Abelian solitons

We describe a new algebraically completely integrable system, whose integral manifolds are co-elliptic subvarieties of Jacobian varieties. This is a multi-periodic extension of the Krichever-Treibich-Verdier system, which consists of elliptic solitons. The goal of this work is to generalize the theory of elliptic solitons, which was developed by A. Treibich and J.-L. Verdier based on earlier works by H. Airault, H.P. McKean and J. Moser [AMcKM], and I.M. Krichever [K]. Elliptic solitons are a subclass of algebrogeometric KP solutions (cf. §1), namely those that are elliptic functions in the x variable. Viewed as Jacobians which contain special configurations of elliptic curves, the elliptic solitons are dense in the classical topology [CPP]. We pose the problem of describing the moduli of abelian solitons, that is, KP-solitons whose first k ≥ 1 flow variables are tangent to some proper abelian subvariety of the Jacobian. We explore several possible constructions of such abelian solitons. Most of these actually fail, cf. the appendix. We do succeed in constructing, for any natural number k and elliptic curve C, an infinite sequence of families of co-elliptic solitons, which are periodic along k-dimensional abelian subvarieties of (k + 1)-dimensional Jacobians, with quotients isogenous to the given elliptic curve C. We show (Theorem 2.1) how these families of coelliptic solitons organize into algebraically integrable systems (aci), which can be identified as nonlinear subsystems of Markman’s system of meromorphic Higgs bundles.