Group Analysis of Differential Equations and Integrable Systems − 2018
Alexander V. Mikhailov (University of Leeds, UK & Centre of Integrable Systems, P.G. Demidov Yaroslavl State University, Russia) Integrable Hamiltonian systems, naturally defined by symmetric powers of algebraic curves Abstract:
We have found $k$ integrable Hamiltonian systems on $\mathbb{C}^{2k}$(or on $\mathbb{R}^{2k}$, if the base field is $\mathbb{R}$), naturally defined by a symmetric power ${\rm Sym}^k (V_g)$ of a plain hyperelliptic curve $V_g$ of genus $g$. When $k=g$ the symmetric power ${\rm Sym }^k (V_g)$ is birationally isomorphic to the Jacobian of the curve $V_g$ and our system is equivalent to a well known Dubrovin's system which has been derived and studied in the theory of finite gap solutions (algebrageometric integration) of the Kortewegde Vrise equation. In the case $k=2$ and $g\ge 1$ we have found the coordinates in which the systems obtained and their Hamiltonians are polynomial \cite{bm1}. For $k=2,\ g=1,2,3$ we present these systems explicitly as well as we discuss the problem of their integration \cite{bm2}. In particular, if $k=2, \ g>2$ the solution of the systems is not a $2g$ periodic Abelian function. Most of the results obtained can be easily extended to a wider class of curves, such as nonhyperelliptic and nonplain. We also sure, but have not proved yet, that for $k>2$ there are natural variables in which the systems obtained and their Hamiltonians are all polynomial. [1]
V.M. Buchstaber and A.V. Mikhailov,
Infinite dimensional Lie algebras determined by the space of
symmetric squares of hyperelliptic curves,
Functional Anal. Appl., 51(1):427, 2017.
