Instability of variable media to long waves with odd dispersion relations

The instability of variable media to a broad class of long waves having dispersion relations that are an odd function of wavenumber is examined. For Hamiltonian media, new necessary conditions for the existence and structure of global modes are obtained. For non-Hamiltonian media, an analysis of the complex WKB branch points yields explicit expressions for the frequency and structure of the global modes, which manifest as spatially oscillatory wave packets or smooth envelope structures. These distinct modes and their locations within the media can be predicted by simply examining the local convergence or divergence of the group velocity in the long wave limit.