Turbulence in a fluid can be thought of as a chaotic dance of waves and currents that scientists have struggled to predict for centuries. This project builds a mathematical model of that dance, using a special class of equations where the waves behave like particles—called solitons—that pass through each other without breaking apart. The core problem is that classical theories of turbulence only work when waves are small and gentle. Real-world turbulence, from ocean currents to airflow over a wing, involves large, fully nonlinear waves that defy simple calculation. By treating these waves as a dense gas of interacting solitons, the researchers can write down an exact kinetic equation describing how the wave field and the background flow exchange energy and momentum. This is fundamental science. It will not directly improve an aeroplane wing or a weather forecast tomorrow. But it provides the first solvable model of a process—wave-mean flow interaction—that appears in every turbulent system. Past work on integrable equations has unexpectedly illuminated phenomena from fibre-optic communications to rogue waves. A rigorous handle on this interaction could eventually help engineers design quieter submarines, more efficient tidal turbines, or better models of ocean mixing that underpin climate predictions.
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Turbulence is a fundamental, ubiquitous and, at the same time, notoriously difficult problem of classical physics. The existing theories of strong (hydrodynamic) and weak (wave) turbulence rely on various approximations and assumptions and, despite many theoretical successes and an enormous literature on the subject, there is no general approach nor exactly solvable scenarios, and turbulence largely remains an unsolved problem of classical physics. It was realised recently that many questions and concepts pertaining to hydrodynamic and wave turbulence can also be meaningfully posed and interpreted in the framework of infinite-dimensional Hamiltonian integrable systems, constituting a new kind of nonlinear multiscale wave motion, dubbed integrable turbulence in the foundational 2009 paper by V. Zakharov. Mathematically, integrable turbulence can be viewed as the theory of multiscale random solutions to integrable partial differential equations such as the Korteweg - de Vries (KdV), nonlinear Schrodinger (NLS) and other equations. Integrability provides a unique set of analytical tools that enable, in principle, a precise description of complex fully nonlinear wave dynamics. Extending these tools to the realm of statistical mechanics of nonlinear waves is one of the major challenges of the modern theory of integrable systems. This project focuses on one key aspect of integrable turbulence: the wave-mean field interaction. In classical turbulence, wave-mean flow is only solvable when the waves are linear. In contrast, the integrable turbulence framework does not involve small amplitude restrictions and allows for the description of fully nonlinear regimes including solitons -- the localised nonlinear waves that exhibit particle-like properties such as elastic, pairwise interactions. Solitons can be viewed as natural “normal modes'' of nonlinear wave fields in integrable systems. This motivates the study of wave-mean field interaction for soliton-dominated integrable turbulence, commonly called soliton gas. The concept of soliton gas as an infinite random ensemble of interacting solitons has recently emerged as a new and powerful tool for understanding nonlinear random wave fields in the language of statistical physics and hydrodynamics. The emergent, macroscopic, dynamics of soliton gases for a broad range of integrable models are universally described by a nonlinear integro-differential kinetic equation which provides a general analytical framework for this project. Project Objectives To derive and analyse a solvable turbulence-mean field interaction model for the KdV and NLS equation (focusing and defocusing cases) in the framework of kinetic theory of soliton gases; To apply the analytical model derived in O.1 to the description of the interaction of soliton gases with rarefaction wave and DSW mean fields initiated by Riemann step data and compare with numerical solutions; To explore physical applications of the developed theory and facilitate experimental realisation of the soliton gas-mean field interaction. The significance of the proposed research lies in the fact that it will be the first solvable model of turbulence-mean interaction with applications to a broad class of nonlinear physical systems modelled by integrable partial differential equations. The project results will benefit broad UK and international research communities involved in nonlinear waves and applied integrable systems research.
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