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:
In collaboration with Victor M. Buchstaber.

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 bi-rationally 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 (algebra-geometric integration) of the Korteweg-de 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 non-hyperelliptic and non-plain. 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):4-27, 2017.
[2] V.M. Buchstaber and A.V. Mikhailov, The space of symmetric squares of hyperelliptic curves and integrable Hamiltonian polynomial systems on $\mathbb{R}^4$, arXiv:1710.00866, 2017.