The hierarchy of rogue wave solutions in nonlinear systems

Oceanic freak waves, optical spikes and extreme events in numerous contexts can arguably be modelled by modulationally unstable solutions within nonlinear systems. In particular, the fundamental nonlinear Schroedinger equation (NLSE) hosts a high-amplitude spatiotemporally localised solution on a plane-wave background, called the Peregrine breather, which is generally considered to be the base-case prototype of a rogue wave. Nonetheless, until very recently, little was known about what to expect when observing or engineering entire clusters of extreme events. Accordingly, this thesis aims to elucidate this matter by investigating complicated structures formed from collections of Peregrine breathers. Many novel NLSE solutions are discovered, all systematically classifiable by their geometry. The methodology employed here is based on the well-established concept of Darboux transformations, by which individual component solutions of an integrable system are nonlinearly superimposed to form a compound wavefunction. It is primarily implemented in a numerical manner within this study, operating on periodically modulating NLSE solutions called breathers. Rogue wave structures can only be extracted at the end of this process, when a limit of zero modulation frequency is applied to all components. Consequently, a requirement for breather asymmetry ensures that a multi-rogue wavefunction must be formed from a triangular number of individual Peregrine breathers (e.g. 1, 3, 6, 10, ...), whether fused or separated. Furthermore, the arrangements of these are restricted by a maximum phase-shift allowable along an evolution trajectory through the relevant wave field. Ultimately, all fundamental high-order rogue wave solutions can be constructed via polynomial relations between origin-translating component shifts and squared modulation frequency ratios. They are simultaneously categorisable by both these mathematical existence conditions and the corresponding visual symmetries, appearing spatiotemporally as triangular cascades, pentagrams, heptagrams, and so on. These parametric relations do not conflict with each other, meaning that any arbitrary NLSE rogue wave solution can be considered a hybridisation of this elementary set. Moreover, this hierarchy of structures is significantly general, with complicated arrangements persisting even on a cnoidal background

The Technological Emergence of AutoML: A Survey of Performant Software and Applications in the Context of Industry

With most technical fields, there exists a delay between fundamental academic research and practical industrial uptake. Whilst some sciences have robust and well-established processes for commercialisation, such as the pharmaceutical practice of regimented drug trials, other fields face transitory periods in which fundamental academic advancements diffuse gradually into the space of commerce and industry. For the still relatively young field of Automated/Autonomous Machine Learning (AutoML/AutonoML), that transitory period is under way, spurred on by a burgeoning interest from broader society. Yet, to date, little research has been undertaken to assess the current state of this dissemination and its uptake. Thus, this review makes two primary contributions to knowledge around this topic. Firstly, it provides the most up-to-date and comprehensive survey of existing AutoML tools, both open-source and commercial. Secondly, it motivates and outlines a framework for assessing whether an AutoML solution designed for real-world application is 'performant'; this framework extends beyond the limitations of typical academic criteria, considering a variety of stakeholder needs and the human-computer interactions required to service them. Thus, additionally supported by an extensive assessment and comparison of academic and commercial case-studies, this review evaluates mainstream engagement with AutoML in the early 2020s, identifying obstacles and opportunities for accelerating future uptake

Infinite hierarchy of nonlinear Schrödinger equations and their solutions

We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that 'even' order equations in the set affect phase and 'stretching factors' in the solutions, while 'odd' order equations affect the velocities. Hence 'odd' order equation solutions can be real functions, while 'even' order equation solutions are always complex

Computational Infrared Spectroscopy of 958 Phosphorus-Bearing Molecules

Phosphine is now well-established as a biosignature, which has risen to prominence with its recent tentative detection on Venus. To follow up this discovery and related future exoplanet biosignature detections, it is important to spectroscopically detect the presence of phosphorus-bearing atmospheric molecules that could be involved in the chemical networks producing, destroying or reacting with phosphine. We start by enumerating phosphorus-bearing molecules (P-molecules) that could potentially be detected spectroscopically in planetary atmospheres and collecting all available spectral data. Gaseous P-molecules are rare, with speciation information scarce. Very few molecules have high accuracy spectral data from experiment or theory; instead, the best current spectral data was obtained using a high-throughput computational algorithm, RASCALL, relying on functional group theory to efficiently produce approximate spectral data for arbitrary molecules based on their component functional groups. Here, we present a high-throughput approach utilizing established computational quantum chemistry methods (CQC) to produce a database of approximate infrared spectra for 958 P-molecules. These data are of interest for astronomy and astrochemistry (importantly identifying potential ambiguities in molecular assignments), improving RASCALL's underlying data, big data spectral analysis and future machine learning applications. However, this data will probably not be sufficiently accurate for secure experimental detections of specific molecules within complex gaseous mixtures in laboratory or astronomy settings. We chose the strongly performing harmonic ωB97X-D/def2-SVPD model chemistry for all molecules and test the more sophisticated and time-consuming GVPT2 anharmonic model chemistry for 250 smaller molecules. Limitations to our automated approach, particularly for the less robust GVPT2 method, are considered along with pathways to future improvements. Our CQC calculations significantly improve on existing RASCALL data by providing quantitative intensities, new data in the fingerprint region (crucial for molecular identification) and higher frequency regions (overtones, combination bands), and improved data for fundamental transitions based on the specific chemical environment. As the spectroscopy of most P-molecules have never been studied outside RASCALL and this approach, the new data in this paper is the most accurate spectral data available for most P-molecules and represent a significant advance in the understanding of the spectroscopic behavior of these molecules.</jats:p

The Challenge of Quickly Determining the Quality of a Single-Photon Source

Novel methods for rapidly estimating single-photon source (SPS) quality, e.g. of quantum dots, have been promoted in recent literature to address the expensive and time-consuming nature of experimental validation via intensity interferometry. However, the frequent lack of uncertainty discussions and reproducible details raises concerns about their reliability. This study investigates one such proposal on eight datasets obtained from an InGaAs/GaAs epitaxial quantum dot that emits at 1.3 {\mu}m and is excited by an 80 MHz laser. The study introduces a novel contribution by employing data augmentation, a machine learning technique, to supplement experimental data with bootstrapped samples. Analysis of the SPS quality metric, i.e. the probability of multi-photon emission events, as derived from efficient histogram fitting of the synthetic samples, reveals significant uncertainty contributed by stochastic variability in the Poisson processes that describe detection rates. Ignoring this source of error risks severe overconfidence in both early quality estimates and claims for state-of-the-art SPS devices. Additionally, this study finds that standard least-squares fitting is comparable to the studied counter-proposal, expanding averages show some promise for early estimation, and reducing background counts improves fitting accuracy but does not address the Poisson-process variability. Ultimately, data augmentation demonstrates its value in supplementing physical experiments; its benefit here is to emphasise the need for a cautious assessment of SPS quality

Infinite hierarchy of nonlinear Schrödinger equations and Infinite hierarchy of nonlinear Schrödinger equations and

We study the infinite integrable nonlinear Schrödinger equation (NLSE) hierarchy beyond the Lakshmanan-Porsezian-Daniel equation which is a particular (fourth-order) case of the hierarchy. In particular, we present the generalized Lax pair and generalized soliton solutions, plane wave solutions, AB breathers, Kuznetsov-Ma breathers, periodic solutions and rogue wave solutions for this infinite order hierarchy. We find that even order equations in the set affect phase and stretching factors in the solutions, while odd order equations affect the velocities. Hence odd order equation solutions can be real functions, while even order equation solutions are always complex

Rogue Wave Modes for the Long Wave-Short Wave Resonance Model

The long wave-short wave resonance model arises physically when the phase velocity of a long wave matches the group velocity of a short wave. It is a system of nonlinear evolution equations solvable by the Hirota bilinear method 1 and also possesses a Lax pair formulation. ‘Rogue wave ’ modes, algebraically localized entities in both space and time, are constructed from the breathers by a singular limit involving a ‘coalescence ’ of wavenumbers in the long wave regime. In contrast with the extensively studied nonlinear Schrödinger case, the frequency of the breather cannot be real and must satisfy a cubic equation with complex coefficients. The same limiting procedure applied to the finite wavenumber regime will yield mixed exponential-algebraic solitary waves, similar to the classical ‘double pole ’ solutions of other evolution systems