EPSRC Maths Foresees Network Focused mathematics workshop report ·...

EPSRC Maths Foresees Network Focused mathematics workshop report

Workshop title: The onset of rogue waves

Workshop organisers: Sara Lombardo, Mathematics, Loughborough University (Northumbria University until31 Oct 2017); Matteo Sommacal, Benoit Huard, Fabio Briscese, Mathematics, Northumbria University.

Other participants: The workshop was attended by fourty-three participants on both days and it attractedtwenty-five early career researchers, PhD students and Post-doctoral researchers, mainly from institutions in theNorth East (notably, Newcastle, Durham, Edinburgh and Glasgow). See attached poster for a list of speakers.Support for early career researchers was possible also thanks for the additional funds by Nortumbria University.

Figure 1: Some of the workshop speakers: from the left Sergei Annenkov (Mathematics, Keele University, UK), Miguel

Onorato (Physics, University of Turin, Italy), Floriane Gidel (Applied Mathematics, University of Leeds, UK and Maritime

Research Institute of Netherlands (MARIN), NL), Benoit Huard (Mathematics, Northumbria University, UK), Sara Lom-

bardo (Mathematics, Northumbria University and Loughborough University, UK), Amin Chabchoub (Hydrodynamics, Aalto

University, Finland & The University of Sidney, Australia), Antonio Degasperis (Physics, University of Rome La Sapienza,

Italy), Fabio Baronio (Fabio Baronio, Photonics, University of Brescia, Italy), Saliya Coulibaly (Physics, PhLAM, Universite

de Lille 1, France), Yiping Ma (Mathematics, Northumbria University, UK)

Background and objectives: The two day workshop took place in April 2017, and it was aimed at bringingtogether scientists working on the problem of formation, description and prediction of rogue wave phenomena,mainly in the ocean, but not limited to. More specifically, it allowed interactions between the Maths Foresee com-munity and the Extreme Environment group at Northumbria University, a multidisciplinary research team, with agreat overlap of interests (see https://www.northumbria.ac.uk/research/research-themes/theme-3/ for a listof participants). The format of the workshop was a mix of traditional 45 minutes talks and round tables discussions(2); see workshop webpage http://group28.northumbria.ac.uk/Rogue/ and attached Book of abstracts for moredetails.

Key outcomes: The feedback from participants has been extremely positive; the round tables in particular gen-erated lively discussions, both on the more theoretical aspects of the workshop, as well as on the modelling side.

Potential for initiating or developing future multidisciplinary collaborations: We are submitting a pro-posal for a special volume around the topics of the workshop, notably on Rogue Waves in Resonant-InteractionsModels. A preliminary meeting was held at the end workshop and possible outlets were identified.

https://www.northumbria.ac.uk/research/research-themes/theme-3/

http://group28.northumbria.ac.uk/Rogue/

LONDON MATHEMATICAL SOCIETY

GOOD PRACTICESCHEME

UNIVERSITY NEWCASTLEnorthumbria

The Onset of Rogue Waves

The workshop is funded by the Maths Foresees Network (Leeds) and supported by the Extreme Environments theme (Northumbria)

Scientific CommitteeFabio BaronioAmin ChabchoubSara LombardoMiguel OnoratoMatteo Sommacal

Organising CommitteeFabio BrisceseBenoit HuardSara Lombardo Northumbria University

Sergei Annekov, Applied Mathematics, Keele University, UKFabio Baronio, Information Engineering, Università di Brescia, ITElzbieta Bitner-Gregersen (*), Ocean Engineering, DNV GL AS, Høvik, NOLuigi Cavaleri (*), Oceanography, ISMAR, Venice, IT Amin Chabchoub, Hydrodynamics, Aalto University, FISaliya Coulibaly, Physics, PhLAM, Université de Lille 1, FRAntonio Degasperis, Physics, Sapienza, Rome, ITFloriane Gidel, Applied Mathematics, University of Leeds, UK + MARIN, NLOleg Kirillov, Mathematics, Northumbria, UKSara Lombardo, Mathematics, Northumbria, UKYiping Ma, Mathematics, Northumbria, UK Miguel Onorato, Physics, Università di Torino, Turin, ITNicholas Parker, Applied Mathematics, Newcastle University, UK

7th & 8th April 2017Northumbria University

City Campus

offic

ine

rom

ane

There is no registration fee; if you would like to attend please contact Dr Fabio Briscese ([email protected]) or Dr Benoit Huard ([email protected]).

Book of Abstracts


A workshop sponsored by

Department of Mathematics, Physics and Electrical Engineering

Northumbria University, Newcastle upon Tyne

7-8 April 2017

Foreword

The two-day workshop aims to bring together scientists working on the prob-

lem of formation, description and prediction of rogue wave phenomena and

coherent structures in several contexts. The workshops thus spans across

a broad range of fields in applied mathematics and physics. It also aims to

bring the Maths Foresees community in touch with the Extreme Environments

group at Northumbria University, one of the multidisciplinary research teams

at Northumbria. The format will be a mix of traditional 45/50 minutes talks

and round tables discussions.

This workshop is made possible thanks to the EPSRC through the Maths

Foresees Network at the University of Leeds, and thanks to the support of

the Extreme Environments theme at Northumbria University.

Fabio Briscese,

Benoit Huard,

Sara Lombardo.

Department of Mathematics, Physics and Electrical Engineering

Northumbria University, Newcastle upon Tyne

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Reaching the workshop

If you are staying in hotel, you will be at the County Hotel, near the CentralStation.

Thistle County Hotel – http://www.countyhotel.co.uk

Neville Street, Newcastle upon Tyne, NE1 5DF, United KingdomTel: +44 (0) 191 731 6670Email: [email protected]

Northumbria University City Campus can be reached with a 10/15 minuteswalk, following the dotted path below.

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http://www.countyhotel.co.uk

Campus map

The workshop will be held in rooms EBA003 (Friday, am), EBA113 (Friday,pm) and EBA101 (Saturday), in Block A of Ellison Building (no 8 on themap). EBA003 is on the ground floor; EBA113 and EBA101 on the first floor.

Mathematics is located in Ellison Building, Block D.

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EllisonBuilding

NorthumberlandBuilding

McKenna’sTEL:01912427242

SIXBALTICTEL:01914404948

THISTLECOUNTYHOTELTEL+44(0)01917316670

BlackfriarsTEL:01912615945

Restaurants

• Blackfriars (British)Friars Street, TEL: 0191 261 5945

• McKenna’s BistroNorthern Stage, Newcastle, TEL: 0191 242 7242

• The Broad Chare (British)25 Broad Chare, TEL: 0191 211 2144

• SIX Restaurant (British)BALTIC Centre for Contemporary Art, Gateshead Quays, TEL: 0191 440 4948

• Mantra Thai (Thai)29 Forth Banks, TEL: 0191 232 6080

• Akbar’s (Indian)Unit 1, City Quadrant, Westmorland Rd, TEL: 0191 232 3234

• Paradiso (Mediterranean)1 Market Lane, TEL: 0191 221 1240

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Timetable

Time Friday, Apr 7 Saturday, Apr 8

Room EBA003(am)/EBA113(pm) EBA101

9:00-9:45 Welcome Co↵ee & Opening Co↵ee & Networking

9:45-10:30 Chabchoub Kirillov

10:30-11:15 Gidel Degasperis

11:15-11:45 Co↵ee Break Co↵ee Break

11:45-12:30 Onorato Coulibaly

12:30-14:00 Lunch Lunch

14:00-14:45 Annekov Baronio

14:45 -15:15 Co↵ee break Co↵ee break

15:15-16:00 Ma Round Table(⇤) – Lombardo

16:00-16:45 Parker Round Table(⇤) – Lombardo

16:45-17:15 Break Break

17:15-18:00 Networking Editorial board(⇤⇤) - EBD201

19:00 Dinner Dinner

(⇤) Round Table discussion topic: RWs in multicomponent waves equations: advantages/limitations,experiments(⇤⇤) Baronio, Chabchoub, Lombardo, Onorato

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List of contributions

Modelling of Kurtosis Evolution for Wind Waves

Sergei Annenkov

Akhmediev Breathers and Peregrine Solitary Waves in a Quadratic Medium

Fabio Baronio

Rogue Waves - Impact on Ships and O↵shore Structures

Elzbieta M Bitner-Gregersen

Rogue Waves on the Water Surface: from Deterministic to Spontaneous Emer-

gence

Amin Chabchoub

Spatiotemporal Chaos-induced Dissipative Extreme Events

Saliya Coulibaly

Two Coupled Nonlinear Schrodinger Equations and Wave Instabilities

Antonio Degasperis

Mathematical and Numerical Modelling of Rogue-type Waves in Deep and Shallow

Water

Floriane Gidel

Enhancing and Inhibiting Modulation Instability by Dissipation via singularities of

Stability Boundary

Oleg Kirillov

Traveling Edge Waves in Photonic Graphene

Yiping Ma

On the Origin of Heavy-tail Statistics in Equations of the Nonlinear Schrodinger

Type

Miguel Onorato

Solitons in Ultracold Atomic Bose-Einstein Condensates

Nicholas Parker

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Modelling of Kurtosis Evolution for Wind Waves

Sergei Annenkov a Victor I. Shrira a

a. School of Computing and Mathematics, Keele University, UK

Email. [email protected]

Predicting probability distribution of wave heights, in particular ocean wave extremes,requires the knowledge of higher statistical moments of wave fields (skewness and kurtosis).While the skewness of a gravity wave field is completely determined by its spectrum, thekurtosis consists of two components: the ”bound harmonics” one, which can be obtainedfrom the spectrum, and the “dynamical” one, which is linked to wave interactions. Atpresent, the knowledge of the latter component is very limited. Reliable theoretical, numericaland experimental data exist only for the case of very narrow directional wave spectra, wherethe evolution of the dynamical component of the kurtosis crucially depends on the value ofthe Benjamin-Feir index (BFI). When BFI exceeds 1, large (order 1) positive values of thedynamical kurtosis are attained during the evolution. However, experiments in wind wavetanks often show negative total kurtosis, which is possible only if the dynamical kurtosis isnegative and not small. At present there is no explanation of these observations within theframework of weakly nonlinear theory. In the current wave forecasting practice, the dynamicalcomponent of the kurtosis is either ignored (and then the total kurtosis is assumed equal tothe bound harmonics component, estimated using the existing second-order theory, which is,strictly speaking, valid only in the narrowband limit), or an extension of the one-dimensionaltheory is used, although its applicability to broadband wave fields is questionable. A unifiedpicture of the kurtosis evolution for random water wave fields is still missing.

In this work, we propose a new conceptual and numerical framework for modelling andforecasting of the higher moments of wave fields of arbitrary spectral width and, hence, ofprobability distribution of wave heights. The evolution of wave spectra and higher momentsis modelled by direct numerical simulation (DNS) based on the Zakharov equation. Thisis the only existing DNS approach that allows to study long-term spectral evolution. Atthe same time, it allows to obtain the dynamical kurtosis directly by calculating higher-order moments of the wave field, without the need to reconstruct the water surface. Weperform numerical simulations of the evolution of JONSWAP spectra of various spectral andangular width, with and without wind. For a subset of the initial spectra, experimental andalternative DNS data for the evolution without wind are used for validation of the approach.In the simulations without wind, we find the much discussed large positive maximum of thekurtosis occurring during the first few dozen wave periods of the evolution of narrow spectra.For initial spectra of intermediate angular width (about the typical width of sea swell) thedynamical kurtosis evolution is qualitatively di↵erent, with a pronounced negative maximum.For wide directional spectra, typical of sea conditions, the dynamical kurtosis is negative and

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mailto:[email protected]

small. The presence of moderate and strong wind considerably reduces the positive kurtosisspike of initially narrow directional wave fields, and enhances the negative maximum of thedynamical kurtosis for intermediate directional width. We also present the results for thekurtosis evolution of the same spectra obtained with the generalised kinetic equation (gKE)and discuss the role of the statistical closure.

Akhmediev Breathers and Peregrine Solitary Waves ina Quadratic Medium

Fabio Baronio a

a. Dipartimento di Ingegneria dell’Informazione, University of Brescia, IT


I investigate the formation of optical localized nonlinear structures, evolving upon a non-zero background plane wave, in a dispersive quadratic medium. I show the existence ofquadratic Akhmediev breathers and Peregrine solitary waves, in the regime of cascadingsecond-harmonic generation. This finding opens a novel path for the excitation of extremerogue waves and for the description of modulation instability in quadratic nonlinear optics.

Rogue Waves - Impact on Ships and O↵shoreStructures

Elzbieta M Bitner-Gregersen a Odin Gramstad a

a. DNV GL AS, NO-1322 Høvik, NO


DNV GL SR&I Position Paper on Rogue Waves – Impact on ships and o↵shore structuresis an open DNV GL publication (by Elzbieta M. Bitner-Gregersen and Odin Gramstad). Itsummarizes the DNV GL AS involvement in research on rogue waves and giving recommen-dations for future research in this field, having in mind engineering applications. DNV GLhas started already to implement rogue waves in DNV GL wave-structure interaction codes,thus the importance of rogue waves for satisfactory prediction of loads and responses isrecognized now.

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Rogue Waves on the Water Surface: fromDeterministic to Spontaneous Emergence

Amin Chabchoub a

a. Department of Mechanical Engineering, Aalto University, FI


Modulation instability is a universal mechanism that describes the significant amplifica-tion of a weak perturbation on top of a plane wave background and it has been extensivelystudied in the context of an integrable weakly nonlinear evolution equation, namely, thenonlinear Schrodinger equation (NLSE). This instability has been further suggested as apossible generation mechanism of rogue waves in the oceans. Here, we report on a detailedexperimental study in which the modulation instability dynamics has been triggered throughexact solutions of the NLSE or has been seeded by random noise added to a Stokes wave.Similarly to experiments recently performed in Optics, we show that the development of suchspontaneous modulation instability in a water wave tank leads to the occurrence of highlylocalized structures that can be interpreted as analytical NLSE breathers. Numerical simu-lations show very good qualitative agreement with the laboratory experiments. Moreover,general physical properties of observed hydrodynamic extreme events will be discussed indetail.

Spatiotemporal Chaos-induced Dissipative ExtremeEvents

Saliya Coulibaly a

a. Physics, PhLAM, Universite de Lille 1, FR


We discuss the route to spatiotemporal chaos combining experimental measurementswith numerical simulations in an optical dissipative system. A detailed and rigorous char-acterization of this spatiotemporal chaotic behavior is addressed together with statisticalindicators such as time di↵erences between successive extreme events.

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Two Coupled Nonlinear Schrodinger Equations andWave Instabilities

Antonio Degasperis a Sara Lombardo b Matteo Sommacal b

a. Physics Department, University of Rome Sapienza, IT

b. Department of Mathematics, Physics and Electrical Engineering, Northumbria University, UK


The Benjamin-Feir instability of periodic waves, which occurs in water dynamics andoptics, is investigated for periodic waves which are modelled by two coupled NonlinearSchrodinger equations. Because of the coupling the linear (short time) instability dependson several parameters. This makes unstable behavior far richer than that described by asingle Nonlinear Schrodinger equation where the main mechanism is due to self-focusing. Toillustrate a variety of processes we consider the stability properties of two coupled continuouswaves with di↵erent wave numbers, amplitudes and coupling constants. In contrast to thestandard approach which relies on Fourier analysis, our method makes use of the integrabilityof the wave propagation model, and of the spectral properties of the Lax pair. This methodcan be generalized so as to be applicable to quite a large number of integrable wave models.

Mathematical and Numerical Modelling of Rogue-typeWaves in Deep and Shallow Water

Floriane Gidel a,b Onno Bokhove a Mark Kelmanson a

a. Applied Mathematics, University of Leeds, UK

b. Maritime Research Institute of Netherlands (MARIN), NL


O↵shore installations are designed to resist the load and stress constantly applied bywaves. However, the occurrence of many accidents in recent years have shown that ships,o↵shore platforms and wind turbines are often not robust enough to face extreme waves suchas Rogue waves. The simulation of a Rogue wave’s impact on o↵shore bodies is a crucialstep in the design of reliable and durable structures. In that perspective, mathematicalmodels of nonlinear dispersive waves in both deep and shallow water are developed herein.The variational approach presently used to derive the equations yields to stable numerical

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simulations that ensure conservation of both mass and overall energy, and which can thereforecompute extreme-wave configurations.

In shallow water, the Rogue-type waves are generated from the oblique interaction of twosolitary waves, through a mechanism described by the Benney-Luke equations. Our presentsimulations show that the resulting wave follows Mach reflection theory and may grow up tofour times the amplitude of the initial incident waves. In deep water, waves are generatedfrom a piston wavemaker in a 3D numerical tank with seabed topography. Discretisationstrategies have been derived to deal with moving boundaries at the wavemaker wall and atthe dynamic free surface between water and air. The results are compared to experimentaldata obtained from the Maritime Research Institute of Netherlands (MARIN) in the case of afocussed wave. In both cases, the models may be used to simulate high-amplitude waves in atarget area, where maritime structures can be placed in order to test wave-body interactions.

Enhancing and Inhibiting Modulation Instability byDissipation via singularities of Stability Boundary

Oleg Kirillov a

a. Department of Mathematics, Physics and Electrical Engineering, Northumbria University, UK


A monochromatic plane wave with a finite amplitude propagating in a nonlinear anddispersive medium can be disrupted into a train of short pulses (called solitons) when theamplitude exceeds some threshold. This process develops due to an unbounded increase inthe percentage amplitude modulation of the wave, i.e. instability of the carrier wave withrespect to the modulations. This is the fundamental modulation instability, known as theBenjamin-Feir instability in hydrodynamics and the Bespalov-Talanov instability in nonlinearoptics. Without dissipation, a slowly varying in time envelope of the rapidly oscillating carrierwave is often described by the nonlinear Schrodinger equation (NLS). More realistic modelsof modulated waves take into account dissipative e↵ects such as dispersive and viscous lossesand nonlinear damping (Bridges and Dias (2007)). To study the stability of the modulation,we linearize the dissipative NLS about the basic traveling wave solution. Assuming spatially-periodic perturbations with some wave number, we substitute their Fourier expansions intothe linearized problem. Then, the spatial modes decouple into four-dimensional subspacesfor each harmonic, yielding a four-dimensional real linear di↵erential equation, depending onseveral parameters. For a parametric family of four-dimensional linear dynamical systemsdetermined by a matrix A(p) with A(0) in 1:1 semi-simple resonance, we have establishedthat the central singularity on the stability boundary has codimension 8, i.e. the centralizerunfolding of the family needs eight parameters. By recognizing equivalence classes in thecentralizer unfolding, we reduced the codimension to 5 and finally, by using the homogeneityproperties, to 3. This allowed us to explicitly find the boundary of the stability domain and

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list all its singularities including six self-intersections and four “Whitney umbrellas”. Wehave proposed an algorithm of approximation of the stability boundary near singularities andapplied the results to the study of enhancement/inhibition of the modulation instability withdissipation. This is a joint work with Igor Hoveijn (Groningen).

Traveling Edge Waves in Photonic Graphene

Yiping Ma a

a. Department of Mathematics, Physics and Electrical Engineering, Northumbria University, UK


Photonic graphene, namely an honeycomb array of optical waveguide, has attracted muchinterest in the optics community. Experimentally it was shown that introducing edges andsuitable waveguides in the direction of propagation, unidirectional edge wave propagationat optical frequencies occurs in photonic graphene. Analytically the system is describedby the lattice nonlinear Schrodinger (NLS) equation with a honeycomb potential and apseudo-magnetic field. In certain parameter regimes, these edge waves were found to benearly immune to backscattering, and thus exhibit the hallmarks of (Floquet) topologicalinsulators.

This talk addresses the linear and nonlinear dynamics of traveling edge waves in photonicgraphene, using a tight-binding model derived from the lattice NLS equation, and assumingthat the pseudo-magnetic field varies rapidly. In the presence of nonlinearity, nonlinear edgesolitons can exist due to the balance between dispersion and nonlinearity; these edge solitonsappear to be immune to backscattering when the dispersion relation is topologically nontriv-ial. Two possible mechanisms of generating rogue waves in this system will be introduced:domain geometry including sharp corners, and modulational instability of extended nonlinearedge modes.

On the Origin of Heavy-tail Statistics in Equations ofthe Nonlinear Schrodinger Type

Miguel Onorato a

a. Physics Department, University of Turin, IT


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We study the formation of extreme events in incoherent systems described by the Nonlin-ear Schrdinger type of equations. We consider an exact identity that relates the evolution ofthe normalized fourth-order moment of the probability density function of the wave envelopeto the rate of change of the width of the Fourier spectrum of the wave field. We show that,given an initial condition characterized by some distribution of the wave envelope, an increaseof the spectral bandwidth in the focusing/defocusing regime leads to an increase/decreaseof the probability of formation of rogue waves. Extensive numerical simulations in 1D+1and 2D+1 are also performed to confirm the results.

Solitons in Ultracold Atomic Bose-EinsteinCondensates

Nicholas Parker a

a. Applied Mathematics, Newcastle University, UK


Ultracold atomic Bose-Einstein condensates are a prototype system for studying nonlinearwaves due to the immense experimental control over their purity, geometry, nonlinearity andpotential landscape. Here I will give an overview of dark and bright solitons in these systems,covering their experimental observation, solutions, propagation, and interactions with linearwaves (sound waves). I will also highlight the opportunity to study non-local solitons.

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LONDON MATHEMATICAL SOCIETY

GOOD PRACTICESCHEME

UNIVERSITY NEWCASTLEnorthumbria


The workshop is funded by the Maths Foresees Network (Leeds) and supported by the Extreme Environments theme (Northumbria)

Scientific CommitteeFabio BaronioAmin ChabchoubSara LombardoMiguel OnoratoMatteo Sommacal

Organising CommitteeFabio BrisceseBenoit HuardSara Lombardo Northumbria University

Sergei Annekov, Applied Mathematics, Keele University, UKFabio Baronio, Information Engineering, Università di Brescia, ITElzbieta Bitner-Gregersen (*), Ocean Engineering, DNV GL AS, Høvik, NOLuigi Cavaleri (*), Oceanography, ISMAR, Venice, IT Amin Chabchoub, Hydrodynamics, Aalto University, FISaliya Coulibaly, Physics, PhLAM, Université de Lille 1, FRAntonio Degasperis, Physics, Sapienza, Rome, ITFloriane Gidel, Applied Mathematics, University of Leeds, UK + MARIN, NLOleg Kirillov, Mathematics, Northumbria, UKSara Lombardo, Mathematics, Northumbria, UKYiping Ma, Mathematics, Northumbria, UK Miguel Onorato, Physics, Università di Torino, Turin, ITNicholas Parker, Applied Mathematics, Newcastle University, UK

7th & 8th April 2017Northumbria University

City Campus

offic

ine

rom

ane

There is no registration fee; if you would like to attend please contact Dr Fabio Briscese ([email protected]) or Dr Benoit Huard ([email protected]).

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