Probabilistic study of nonlinear Schrödinger equations on compact domains

The Nonlinear Schrödinger Equation (NLS) is a fundamental model in the study of wave phenomena, particularly in the context of oscillating wave packets. It captures the intricate nonlinear interactions of wave propagation, which often pose challenges for analysis due to their singular nature. In such scenarios, traditional analytical methods often prove inadequate, thus demanding alternative approaches. One such approach is the statistical method employing Gibbs measures, which provides a robust framework to understand the long-term behaviour of solutions to nonlinear PDEs, particularly those demonstrating chaotic or turbulent phenomena. Zakharov's famous question (1983) is a significant example in this regard, seeking to explain the "returning" property of NLS solutions after a rather chaotic evolution. Bourgain's groundbreaking work in the 1990s and recent breakthroughs by Deng-Nahmod-Yue have provided a deep understanding of the Gibbs dynamics of NLS on the flat torus. Nevertheless, challenging open questions remain regarding the Gibbs dynamics of NLS in various geometric contexts. The main difficulty stems from the irregularity of the random initial data, due to the roughness of the support of the Gibbs measure. Over the last three decades, there has been remarkable progress in this field, particularly in the flat torus setting, where the late Fields medalist Bourgain played a leading role. In the past five years, the principal investigator (PI) and his collaborators have made substantial contributions to advancing our theoretical understanding of Gibbs measures and Gibbs dynamics of NLS. See the "Applicant and team capability to deliver" section for further details. This proposal aims to tackle two well-known unresolved issues left open by Zakharov in 1983, Tzvetkov in 2006, and Bourgain and Bulue in 2014. The latter work constructed almost sure global dynamics and confirmed the invariance of the Gibbs measure of NLS posed on certain geometric settings (disc and ball) but did not successfully demonstrate the flow property of the resulting dynamics, which is essential for addressing Zakharov's question. The main aim of this proposal is to tackle these outstanding open problems by introducing a novel approach that leverages random averaging operator theory and random tensor theory. The Nonlinear Schrödinger Equations are essential in understanding various phenomena across scientific disciplines. They are crucial for understanding light propagation in nonlinear optical fibres, the dynamics of Bose-Einstein condensates in quantum field theory, and the modelling of wave packets in various applied sciences. This proposal holds significant value in these fields as it provides a theoretical foundation for advancements in simulations and experimental setups.