Integrable turbulence and rogue waves: semi-classical nonlinear Schrödinger equation framework

Turbulence is one of the most recognisable, and at the same time, one of the most intriguing forms of nonlinear motion that is commonly observed in everyday phenomena such as wind blasts or fast flowing rivers. Despite its widespread occurrence, the mathematical description of turbulence remains one of the most challenging problems of modern science. Physical mechanisms giving rise to turbulent motion can be very different but typically they involve some sort of dissipation, e.g. viscosity.

The current project explores a very different kind of turbulence that does not involve any dissipation but is concerned with dynamics and statistics of random nonlinear waves that are modelled by the so-called integrable partial differential equations (PDEs) such as the Korteweg - de Vries and nonlinear Schroedinger (NLS) equations. These equations are universal mathematical models for a broad spectrum of nonlinear wave phenomena in water waves, optical media, plasmas and superfluids. Owing to their rich mathematical structure and a wide range of physical applications, integrable PDEs have been the subject of incredibly intense research in the last 50 or so years.

The idea of using random (stochastic) solutions to integrable equations for modelling complex nonlinear wave phenomena in the ocean and optical media has been recently put forward by V.E. Zakharov who has coined the term "integrable turbulence". In particular, the integrable turbulence framework can help to explain the formation and evolution of rogue waves - rare events of large amplitude that appear unpredictably on the ocean surface and can be devastating for ships and oil platforms. Rogue waves have also been observed in optical fibres as spontaneous field fluctuations of large amplitude with a number of undesirable implications for high power lasers and optical communications systems.

To date, very few analytical results in integrable turbulence are available with the majority of the developments being numerical. The project will attack this outstanding issue by constructing the first analytical model of integrable turbulence in the framework of the semi-classical limit of the focusing NLS equation, which is a fundamental mathematical model in nonlinear science that applies to a wide range of physical contexts including water waves, plasmas, nonlinear optical fibres and Bose-Einstein condensates. In particular, the so-called breather solutions of the NLS equation have the properties that strongly suggest their links with rogue waves in the ocean and optical media. In the project, the mathematical description of integrable turbulence and the rogue wave formation will be achieved via the asymptotic approach bridging two major techniques in the semi-classical analysis of dispersive PDEs: the Whitham modulation theory and the Riemann-Hilbert problem analysis. This unified approach was recently developed by the PI in collaboration with Prof. A. Tovbis who is also one of the main collaborators in the current project. One of the fundamental mathematical hypotheses to be proved in the project is related to the special thermodynamic structure of the nonlinear spectrum of the developed integrable turbulence, which will then be used for the analysis of its kinetic properties and particularly, the determination of the rogue wave content.

The unique feature of the project is the integrated pathway to impact via the linked PhD project concerned with the fibre optics implementation of the semi-classical NLS approach to integrable turbulence. The particular objectives of the PhD project, which is approved for funding by the Defence Science and Technology Laboratory, are related to the development of practical methods of analysis and control of the rogue wave formation in the partially coherent light propagation through optical fibres. The developed methods will be verified experimentally in the PhLAM optics laboratory at the University of Lille.

The main areas of economic and societal impacts of the project lie in the following areas:

1. Defence and Security.

A major strand of the proposed research is the theoretical and experimental analysis of rogue waves in optical fibres. These rogue waves represent strong spontaneous fluctuations of the optical field, that can damage or even destroy high power lasers or amplifiers. On the other hand, the possibility of the control of their emergence can open the way to a new generation of high power integrated fibre-based sources enabling the development of new specialist components for high-end industries such as security and defence.

The beneficiaries include various defence and security programmes and other government department initiatives which will be informed via the direct involvement in the project of the Defence Science and Technology Laboratory (Dstl). Dstl will also provide funding for the PhD project integrated within this EPSRC research in the framework of UK-France 2017 PhD Programme.

2. Information and Communication Technologies.

The inevitable presence of noise and nonlinearity of the fibre are considered to be the major factors limiting the performance of optical communication networks. The project results will help to develop the methods of noise control in nonlinear light propagation in optical fibres and will thus impact the design of optical communications systems.

The pathways to this impact will be explored via the interactions with Prof. S. Turitsyn's group at the Aston Institute of Photonic Technologies (AIPT) and by communicating the results of the project at AIPT seminars.

3. Coastal and waterway engineering. Ship design.

Along with nonlinear propagation of light in single-mode fibres, the focusing 1D-NLS equation describes at the leading order the evolution of deep water waves. Rogue waves in random sea states can have major detrimental effects on offshore structures and shipping.

Main beneficiaries in this area are experts and organisations involved in the coastal and waterway engineering research. The PI has already established collaboration with Prof. M. Onorato (University of Torino) and Prof. J. Dudley (FEMPTO-ST, Besancon), the leading rogue wave experts involved in the Extreme Seas Programme whose mission is to enable European shipping industry to improve the design of ship structures that are exposed to rough climate, by providing technology and methodology that need to be a part of the design for the ship safety in extreme seas.

4. Knowledge.

The project will have an indirect impact on a broad area of nonlinear science by attracting the attention of pure mathematicians to the subject of integrable turbulence, which could prove highly beneficial for both (theoretical and applied) areas. This pathway is already integrated within the proposal due to the participation of A. Tovbis and M. Bertola, who are the recognised experts in a refined mathematical analysis of integrable systems. Further planned activities in this direction include dissemination of the project results through the existing Leicester-Loughborough-Bristol network "Schroedinger equations: Asymptotics, Integrability and Beyond" which mostly involves experts in applied analysis, integrable systems and geometry and the organisation of the mini-symposium "Shock waves and turbulence in dispersive hydrodynamics" at the International conference Dynamics Days Europe 2018.

5. Educational impact.

The PI currently teaches a PGR course `Nonlinear Waves' in the framework of MAGIC consortium of 21 UK Universities. The MAGIC group runs a wide range of postgraduate-level lecture courses in mathematics via Video Conferencing technology. The lectures are recorded and made available online. One of the lectures is typically devoted to a topic of current research giving the PI a great opportunity to disseminate some of the results obtained within this research project to a broad audience of PhD student