Spectral Stability of Solitary Waves and Undercompressive Shocks

This dissertation establishes spectral stability of traveling waves in two different settings. In the first part, we prove stability of solitary waves in Hamiltonian partial differential equations, notably in the generalized Korteweg-de Vries equation, the generalized Boussinesq equation, and equations which are closely related with both. Under natural and physically meaningful assumptions on the nonlinearity, we establish stability of large- and small-amplitude solitary waves in this context. In the second part, we prove spectral stability of small undercompressive shocks in viscous systems of non-strictly hyperbolic conservation laws. Here, at a point $v_*\in\R^n,$ the nonlinearity $f(v)$ has the property that two of the eigenvalues of $Df(v_*)$ coincide. We show that in this situation, which frequently arises in applications, the essential dynamics are governed by the dynamics of an associated two dimensional system. We finish this thesis with a careful investigation of the latter