Evaluate graduate students' ability to analyze stability and bifurcations of solitary-wave solutions of nonlinear Schrödinger equations by applying spectral (linearization, eigenvalue counts, Evans function), variational (constrained minimization, Vakhitov–Kolokolov, energy–momentum), and modulational (modulation equations, Lyapunov–Schmidt) techniques. Tasks may require deriving spectral stability criteria, computing Morse indices of linearized operators, identifying bifurcation types (saddle-node, pitchfork, Hamiltonian–Hopf) under parameter variation, and formulating modulation dynamics for slow instabilities; state relevant assumptions (decay, symmetry, nondegeneracy), function spaces, and boundary conditions. Problems can combine rigorous arguments, formal asymptotics, and interpretation of numerical spectra.