A Geometric Formulation of the Conservation of Wave Action and Its Implications for Signature and the Classification of Instabilities
Abstract
Action, symplecticity, signature and complex instability are fundamental concepts in Hamiltonian dynamics which can be characterized in terms of the symplectic structure. In this paper, Hamiltonian PDEs on unbounded domains are characterized in terms of a multisymplectic structure where a distinct differential twoform is assigned to each space direction and time. This leads to a new geometric formulation of the conservation of wave action for linear and nonlinear Hamiltonian PDEs, and, via Stokes's theorem, a conservation law for symplecticity. Each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode. The first invariant in this family is classical Krein signature (or energy sign, when the energy is time independent) and the other (spatial) signatures are energy flux signs, leading to a classification of instabilities that includes information about directional spatial spreading of an instability. The theory is applied to several examples: the Boussinesq equation, the waterwave equations linearized about an arbitrary Stokes's wave, rotating shallow water flow and flow past a compliant surface. Some implications for nonconservative systems are also discussed.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 July 1997
 DOI:
 10.1098/rspa.1997.0075
 Bibcode:
 1997RSPSA.453.1365B