Functional spaces : a direct approach
Barakat, Mohamed; Plesken, Wilhelm (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2001, 2002)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2001
In this thesis I define the notion of a functional tensor space, and derive the specific form of the Lie derivative (w.r.t. generalized vector fields) on the constructed spaces, giving them a Lie module structure over the Lie algebra of generalized vector fields. I also construct the Euler complex in a recursive manner using the celebrated Cartan formula, and write down the first three operators explicitly. I call the third one, the Takens operator. Then, in the context of functional spaces, I prove and use the fact, that a Hamiltonian structure of an evolution equation is invariant under the flow of the equation, to derive all Hamiltonian structures (up to a certain order) of KdV equation and the Boussinesq equation. These are well known examples for nonlinear completely integrable evolution equations.