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UNIVERSITY OF BRITISH COLUMBIA MATH 557 SPRING 2010 LINEAR AND NONLINEAR WAVES Description: Our goal is to understand the qualitative behaviour of solutions of linear and nonlinear wave equations which are important in areas such as physics, applied math, and geometry. Our examples will mostly be equations of dispersive type, such as wave, Klein-Gordan, Schrödinger, and KdV type equations, and wave and Schrödinger maps (geometric equations). Some of the fundamental questions we consider: Can solutions be defined locally in time? Do solutions exist for all time, or do they “blow up” at finite time? If the former, how do they “look” after a long time? Do they become trivial? Do they settle down to some interesting configuration? Outline: 1. Linear equations: solution formulas, group velocity, stationary phase, decay and Strichartz estimates. 2. Nonlinear equations: Hamiltonian structure, conserved quantities, and solitary waves. 3. Local wellposedness: Existence and uniqueness of time local solutions, continuous dependence on data. 4. Global wellposedness vs. finite time singularity. 5. Scattering theory. 6. Orbital stability of solitary waves. 7. Introduction to some geometric equations. References: 1. Thierry Cazenave: Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10. 2003. 2. Jalal Shatah and Michael Struwe: Geometric wave equations, Courant Lecture Notes in Mathematics 2. 1998. 3. Walter A. Strauss: Nonlinear wave equations, CBMS Regional Conference Series in Mathematics, 73. American Mathematical Society, Providence, RI, 1989. Prerequisites: Basic properties of Sobolev spaces and Fourier transform will be needed throughout the course, and will be reviewed in the first lecture. Course homepage: http://www.math.ubc.ca/∼ttsai/math557/ Evaluation: Biweekly assignments. Instructor: Tai-Peng Tsai, Math 109, phone 604-822-2591, [email protected]