Assess graduate students' ability to analyze orbital and asymptotic stability of solitary waves for nonlinear Schrödinger and Klein–Gordon equations using spectral and modulation methods. Topics include linearization and spectral decomposition (point vs continuous spectrum), Evans function and eigenvalue bifurcation, Grillakis–Shatah–Strauss criteria, derivation and use of modulation equations, dispersive/Strichartz estimates, and the Fermi Golden Rule; expect proofs of spectral criteria, derivation of modulation dynamics, and arguments establishing radiation damping or instability.