Integrable Partial Differential Equations as Pathfinders in Mathematical Physics

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

Integrable systems have long served as pathfinders in science. Newton transformed our understanding of gravity because he was able to solve the two-body problem (a completely integrable system) and so verify that his inverse-square law of gravity matched empirical observations. That the three-body problem is not solvable in the same sense does not undermine this; indeed, it is precisely what makes the basic two-body paradigm so essential to our understanding! Likewise, the shell model of the atom is born of a completely integrable approximation. The integrable systems at the center of this project are significantly more complicated, but likewise serve as oracles for the full-complexity systems at the forefront of science and engineering. Indeed, all the concrete models studied as part of this project arose initially as effective models of real physical systems (such as water waves, optics, and magnetohydrodynamics), stripped down to reveal the central mechanisms behind the observed phenomena. To fulfill the role just articulated, we must study these integrable systems in large-data non-perturbative regimes - regimes where still simpler models cannot reproduce the observed phenomenology. This is a hallmark of this research project and indeed, of the methods developed in the PI's recent work, more broadly. The project provides significant research training opportunities for graduate students and postdoctoral scholars, who are integrated into every major activity of the project. The primary goal of the project is to advance the low-regularity theory of certain completely integrable dispersive partial differential equations, both as an end onto itself and as a tool for understanding the statistical mechanics of these systems. As a means to this end, the project further develops the commuting flows paradigm and seeks out new and diverse applications for several technical tools developed in support of this approach. Well-posedness problems will be studied, including that of the derivative nonlinear Schrodinger equation. Microscopic conservation laws will be deployed to further elaborate the spacetime structural properties of solutions constructed via commuting flows. The synthesis of renormalization and commuting flows technologies will be investigated, with a view to well-posedness, to the construction of Gibbs states, and to the dynamical properties thereof. Likewise, the continuum limit of discrete integrable models will be investigated both for deterministic initial data and as a means of gaining insight into Gibbs-state dynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.