Varieties of elliptic solitons (Q2716889)

An \((n,s)\)-curve is a curve of the form \(w^n-z^s+\ldots =0\) where \(n\), \(s\) are coprime positive integers; dots indicate lower-order terms in the quasi-homogeneous filtration defined by \(w^n\) and \(z^s\). A set of \((n,s)\)-curves is considered which cover elliptic curves so that the associated \(\sigma\)-function can be factored NEWLINE\[NEWLINE\sigma (t_1,\ldots ,t_{g-1},x)=\prod \sigma_{\text{W}}(x-x_i(t_1,\ldots ,t_{g-1})), \quad t_g=x,NEWLINE\]NEWLINE where \(\sigma _W\) is the Weierstrass elliptic function and \(t_i\) are the `times' of the integrable hierarchy or coordinates of the Jacobi variety of the curve. The functions \(x_i\) are evaluated on the flow variables of the elliptic Calogero-Moser system. NEWLINENEWLINENEWLINESuppose that the curve covers additional tori and that all independent holomorphic differentials are reduced to elliptic ones by some rational substitution. The Jacobi inversion problem for the particles \(x_i\) (which is reduced to the inversion of elliptic integrals) is solved in terms of Kleinian functions. Inversion of the elliptic integrals leads to algebraic varieties whose coordinates are elliptic functions in the `times' \(t_i\) and elliptic functions associated with the torus parallel to the \(x\) flow. An example is given for \((3,4)\)-curves describing the Calogero-Moser dynamics associated with the Boussinesq flow.