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

2025/12/05 by Dragovic, Vladimir, Shramchenko, Vasilisa
#14D07 #14H70 #35B10 #Algebraic Geometry (math.AG) #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2512.06168

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, ∞, x₁, …, x_g, u₁, …, u_g\. The branch point at infinity P_∞ 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_∞, 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(x₁, …, x_g), while x₁, …, x_g are independent parameters. We derive a system of differential equations for the functions u_j(x₁, …, 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.

Citations

• Real-normalized differentials and the elliptic Calogero-Moser system
• The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces
• Reality problems in the soliton theory
• A new hierarchy of integrable systems associated to Hurwitz spaces

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• Isomonodromic deformations in genus zero and one: algebrogeometric solutions and Schlesinger transformations