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Table Of Integrals Series And Products

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Recent observations show that the probability of encountering an extremely large rogue wave in the open ocean is much larger than expected from ordinary wave-amplitude statistics. Although considerable effort has been directed towards understanding the physics behind these mysterious and potentially destructive events, the complete picture remains uncertain. Furthermore, rogue waves have not yet been observed in other physical systems. Here, we introduce the concept of optical rogue waves, a counterpart of the infamous rare water waves. Using a new real-time detection technique, we study a system that exposes extremely steep, large waves as rare outcomes from an almost identically prepared initial population of waves. Specifically, we report the observation of rogue waves in an optical system, based on a microstructured optical fibre, near the threshold of soliton-fission supercontinuum generation--a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input. We model the generation of these rogue waves using the generalized nonlinear Schrodinger equation and demonstrate that they arise infrequently from initially smooth pulses owing to power transfer seeded by a small noise perturbation.

Optical rogue waves

The stability of two-dimensional (2D) layers and membranes is subject of a long standing theoretical debate. According to the so called Mermin-Wagner theorem, long wavelength fluctuations destroy the long-range order for 2D crystals. Similarly, 2D membranes embedded in a 3D space have a tendency to be crumpled. These dangerous fluctuations can, however, be suppressed by anharmonic coupling between bending and stretching modes making that a two-dimensional membrane can exist but should present strong height fluctuations. The discovery of graphene, the first truly 2D crystal and the recent experimental observation of ripples in freely hanging graphene makes these issues especially important. Beside the academic interest, understanding the mechanisms of stability of graphene is crucial for understanding electronic transport in this material that is attracting so much interest for its unusual Dirac spectrum and electronic properties. Here we address the nature of these height fluctuations by means of straightforward atomistic Monte Carlo simulations based on a very accurate many-body interatomic potential for carbon. We find that ripples spontaneously appear due to thermal fluctuations with a size distribution peaked around 70 \AA which is compatible with experimental findings (50-100 \AA) but not with the current understanding of stability of flexible membranes. This unexpected result seems to be due to the multiplicity of chemical bonding in carbon.

Intrinsic ripples in graphene

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.

/pdf/physical-mechanisms-of-the-rogue-wave-phenomenon-iao5kemdkh.pdf

Physical Mechanisms of the Rogue Wave Phenomenon

A numerical solution to the generalized Korteweg-de Vries (K-dV) equation, including horizontal variability and dissipation, is used to model the evolution of an initially sinusoidal long internal wave, representing an internal tide. The model shows the development of the waveform to the formation of shocks and solitons as it propagates shoreward over the continental slope and shelf. The model is run using observed hydrographic conditions from the Australian North West Shelf and results are compared to current meter and thermistor observations from the shelf-break region. It is found from observations that the coefficient of nonlinearity in the K-dV equation changes sign from negative in deep water to positive in shallow water, and this plays a major role in determining the form of the internal tide transformation. On the shelf there is strong temporal variability in the nonlinear coefficient due to both background shear flow and the large amplitude of the internal tide, which distorts the density profile over a wave period. Both the model and observations show the formation of an initial shock on the leading face of the internal tide. In shallow water, the change in sign of the coefficient of nonlinearity causes the shock to evolve into a tail of short period sinusoidal waves. After further propagation a second shock forms on the back face of the wave, followed by a packet of solitons. The inclusion of bottom friction in the model is investigated along with the dependance on initial wave amplitude and variability in the coefficients of nonlinearity and dispersion. Friction is found to be important in limiting the amplitudes of the evolving waves.

A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf

This short editorial contains the introductory remarks for the special issue “discussions and debates” on the subject of “rogue waves”. This issue is the first of its kind, in the sense that the authors have the chance to discuss the basic concepts of an emerging topic in science. Based on these discussions, an attempt to give a “definition” of a rogue wave is made.

/pdf/editorial-introductory-remarks-on-discussion-debate-rogue-tygucakthy.pdf

Editorial – Introductory remarks on “Discussion & Debate: Rogue Waves – Towards a Unifying Concept?”

A nonlinear model is developed, based on the rotated-modified extended Korteweg-de Vries (reKdV) equation, of the evolution of an initially sinusoidal long wave in the coastal zone, representing an internal tide, into nonlinear waves including internal solitary waves. The coefficients of the basic equation are calculated using observed conditions for the north west shelf (NWS) of Australia. The roles of both quadratic and cubic nonlinearity, the Earth's rotation, and frictional dissipation are discussed. The combined action of nonlinearity and rotation leads to a number of intersting features in the wave form including solitons of both polarities, “thick” solitons, and sharp waves with steep fronts. It is shown that rotation is important for modelling the evolution of the internal tide, even for the relatively low latitude on the NWS of 20°S. Rotation increases the phase speed of the long internal tide, reduces the number of internal solitary waves that form from a long wave, and changes the form of the waves. The effects of nonlinearity on the vertical modal structure of the internal waves are also discussed. Results of numerical simulations are compared with current and temperature observations of the internal wave field on the NWS which show many of the features produced by the generalized KdV model.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/1999JC900144

A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone

Preface.- Freak Waves: Peculiarities of Numerical Simulations.- Rogue Waves in High-Order Nonlinear Schrodinger Models.- Non-Gaussian Properties of Shallow Water Waves in Crossing Seas.- Modeling of Rogue Wave Shapes in Shallow Water.- Runup of Long Irregular Waves on Plane Beach.- Symbolic Computation for Nonlinear Wave Resonances.- Searching for Factors that Limit Observed Extreme Maximum Wave Height Distributions in the North Sea.- Extremes and Decadal Variations of the Northern Baltic Sea Wave Conditions.- Extreme Waves Generated by Cyclones in Guadeloupe.- An Analytical Model of Large Amplitude Internal Solitary Waves.

Efim Pelinovsky

Papers

Physical Mechanisms of the Rogue Wave Phenomenon

A Nonlinear Model of Internal Tide Transformation on the Australian North West Shelf

Editorial – Introductory remarks on “Discussion & Debate: Rogue Waves – Towards a Unifying Concept?”

A generalized Korteweg-de Vries model of internal tide transformation in the coastal zone

Extreme ocean waves