This award will suuport Nahmod's research on geometric partial differential equations and also on the analysis of multilinear pseudodifferential operators. Of special interest are the short and long time behavior of nonlinear waves arising in geometry, ferro-magnetism and gauge field theories; and the development of the time frequency analysis techniques so successfully used to study multi-linear singular operators in one dimension to both develop the $x$-dependent and non-tensorial higher dimensional situations. These two areas come together by way of wave-packet decompositions and frequency interactions estimates needed in the study of nonlinear partial differential equations. The geometric Hamiltonian PDEs to be investigated include the Schroedinger map equation and the Ishimori system - both of which may be characterized variationally and model specific wave-like phenomena. Nahmod will attempt to prove existence results for the Cauchy problem for these systems and also to study stability and blow-up questions of special solutions exploiting the geometric features of the systems. The second topic is to study multi-linear pseudo-differential operators. Their treatment departs from the classical multi-linear theory for in the present situation, the symbols' behavior may be governed by a variety that's allowed to change at each spatial point.
The dynamics of many real world physical systems can be described by geometric evolution equations, in particular geometric Hamiltonian partial differential equations. The dynamics of the magnetization field in a ferromagnetic material for example, is described by the Landau-Lifshitz equation. The theory of Schroedinger maps is to model the long wave length limit of an isotropic Heisenberg ferromagnetic lattice. PDEs are the mathematical models of the laws governing much of the phenomena in the physical world. Wave equations model the propagation of different kind of waves - such as light waves- in homogeneous media. Nonlinear models of conservative type arise in quantum mechanics while other variants appear for example, in the study of vibrating systems and semiconductors. The nonlinear Schroedinger equations are fundamental physical equations for they govern the motion of quantum particles, such as electrons. All of these equations have applications to diverse physical problems, e.g. the dynamics of nonlinear waves through optical fibers in which the index of refraction is sensitive to the wave amplitude, and waves at the free surface of an ideal fluid or plasma. Nonlinear Fourier analysis and in particular adapted phase analysis - "wave-packet or time-frequency analysis"- consists in decomposing complex structures into basic building blocks - which are localized and easier to understand - via modulated waveforms. And then piecing them back together in a straightforward manner. It works very similarly to a musical score. These modulated waveforms possess capture amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The objects could be speech, radar signals, as well as oscillatory expressions arising in optics, wave propagation and other phenomena of nonlocal nature.
