arXiv subjects

Michelle D. Maiden

Publications and source records attributed to Michelle D. Maiden.

5 recordsLinked to original sources

arXiv2020

Evolution of truncated and bent gravity wave solitons: the Mach expansion problem

The dynamics of initially truncated and bent line solitons for the Kadomtsev-Petviashvili (KPII) equation modelling internal and surface gravity waves are analysed using modulation theory. In contrast to previous studies on obliquely interacting solitons that develop from acute incidence angles, this work focuses on initial value problems for the obtuse incidence of two or three partial line solitons, which propagate away from one another. Despite counterpropagation, significant residual soliton interactions are observed with novel physical consequences. The initial value problem for a truncated line soliton-describing the emergence of a quasi-one-dimensional soliton from a wide channel-is shown to be related to the interaction of oblique solitons. Analytical descriptions for the development of weak and strong interactions are obtained in terms of interacting simple wave solutions of modulation equations for the local soliton amplitude and slope. In the weak interaction case, the long-time evolution of truncated and large obtuse angle solitons exhibits a decaying, parabolic wave profile with temporally increasing focal length that asymptotes to a cylindrical Korteweg-de Vries soliton. In contrast, the strong interaction case of slightly obtuse interacting solitons evolves into a steady, one-dimensional line soliton with amplitude reduced by an amount proportional to the incidence slope. This strong interaction is identified with the "Mach expansion" of a soliton with an expansive corner, contrasting with the well-known Mach reflection of a soliton with a compressive corner. Interestingly, the critical angles for Mach expansion and reflection are the same. Numerical simulations of the KPII equation quantitatively support the analytical findings.

nlin.PS↗

arXiv2019

Controlling Dispersive Hydrodynamic Wavebreaking in a Viscous Fluid Conduit

The driven, cylindrical, free interface between two miscible, Stokes fluids with high viscosity contrast have been shown to exhibit dispersive hydrodynamics. A hallmark feature of dispersive hydrodynamic media is the dispersive resolution of wavebreaking that results in a dispersive shock wave. In the context of the viscous fluid conduit system, the present work introduces a simple, practical method to precisely control the location, time, and spatial profile of wavebreaking in dispersive hydrodynamic systems with only boundary control. The method is based on tracking the dispersionless characteristics backward from the desired wavebreaking profile to the boundary. In addition to the generation of approximately step-like Riemann and box problems, the method is generalized to other, approximately piecewise-linear dispersive hydrodynamic profiles including the triangle wave and N-wave. A definition of dispersive hydrodynamic wavebreaking is used to obtain quantitative agreement between the predicted location and time of wavebreaking, viscous fluid conduit experiment, and direct numerical simulations for a range of flow conditions. Observed space-time characteristics also agree with triangle and N-wave predictions. The characteristic boundary control method introduced here enables the experimental investigation of a variety of wavebreaking profiles and is expected to be useful in other dispersive hydrodynamic media. As an application of this approach, soliton fission from a large, box-like disturbance is observed both experimentally and numerically, motivating future analytical treatment.

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arXiv2018

Solitonic dispersive hydrodynamics: theory and observation

Ubiquitous nonlinear waves in dispersive media include localized solitons and extended hydrodynamic states such as dispersive shock waves. Despite their physical prominence and the development of thorough theoretical and experimental investigations of each separately, experiments and a unified theory of solitons and dispersive hydrodynamics are lacking. Here, a general soliton-mean field theory is introduced and used to describe the propagation of solitons in macroscopic hydrodynamic flows. Two universal adiabatic invariants of motion are identified that predict trapping or transmission of solitons by hydrodynamic states. The result of solitons incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves is the same, an effect termed hydrodynamic reciprocity. Experiments on viscous fluid conduits quantitatively confirm the soliton-mean field theory with broader implications for nonlinear optics, superfluids, geophysical fluids, and other dispersive hydrodynamic media.

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arXiv2016

Modulations of viscous fluid conduit periodic waves

In this work, modulation of periodic interfacial waves on a conduit of viscous liquid is explored utilizing Whitham theory and Nonlinear Schrödinger (NLS) theory. Large amplitude periodic wave modulation theory does not require integrability of the underlying model equation, yet in practice, either integrable equations are studied or the full extent of Whitham (wave-averaging) theory is not developed. The governing conduit equation is nonlocal with nonlinear dispersion and is not integrable. Via a scaling symmetry, periodic waves can be characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the NLS equation are derived, depending on the wavenumber. Dark and bright envelope solitons are found to persist in the conduit equation. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted. In the large amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity, and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. These results have implications for dispersive shock waves, recently observed experimentally.

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arXiv2016

Observation of dispersive shock waves, solitons, and their interactions in viscous fluid conduits

Dispersive shock waves and solitons are fundamental nonlinear excitations in dispersive media, but dispersive shock wave studies to date have been severely constrained. Here we report on a novel dispersive hydrodynamics testbed: the effectively frictionless dynamics of interfacial waves between two high contrast, miscible, low Reynolds' number Stokes fluids. This scenario is realized by injecting from below a lighter, viscous fluid into a column filled with high viscosity fluid. The injected fluid forms a deformable pipe whose diameter is proportional to the injection rate, enabling precise control over the generation of symmetric interfacial waves. Buoyancy drives nonlinear interfacial self-steepening while normal stresses give rise to dispersion of interfacial waves. Extremely slow mass diffusion and mass conservation imply that the interfacial waves are effectively dissipationless. This enables high fidelity observations of large amplitude dispersive shock waves in this spatially extended system, found to agree quantitatively with a nonlinear wave averaging theory. Furthermore, several highly coherent phenomena are investigated including dispersive shock wave backflow, the refraction or absorption of solitons by dispersive shock waves, and the multi-phase merging of two dispersive shock waves. The complex, coherent, nonlinear mixing of dispersive shock waves and solitons observed here are universal features of dissipationless, dispersive hydrodynamic flows.

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