Normal forms for integrable PDEs and billiards

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is concerned with a variety of problems related to the construction of normal forms and their various applications to well-posedness of solutions and perturbation theory for classes of integrable nonlinear partial differential equations. It also explores potential applications of these ideas to the dynamics and the spectral geometry of a class of integrable billiard systems. More specifically, it is proposed to study the focusing and defocusing nonlinear Schrodinger equation and the focusing modified Korteweg-de Vries equation. The principal investigator aims to construct various normal forms for these equations in the phase space of square-integrable, 1-periodic functions and in certain spaces of distributions. The analysis of the corresponding frequency maps would eventually lead to new well-posedness results and to various applications to perturbation theory. The principal investigator proposes to establish spectral rigidity of a class of Riemannian manifolds with boundary (known as "Liouville billiard tables") by introducing new techniques involving quasimodes and a generalization of the Radon transform. The dynamical properties of the corresponding discrete billiard system will play a crucial role in the proof of the claimed spectral rigidity. The scope of the project lies in the development of new methods coming from the theory of Hamiltonian systems to fundamental problems in analysis, geometry, and the theory of integrable nonlinear partial differential equations. Many of the results obtained by these methods cannot be obtained by the traditional and more general methods for solving such equations. Beyond applications to well-posedness and perturbation theory, normal forms provide a very detailed description of the solutions via so-called Birkhoff coordinates. The proposed activity would lead to a better understanding of the dynamics and the stability properties of the solutions of a class of nonlinear evolution equations that arise in various physical systems such as water waves, plasma physics, solid-state physics, nonlinear optics, and fluid mechanics. Recent applications of the activity include new results in statistical mechanics (preservation of white noise) and applications to astrophysics, namely, the problem of studying light rays moving on black hole backgrounds in the presence of a cosmological constant.