DART XII

Kassel (Germany) April 9^{th} - 12^{th} 2024

XII. Conference on Differential Algebra and Related Topics

April 9^{th} - 12^{th} 2024

Kassel University, Germany

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# Maria-Angeles Zurro: *A Galoisian approach to spectral problems for third-order ODOs*

The algebraic study of the spectral problem $Ly=\lambda y$ for a third-order ordinary differential operator $L$ is closely related to the study of the centralization of $L$ in the ring of differential operators. I will present the study of a factorization of the operator $L-\lambda$ in the case of having a non-trivial centralizer, that is, in the case that its centralizer is not a ring of polynomials in $L$. An algebraic curve is needed to perform such a parametric factorization, the spectral curve. The techniques used extend those built for the study of the second-order algebrogeometric operator, see [1] and [2]. As an application, I will show a Galoisian study of the variational equation of the Korteweg-de Vries equation around a cnoidal wave.

The work presented is essentially based on a joint work with Sonia Luisa Rueda. If time permits, I will also mention a joint work in progress with Juan J. Morales-Ruiz and Jean-Pierre Ramis.

[1] J.J. Morales-Ruiz, S.L. Rueda, and M.A. Zurro, Factorization of KdV SchrÃ¶dinger operators using differential subresultants, Adv. Appl. Math. 120, 102065 (2020)

[2] J.J. Morales-Ruiz, S.L. Rueda, and M.A. Zurro, Spectral Picard-Vessiot fields for Algebro-geometric SchrÃ´dinger operator}, Annales de l'Institut Fourier 71:3, 1287-1324 (2021)