Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy

New extensions of the KP and modified KP hierarchies with self-consistent sources are proposed. The latter provide new generalizations of $(2+1)$-dimensional integrable equations, including the DS-III equation and the $N$-wave problem. Furthermore, we recover a system that contains two types of the KP equation with self-consistent sources as special cases. Darboux and binary Darboux transformations are applied to generate solutions of the proposed hierarchies

New coherent structures of the Vakhnenko–Parkes equation

AbstractA variable separation solution with two arbitrary functions is obtained for the Vakhnenko–Parkes equation. New coherent structures such as the soliton-type, instanton-type and rogue wave-type structures are presented

Ocean swell within the kinetic equation for water waves

Effects of wave-wave interactions on ocean swell are studied. Results of extensive simulations of swell evolution within the duration-limited setup for the kinetic Hasselmann equation at long times up to $10^6$ seconds are presented. Basic solutions of the theory of weak turbulence, the so-called Kolmogorov-Zakharov solutions, are shown to be relevant to the results of the simulations. Features of self-similarity of wave spectra are detailed and their impact on methods of ocean swell monitoring are discussed. Essential drop of wave energy (wave height) due to wave-wave interactions is found to be pronounced at initial stages of swell evolution (of order of 1000 km for typical parameters of the ocean swell). At longer times wave-wave interactions are responsible for a universal angular distribution of wave spectra in a wide range of initial conditions.Comment: Submitted to Journal of Geophysical Research 18 July 201

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

Modelling of tides, waves and currents in coastal waters, hydrodynamics of the Firth of Clyde and of the east coast of Scotland

The thesis here presented is based on three main parts. The first part of the thes is based on modelling the water circulation of the Clyde Sea, in order to understand the dynamics of the dispersion of the Neprophs larvae. Previous researches in this area of Scotland highlighted the importance of the temperature- and saline-driven circulation in the Clyde Sea. However, few researches were focused on the surge dynamics, that are governing the dynamics of the water level in winter.;A three-dimensional finite-volume model was used to simulate the surge propagation, while a historical re-analysis was applied to understand the pattern and the propagation of the surge wave in the Clyde.The results highlighted that the largest storms that hit the Clyde in the past 30 years were mostly generated in the North Atlantic. Most interestingly, the results also suggest that severe surges are not only caused by extreme surge events, but also by the coupling of spring high tides with moderate surges.;In the second part the coupled dynamics of waves, tides and wind-driven circulation in the east coast of Scotland are studied. Wave-Current Interactions (WCI) are particularly relevant close to the coastline, where the effect of the spectral dispersion and wave breaking are more important and where the currents are stronger. The results showed that the coupling of strong currents with large waves travelling inopposite direction could enhance in east coast of Scotland the significant wave height (Hs) up to 3 m, threatening potentially infrastructures and ships near the coastline.;The last part of the thesis was dedicated to an experimental study of rogue waves in crossing sea. Crossing sea is one of the most common state in world seas, and occurs when a wind-generated wave train interact with another train of waves, that can be swell waves or another wind-generated wave train, maybe caused by a rapidly turning wind direction. Some numerical studies showed that this interaction can lead to the mechanism of modulation instability and, consequently, to the formation of rogue waves. A water tank experiment was carried out to confirm this theory.;The results shows that the angle of the interaction is a fundamental variable that can decrease or increase the instability of the wave train. However, most interestingly, in the sameconditions, the monochromatic sea state was showing a larger number of rogue waves than the crossing sea.The thesis here presented is based on three main parts. The first part of the thes is based on modelling the water circulation of the Clyde Sea, in order to understand the dynamics of the dispersion of the Neprophs larvae. Previous researches in this area of Scotland highlighted the importance of the temperature- and saline-driven circulation in the Clyde Sea. However, few researches were focused on the surge dynamics, that are governing the dynamics of the water level in winter.;A three-dimensional finite-volume model was used to simulate the surge propagation, while a historical re-analysis was applied to understand the pattern and the propagation of the surge wave in the Clyde.The results highlighted that the largest storms that hit the Clyde in the past 30 years were mostly generated in the North Atlantic. Most interestingly, the results also suggest that severe surges are not only caused by extreme surge events, but also by the coupling of spring high tides with moderate surges.;In the second part the coupled dynamics of waves, tides and wind-driven circulation in the east coast of Scotland are studied. Wave-Current Interactions (WCI) are particularly relevant close to the coastline, where the effect of the spectral dispersion and wave breaking are more important and where the currents are stronger. The results showed that the coupling of strong currents with large waves travelling inopposite direction could enhance in east coast of Scotland the significant wave height (Hs) up to 3 m, threatening potentially infrastructures and ships near the coastline.;The last part of the thesis was dedicated to an experimental study of rogue waves in crossing sea. Crossing sea is one of the most common state in world seas, and occurs when a wind-generated wave train interact with another train of waves, that can be swell waves or another wind-generated wave train, maybe caused by a rapidly turning wind direction. Some numerical studies showed that this interaction can lead to the mechanism of modulation instability and, consequently, to the formation of rogue waves. A water tank experiment was carried out to confirm this theory.;The results shows that the angle of the interaction is a fundamental variable that can decrease or increase the instability of the wave train. However, most interestingly, in the sameconditions, the monochromatic sea state was showing a larger number of rogue waves than the crossing sea

Numerical Modelling of Extreme Waves: The Role of Nonlinear Wave-Wave Interactions

The real monsters of the ocean, extreme waves, haunted mariners since the early days of human activities in the sea. Despite having caused numerous accidents and casualties, their systematic study began only in 2000s. Many mechanisms have been proposed to simulate these rare but catastrophic events, with the most prominent being wave focusing. This is connected to the NewWave theory, which has been used extensively in experimental and numerical modelling. However, the majority of the studies fail to capture the distinguishing characteristics of extreme waves, due to the inherent high nonlinearity of the problem and shortcomings of the modelling practice, but also due to inadequate knowledge of the underlying physics. Overcoming these issues is unquestionably necessary for understanding extreme waves and including them in the engineering design practice. The nonlinearity of the problem lies upon the nonlinear wave-wave interactions, which violate the fundamental linear assumptions of NewWave and pose challenges to numerical models. The present work aims at contributing in both understanding the nature of nonlinear wave-wave interactions during the formation of extreme wave events, and examining the applicability and performance of numerical solvers via their systematic validation with state-of-the-art techniques that give new insights into the problem. A range of phase-resolving and phase-averaged models are employed to cover different scales and examine the undergoing physical processes. Through the study of limiting breaking unidirectional dispersive wave groups in finite water depth, it is demonstrated that the free-wave spectrum undergoes considerable transformation and a large portion of energy is transferred to higher and lower harmonics. These effects can be attributed to the action of near-resonant and bound nonlinearities, which have however robust mathematical description. As such, a large part of the thesis is devoted to analytical methods towards establishing an efficient integrated framework for estimating extreme wave profiles, going beyond the classic NewWave. Overall, the present work is a balance of physics and numerics to tackle parts of the challenging problem of extreme waves and improve safety at sea

Self-similarity of wind-driven seas

International audienceThe results of theoretical and numerical study of the Hasselmann kinetic equation for deep water waves in presence of wind input and dissipation are presented. The guideline of the study: nonlinear transfer is the dominating mechanism of wind-wave evolution. In other words, the most important features of wind-driven sea could be understood in a framework of conservative Hasselmann equation while forcing and dissipation determine parameters of a solution of the conservative equation. The conservative Hasselmann equation has a rich family of self-similar solutions for duration-limited and fetch-limited wind-wave growth. These solutions are closely related to classic stationary and homogeneous weak-turbulent Kolmogorov spectra and can be considered as non-stationary and non-homogeneous generalizations of these spectra. It is shown that experimental parameterizations of wind-wave spectra (e.g. JONSWAP spectrum) that imply self-similarity give a solid basis for comparison with theoretical predictions. In particular, the self-similarity analysis predicts correctly the dependence of mean wave energy and mean frequency on wave age Cp / U10. This comparison is detailed in the extensive numerical study of duration-limited growth of wind waves. The study is based on algorithm suggested by Webb (1978) that was first realized as an operating code by Resio and Perrie (1989, 1991). This code is now updated: the new version is up to one order faster than the previous one. The new stable and reliable code makes possible to perform massive numerical simulation of the Hasselmann equation with different models of wind input and dissipation. As a result, a strong tendency of numerical solutions to self-similar behavior is shown for rather wide range of wave generation and dissipation conditions. We found very good quantitative coincidence of these solutions with available results on duration-limited growth, as well as with experimental parametrization of fetch-limited spectra JONSWAP in terms of wind-wave age Cp / U10

Solitons, Breathers and Rogue Waves in Nonlinear Media

In this thesis, the solutions of the Nonlinear Schrödinger equation (NLSE) and its hierarchy are studied extensively. In nonlinear optics, as the duration of optical pulses get shorter, in highly nonlinear media, their dynamics become more complex, and, as a modelling equation, the basic NLSE fails to explain their behaviour. Using the NLSE and its hierarchy, this thesis explains the ultra-short pulse dynamics in highly nonlinear media. To pursue this purpose, the next higher-order equations beyond the basic NLSE are considered; namely, they are the third order Hirota equation and the fifth order quintic NLSE. Solitons, breathers and rogue wave solutions of these two equations have been derived explicitly. It is revealed that higher order terms offer additional features in the solutions, namely, ‘Soliton Superposition’, ‘Breather Superposition’ and ‘Breather-to-Soliton’ conversion. How robust are the rogue wave solutions against perturbations? To answer this question, two types of perturbative cases have been considered; one is odd-asymmetric and the other type is even-symmetric. For the odd-asymmetric perturbative case, combined Hirota and Sasa-Satsuma equations are considered, and for the latter case, fourth order dispersion and a quintic nonlinear term combined with the NLSE are considered. Indeed, this thesis shows that rogue waves survive these perturbations for specific ranges of parameter values. The integrable Ablowitz-Ladik (AL) equation is the discrete counterpart of the NLSE. If the lattice spacing parameter goes to zero, the discrete AL becomes the continuous NLSE. Similar rules apply to their solutions. A list of corresponding solutions of the discrete Ablowitz-Ladik and the NLSE has been derived. Using associate Legendre polynomial functions, sets of solutions have been derived for the coupled Manakov equations, for both focusing and defocusing cases. They mainly explain partially coherent soliton (PCS) dynamics in Kerr-like media. Additionally, corresponding approximate solutions for two coupled NLSE and AL equations have been derived. For the shallow water case, closed form breathers, rational and degenerate solutions of the modified Kortweg-de Vries equation are also presented