Hyperelliptic reduction of the Benney moment equations
Author(s)
Baldwin,S.
Gibbons,J.
Type
Thesis
Abstract
We consider N-parameter reductions of the Benney moment equations. These were shown in Gibbons and Tsarev (1996 Physics Letters A 211 19, 1999 Physics Letters A 258 263 ) to correspond to N−parameter families of conformal maps and to satisfy a particular system of PDE. A specific known example of this, the (N = 2) elliptic reduction (L Yu and J Gibbons 2000 Inverse Problems 16 605) is described. We then consider an analogous reduction for a genus 2 hyperelliptic curve (N = 3). The mapping function λ is given by the inversion of a 2nd kind Abelian integral on the Θ−divisor, Θ1. This is found explicitly following a method given by Enolskii, Pronine and Richter (2003 J. Nonlinear Science 13 157). Key to this is the identification of PDE satisfied on Θ1. We then consider the general case for N > 3. Again, the mapping function λ is calculated explicitly by inverting a second kind Abelian integral on the stratum Θ1 of the genus g Jacobi variety. This is done using a method based on the result of Jorgenson (1992 Isr. J Math. 77 273). The approach used for the hyperelliptic reductions is then applied to a genus 4 trigonal reduction and the function λ is calculated explicitly. A secondary result of this calculation is an expansion of the leading terms of the σ function for this family of curves.
Version
Imperial Users only
Date Issued
2003
Date Awarded
2006-01
Citation
Journal of Physics A - Mathematical and General, 2003, 31, 36, pp.8393-8418
ISSN
0305-4470
Start Page
8393
End Page
8418
Journal / Book Title
Journal of Physics A - Mathematical and General
Format Extent
905251 bytes
Volume
36
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
Gibbons, John
Creator
Baldwin, Sadie A
Publisher Department
Mathematics
Publisher Institution
University of London - Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)
Edition
31