Deformations of the Hill curves and isoperiodicity in the KdV and the sine-Gordon equations

TL;DR

This paper studies how hyperelliptic curves and associated differentials deform while preserving periods, providing differential equations for these deformations, with applications to maintaining periodic solutions in integrable PDEs like KdV and sine-Gordon.

Contribution

It introduces a system of rational coefficient differential equations describing isoperiodic deformations of hyperelliptic curves, with implications for preserving periodic solutions in integrable systems.

Findings

Derived differential equations for isoperiodic deformations with rational coefficients.

Established conditions for existence and uniqueness of these deformations.

Applied results to maintain periodicity in KdV and sine-Gordon solutions.

Abstract

We consider a family of genus $g$ hyperelliptic curves as double ramified coverings over the Riemann sphere with the set of branch points of the form $\{0, \infty, x_1, \dots, x_g, u_1, \dots, u_g\}$. The branch point at infinity $P_\infty$ is selected to be a marked point on the Riemann surfaces. A meromorphic differential $\Omega$ with a unique pole being of order two at $P_\infty$, is completely defined by the values of half of its periods, the $a$-periods. Fixing values of $a$-periods of $\Omega$, we then find a continuous subfamily in the considered family of hyperelliptic curves along which all the periods of $\Omega$ are constant. This subfamily is defined by the functions $u_j(x_1, \dots, x_g)$, while $x_1, \dots, x_g$ are independent parameters. We derive a system of differential equations for the functions $u_j(x_1, \dots, x_g)$, which, remarkably, has rational coefficients. We call this subfamily the isoperiodic deformations of the hyperelliptic curves relative to the given differential of the second kind $\Omega.$ We deduce necessary and sufficient conditions for the existence and uniqueness of isoperiodic deformations. We discuss reality conditions as well. Using the obtained results, we solve the following problem for the Korteweg-de Vries and sine-Gordon equations: starting from an algebro-geometric data which generate a real periodic solution of a period $T$, how to deform the data, so that the associated solutions remain periodic with the same period $T$.