Solitary waves are critical points of Hamiltonians in the constrained variational principle. Spectral stability of solitary waves follows from the convexity of the second variation of the constrained Lyapunov functional. Spectral instabilities occur when the second variation is not sign-definite. Alternatively, spectral stability and instability theory is developed from analysis of eigenvalues of non-self-adjoint operators, which are compositions of symplectic and self-adjoint operators.

Using the Hamiltonian formalism of the nonlinear PDEs, we study eigenvalues of the spectral stability problem for solitary waves in the NLS, Dirac, and KdV-type equations. Using the spectral methods in Hilbert spaces, we find rigorous bounds on the number of unstable eigenvalues in the stability problems. Using dynamical system methods, we derive normal forms for nonlinear instability-induced dynamics of solitary waves. Applications include optical solitons, Bose-Einstein condensates, photonic materials, and water wave solitons.

Vitali Vougalter

• Sylvester's inertial laws and stability theorems in the coupled NLS equations

• Completeness of negative index for energy of standing waves in 3D-NLS

• Nonlinear instability of standing waves of minimal energy