Integrable Partial Differential Equations as Pathfinders in Mathematical Physics

可积系统长期以来一直是科学的探路者。牛顿改变了我们对引力的理解，因为他能够解决二体问题（一个完全可积的系统），从而验证了他的引力平方反比定律与经验观察相匹配。三体问题在同样意义上是不可解决的，但这并不能削弱这一点。事实上，这正是基本的二体范式对于我们的理解如此重要的原因！同样，原子的壳层模型也是由完全可积的近似产生的。该项目中心的可积系统要复杂得多，但同样可以作为科学和工程前沿的全复杂性系统的预言机。事实上，作为该项目一部分研究的所有具体模型最初都是作为真实物理系统（例如水波、光学和磁流体动力学）的有效模型而产生的，经过简化以揭示观察到的现象背后的中心机制。为了履行刚刚阐明的角色，我们必须在大数据非微扰体系中研究这些可积系统，在这种体系中，更简单的模型无法重现观察到的现象学。这是该研究项目的一个标志，实际上也是 PI 最近工作中开发的更广泛方法的一个标志。该项目为研究生和博士后学者提供了重要的研究培训机会，他们融入到该项目的每一项主要活动中。该项目的主要目标是推进某些完全可积色散偏微分方程的低正则理论，既作为其本身的目的，又作为理解这些系统统计力学的工具。作为实现这一目标的手段，该项目进一步开发了通勤流程范式，并为支持这种方法而开发的几种技术工具寻找新的、多样化的应用。将研究适定性问题，包括导数非线性薛定谔方程的适定性问题。将运用微观守恒定律来进一步阐述通过通勤流构建的解决方案的时空结构特性。将研究重整化和通勤流技术的综合，以适定性、吉布斯态的构造及其动力学特性为目的。同样，将研究离散可积模型的连续极限，以获得确定性初始数据，并作为深入了解吉布斯态动力学的一种手段。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.