The main accomplishments of this monograph are (a) that the theory is developed for an abstract class of Hamiltonian PDEs. Since almost all Hamiltonian PDEs known to the authors can be multisymplectified, this class of PDEs is quite general. (b) Alternatively, an entire abstract theory can be developed purely from geometry, building the PDEs starting from the geometry of space-time, and manifolds built on space-time. (c) The basic states are constructed based on a two-parameter variational principle for oblique line solitary waves, and the parameter structure of the variational principle feeds into the linear stability problem. (d) New algebraic structure, in the form of three independent symplectic forms, of the Evans function in \(2+1\) is constructed. (e) A general instability condition is proved that uses only the geometric properties of the family of existing solitary wave. (f) The Maslov index arises in the linear instability condition in a surprising way. (g) An explicit example is constructed which illustrates the theory as well as the geometry of the linear stability problem.