F. Gesztesy, H. Holden
Dubrovin equations and
integrable systems on
hyperelliptic curves
(95K, LATeX 2e)
ABSTRACT. We introduce the most general version of Dubrovin-type
equations for divisors on a hyperelliptic curve
of arbitrary genus, and provide a new argument for linearizing
the corresponding completely integrable flows. Detailed
applications to completely integrable systems, including the
KdV, AKNS, Toda, and the combined sine-Gordon and mKdV
hierarchies, are made. These investigations uncover a new
principle for $1+1$-dimensional integrable soliton equations
in the sense that the Dubrovin equations, combined with
appropriate trace formulas, encode all hierarchies of soliton
equations associated with hyperelliptic curves. In other words,
completely integable hierarchies of soliton equations
determine Dubrovin equations and associated trace
formulas and, vice versa, Dubrovin-type equations combined
with trace formulas permit the construction of
hierarchies of soliton equations.