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

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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.