Isoperiodic deformations of Abelian differentials of the second kind over elliptic curves and the Boussinesq equation
Vladimir Dragovic, Vasilisa Shramchenko
TL;DR
The work develops a general isoperiodic deformation theory for genus-one Riemann surfaces with a second-kind Abelian differential, deriving a second-order ODE for the pole position $y_0(x)$ that governs period-preserving deformations. Central to the method are Rauch variational formulas and Bell polynomials, which organize derivatives and yield rational-coefficient equations parameterized by the pole order $n$. The authors provide explicit ODEs for $n=0$ and $n=1$, connect the deformations to genus-one Boussinesq solutions via Baker–Akhiezer constructions, and establish conditions under which isoperiodic deformations yield periodic solutions in the Boussinesq context, with precise relations between the wave number and frequency through elliptic data and theta-functions. The results extend prior work on isoperiodic deformations to higher-order poles and illuminate the link between isoperiodicity and integrable PDEs in the genus-one setting, offering a framework for generating families of periodic solutions. The work is relevant for both the theory of Riemann-surface deformations and the construction of explicit, period-preserving solutions to 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.