Isoperiodic deformations of Abelian differentials of the second kind over elliptic curves and the Boussinesq equation

TL;DR

This paper investigates isoperiodic deformations of Abelian differentials on elliptic curves, deriving differential equations governing pole variations, and applies these findings to genus one solutions of the Boussinesq equation.

Contribution

It introduces a second order differential equation characterizing isoperiodic deformations of Abelian differentials of the second kind on elliptic curves and connects these deformations to solutions of the Boussinesq equation.

Findings

Derived a differential equation with rational coefficients for pole variations.

Characterized solutions corresponding to isoperiodic deformations.

Applied results to genus one solutions of the Boussinesq equation.

Abstract

We study deformations of a genus one Riemann surface and of a second order Abelian differential on the surface which preserve the periods of the differential with respect to a chosen canonical homology basis of the surface. We call these deformations isoperiodic. We derive a second order ordinary differential equation with rational coefficients governing the variations of the position of the unique pole of the differential under the isoperiodic deformations. The obtained equation depends on the order of the pole of the differential. We characterize the solutions of the obtained ordinary differential equations that correspond to the isoperiodic deformations. We apply these results to the theory of genus one solutions to the Boussinesq equation.