Transformation theory of Hamiltonian PDE and the problem of water waves

Abstract

This set of lecture notes gives (i) a formal theory of Hamiltonian systems posed in infinite dimensions, (ii) a perturbation theory in the presence of a small parameter, adapted to reproduce some of the well-known formal computations of fluid mechanics, and (iii) a transformation theory of Hamiltonian systems and their symplectic structures. A series of examples is given, starting with a rather complete description of the problem of water waves, and, following a series of scaling and other simple transformations placed in the above context, a derivation of the well known equations of Boussinesq and Korteweg deVries. 1 Hamiltonian systems A Hamiltonian system is given in terms of a Hamiltonian function H : M → R, where M is the phase space. We will restrict ourselves to phase spaces which are Hilbert spaces, denoting the inner product between two vectors V1,V2 ∈ T (M) by � V1|V2�. The symplectic structure is as usual given by a two-form ω on (M), which can be represented by the inner product, namely ω(V1,V2 )= � V1|J −1 V2�, where, because of the antisymmetry of two-forms, the operator J satisfies J −T = −J −1 . The Hamiltonian vector field XH is defined through the relation dH(V )= ω(V, XH ) which is asked to hold for all V ∈ T (M). The system of equations that we study, known as Hamilton's canonical equations, is given by