Mapping properties of normal forms transformations for water waves

Abstract

It is a well known fact, dating from Zakharov (J Appl Mech Tech Phys 9:190–194, 1968), that the equations for free surface water waves can be posed as a Hamiltonian system. It is also known that the Hamiltonian involves no three-wave resonances, and that in the case of dimension \(n=2\) and in infinite depth, the four-wave resonances have a special integrable structure (Dyachenko and Zakharov, Phys Lett A 190:144–148, 1994; Craig and Worfolk, Phys D 84:513–531, 1995). The implication, at least on a formal level, is that cubic terms and nonresonant quartic terms can be removed from the Hamiltonian by suitable canonical transformations, and thus these terms do not enter the equations of motion. The resulting Hamiltonian is said to be in third order (respectively, fourth order) Birkhoff normal form. In this article we study the function space mapping properties of the third and fourth order canonical transformations to Birkhoff normal form, in the case of spatially periodic data in dimension \(n=2\). We find that the third order normal forms transformation is defined in a neighborhood of zero in the Hilbert space \(H^r\times H^r\) for any \(r > 3/2\), and somewhat surprisingly the transformation is based on the time-one flow of Burger’s equation. For the fourth order Birkhoff normal form, we derive a clean expression for the formal normal form, but also we find that the vector field of the Hamiltonian flow that would give rise to this transformation is not bounded on any Sobolev space of the form \(H^r\times H^s\). Alternatively, we give a partial normal form, which results in the elimination of many, but not all, of the nonresonant quartic terms of the Hamiltonian.