-component system of nonlinear equations which model two-dimensional shallow water waves with constant vorticity. Then we prove well-posedness of this equation using a geometrical framework which allows us to recast this equation as a geodesic flow on an infinite dimensional manifold. Finally, we provide a criteria for global existence.

math-ph↗
arXiv2012

Integrable models for shallow water with energy dependent spectral problems

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependance on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup-Bousinesq equation. We formulate the inverse problem as a Riemann-Hilbert problem with a Z2 reduction group. The soliton solutions are explicitly obtained.

nlin.SI↗
arXiv2012

Dark solitons of the Qiao's hierarchy

We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Z. Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called 'dark solitons'.

nlin.SI↗