Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 047, 40 pages      arXiv:1808.00743
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Rational KdV Potentials and Differential Galois Theory

Sonia Jiménez a, Juan J. Morales-Ruiz b, Raquel Sánchez-Cauce c and María-Ángeles Zurro c
a) Junta de Castilla y León, Salamanca, Spain
b) Departamento de Matemática Aplicada, E.T.S. Edificación, Universidad Politécnica de Madrid, Madrid, Spain
c) Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

Received September 11, 2018, in final form May 29, 2019; Published online June 25, 2019

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schrödinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schrödinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

Key words: differential Galois theory; KdV hierarchy; Schrödinger operator; Darboux transformations; spectral curves; rational solitons.

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