On the Conserved Quantities of the Vortex Filament Equation
Fenici, Elena Cristina
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In this Master Thesis, we take a closer look at the family of conserved quantities of the Vortex Filament Equation (VFE), a model of vortex filament dynamics in an ideal fluid and a well-known example of completely integrable partial differential equation (PDE). We use a recursion scheme proposed by Joel Langer to generate the VFE conserved quantities, and to compare them with the conserved quantities for the Nonlinear Schrodinger equation, an equivalent completely integrable PDE for a complex function of the curvature k and torsion tau of the filament. Next, we study the Brylinski Beta Function, a quantity that associates a meromorphic function to any knot, which generalizes the idea of the well-known knot energy functional. We show that the integrands of its residues are linear combinations of monomials in k, tau and their derivatives, all of the same weight, matching the monomials that appear in expression of the even-numbered conserved densities of the VFE. In searching for a Brylinski-type function whose residues coincide with the VFE conserved quantities (i.e. a generating functional for the conserved quantities), we propose a new meromorphic function that generalizes the self-linking number of a knot. We show that the integrands of its residues are linear combinations of differential monomials in k and tau, all of the same weight, matching the monomials in the expression of the odd-numbered conserved quantities of the VFE.