Dynamics of singular stochastic nonlinear dispersive PDEs

Dispersion exists ubiquitously in nature. The most famous example of dispersion is seen in a rainbow, where dispersion effect separates the white light spatially into components of different wavelengths (different colours). Nonlinear dispersive partial differential equations (PDEs), such as nonlinear Schrodinger equations (NLS) and nonlinear wave equations (NLW), appear naturally in models describing wave phenomena in the real world. In the past thirty years, the study of deterministic nonlinear dispersive PDEs has seen significant development, in which harmonic analysis has played a fundamental role, led by Kenig, Bourgain and Tao, among others. In recent years, a combination of deterministic analysis with probability theory has played an increasingly important role in the field. This probabilistic perspective allows us to go beyond the limits of deterministic analysis. More importantly, it is also essential to understand the effect of stochastic perturbation in practice since such stochastic perturbation is ubiquitous.

The main objective of this research is to develop novel mathematical ideas and techniques to clarify long-standing fundamental questions in the study of stochastic nonlinear dispersive PDEs, with primary examples given by stochastic NLS and stochastic NLW. In the field of singular stochastic parabolic PDEs, significant progress has been taking place led by Hairer and Gubinelli with their collaborators. This has enabled striking theories which are changing the landscape of the study in this field. However, their new theories are designed to handle parabolic problems, and it is not a priori clear on how to adapt them to solve dispersive equations. Despite some exciting recent progress, our understanding of stochastic dispersive PDEs is still very far from satisfactory.

In these proposed projects, the principal investigator (PI) will study several open problems in the field of stochastic dispersive PDEs. More specifically, the PI will focus on studying the properties of invariant measures and the local and global-in-time solutions to stochastic NLS and NLW in periodic domains. The PI plans to address these problems by combining tools from dispersive PDEs, stochastic analysis, probability theory and harmonic analysis with recent progress.

This research lies at the interface of dispersive partial differential equations (PDEs) and stochastic analysis and aims to make substantial progress towards resolving long-standing open problems in the field of stochastic dispersive PDEs. More precisely, it concerns the dynamics of nonlinear Schrodinger equations and nonlinear wave equations under singular random perturbations. These equations appear naturally in various models in natural science. Therefore this research is expected to have a substantial impact within mathematics and beyond.

Within Mathematics:

This research will substantially forward our understanding of stochastic dispersive PDEs, such as the stochastic nonlinear Schrodinger and wave equations arising from various settings. With his collaborators, The PI has made exciting progress recently, by incorporating ideas from harmonic analysis, dynamical systems and even number theory in an innovative way, placing this research at the cutting edge position in this field. Thus proposal will facilitate the PI to keep ahead by forming and leading his own group, and consequently, this proposal will enhance the study of this subject and thus strengthen the UK's leading position on a global scale.

Beyond Mathematics:

Dispersive PDEs such as the nonlinear Schrodinger equations and nonlinear wave equations appear naturally in models describing wave phenomena in applied mathematics, physics and engineering. The proposal aims to develop systematic methods to understand the properties of stochastic nonlinear dispersive PDEs, underpinning applications in various areas. The nonlinear Schrodinger equation (NLS) in Project A is a fundamental model in nonlinear optics and thus plays a key role in optical fibre communication systems. The stochastic nonlinear wave equation in Project B, also known as canonical stochastic quantisation, together with the sine-Gordon medal and Liouville models in Project C, play important roles in quantum field theory. The mathematical analysis of these singular dispersive models will potentially provide solid foundations for these fields and thus promote natural science and technology advances, in particular, in fibre communication systems and quantum field theory.

Impact on people:

This proposal supports early-career researchers. A postdoctoral research assistant (PDRA) will work under this project for 24 months and gain expertise in this active research area under the PI's supervision. Through the travel fund provided by this project, the PDRA can attend conferences and meet experts in this field, thus helping to build their own research network. The successful delivery of this research will attract young talents to pursue mathematical research through the public lecture that the PI plans to deliver. The School has promised to provide a full scholarship for a PhD to work with the PI on the successful award of the grant, which will help in nurturing homegrown talent and change the situation of lack of PhDs in analysis in the UK. The PI also foresees that the influence of this research will go beyond academia. Dispersive PDEs are fundamental models in optical fibres and has important applications in the telecommunications industry. Therefore in the longer term, it is envisaged that this research will have a potential economic impact in the telecommunications industry, thus helping to create high-tech jobs.