Deformations of the Hill curves and isoperiodicity in the KdV and the sine-Gordon equations
Vladimir Dragovic, Vasilisa Shramchenko
TL;DR
This work develops a general theory of isoperiodic deformations for families of hyperelliptic curves carrying a meromorphic differential of the second kind with a single second-order pole at the marked infinity. By exploiting Rauch variation formulas and a carefully chosen basis of holomorphic differentials, the authors derive a rational-coefficient system for the dependent branch points that preserves all periods of the differential, including explicit genus-one and higher-genus cases. The framework yields concrete applications to integrable systems: deformations that preserve the period of finite-gap KdV and sine-Gordon solutions, as well as realizations for Hill curves, two-gap potentials, and Neumann problems, with reality conditions carefully treated. A second, combinatorial viewpoint ties the deformations to comb-region dynamics and Loewner-type conformal maps, illustrating the broad reach of isoperiodicity in mapping algebro-geometric data to physically meaningful, periodic solutions. Overall, the paper provides a rigorous, constructive approach to maintaining periodicity while varying spectral data, enriching the toolbox for spectral theory, integrable systems, and real-algebro-geometric analysis.
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 $Ω$ 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 $Ω$, we then find a continuous subfamily in the considered family of hyperelliptic curves along which all the periods of $Ω$ 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 $Ω.$ 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 $T$.