Rational KdV potentials and Differential Galois Theory

Sonia Jimenez, Juan J. Morales-Ruiz, Raquel Sanchez-Cauce, Maria-Angeles Zurro

Published 2018-08-02Version 1

In this work we study the spectral problem $-\partial^2 +u=E $ where $u$ is a rational solution of the KdV hierarchy and $E$ is a complex parameter. We use some differential Galois techniques by means of Darboux tramsformations. First we extend the family of Adler-Moser polynomials $\theta_n$, to a family of rational functions $Q_{n}^{\pm}$, that seems to be new as far as we know. For the first time, we find a fundamental matrix for these Schr\"odinger operators $-\partial^2 +u-E $ with non zero energies $E$ in terms of $Q_{n}^{\pm}$. Finally we construct the sheaves of differential Galois groups over their corresponding stationary spectral curve. We prove the invariance of the Galois groups with respect to the time, generic values of the spectral parameter and Darboux transformations, for these potentials.