Assess graduate-level ability to analyze spectral and nonlinear stability of solitary waves for the nonlinear Schrödinger and Korteweg–de Vries equations by (i) formulating the linearized spectral problem, (ii) constructing and applying the Evans function to locate eigenvalues and determine spectral stability, (iii) using Grillakis–Shatah–Strauss theory (including Vakhitov–Kolokolov-type conditions and coercivity of constrained Hessians) to infer orbital stability, and (iv) deriving modulation equations to study long-time dynamics and asymptotic stability; questions will require derivations, spectral counts, and verification of hypotheses for example solitary-wave families.