Description
In this reading seminar we plan to discuss various aspects of wave motions begining with physical background and motivations. We will introduce basic mathematical techniques and estimates along the way. We will study existence, uniqueness, singularity formation, asymptotics, multi-scaling through reading basic texts and research articles. A particular focus is the phenomena of singularity formation such as blow-up, discontinuity and caustics.

Pre-requisite
Elementary knowledge of partial differential equations.

Basic Texts

• Nonlinear Waves
  by W. Strauss, AMS 1989.
  
• Linear and nonlinear Waves
  G.B. Whitham, 1974.
  

Research articles

• R. Glassey: On the blowing up of solutions to the Cauchy problem for nonlinear Schrodinger equations J. Math. Phys. 18 (1977), 1794-1797.
• T. Kato: Blow-up solutions of some nonlinear hyperbolic equations Commun. Pure Appl. 33 (1980), 501-505.
• T. Kato: Nonlinear equations of evolution in Banach spaces Proc. Sympos. Pure Math. 45 (1986), 9-23.
  
• S. Klainerman: Geometric and Fourier Methods for Nonlinear Wave Equations
• Kirrmann, Schneider and Mielke Validity of modulation equation for extented systems with cubic nonlinearity
• Struwe: Semilinear Wave equations
• B. White: Stochastic Caustic

Lectures

• Lecture 1 (J. Hunter): Overview of wave motions, physical motivations.
• Lecture 2 (J. Hunter): Introduction to hyperbolic conservation laws.
  • Incompressible Euler equations
  • Isentropic gas dynamics
  • Symmetric hyperbolic systems
  • Well-posedness for small and large data
  • Open problems: N X N systems in one dimension, N> 2; Conservation laws in two or higher dimensions
• Lecture 3 (J. Hunter): Introduction to dispersive waves.
• Lecture 4 (J. Hunter): derivation of nonlinear Schrodinger equation from the nonlinear Klein-Gorden equation.
• Lecture 5 (A. Fannjiang): parabolic approximation from Maxwell equation for dielectric materials .
• Lecture 6 (A. Fannjiang): Singularity formation in supercritical self-focusing nonlinear Schrodinger equations.
• Lecture 7 (Wenbin Chen): From Maxwell equations to integrated circuits: modeling and model-reduction.
• Lecture 8 (Brian Wissman): General relativity and gravitational wave equation.

Credits: 1-3 units
Students who make presentation can earn three units.