# Registered Data

## [00234] Differential Galois Theory and Integrability of Dynamical Systems

**Session Date & Time**:- 00234 (1/2) : 2D (Aug.22, 15:30-17:10)
- 00234 (2/2) : 2E (Aug.22, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: The main objective of this minisymposium is to bring together researchers working on differential Galois theory and integrability of dynamical systems and to discuss recent results on the related topics containing the following: - Developments of differential Galois theory in dynamical systems - Integrability of dynamical Systems and PDE’s - Integrability in quantum mechanics and spectral theory - Galois approach to nonintegrability**Organizer(s)**: Kazuyuki Yagasaki**Classification**:__34M15__,__37J30__,__12H05__,__37J35__,__37K10__**Speakers Info**:- Xiang Zhang (Shanghai Jiaotong University)
- Juan Jose Morales-Ruiz (Universidad Politecnica de Madrid)
- Holger Dullin (The University of Sydney)
- Maria-Angeles Zurro (Universidad Autonóma de Madrid)
- Shoya Motonaga (Ritsumeikan University)
**Kazuyuki Yagasaki**(Kyoto University)- Zbigniew Hajto (Jagiellonian University)
- Thierry Combot (University of Burgundy)

**Talks in Minisymposium**:**[03161] Non-integrability of a model of two tethered satellites****Author(s)**:**Thierry Combot**(Universite de Bourgogne)

**Abstract**: We study the integrability of a model of two tethered satellites whose centre of mass moves in a circular Keplerian orbit around a gravity centre. When tether rest length is zero, the model is integrable and even superintegrable for selected values of the parameters. For positive rest length, the system is non-integrable. Obstructions to integrability are obtained through study of the differential Galois group of an irreducible symplectic variational equation in dimension 4.

**[03529] Real Liouvillian extensions of partial differential fields****Author(s)**:**Zbigniew Hajto**(Faculty of Mathematics and Computer Science UJ)

**Abstract**: I will present Galois theory for partial differential systems defined over formally real differential fields with a real closed field of constants and over formally $p$-adic differential fields with a $p$-adically closed field of constants. For an integrable partial differential system, there exists a formally real (resp. formally $p$-adic) Picard-Vessiot extension. I will explain the applications of this theorem.

**[03566] Local integrability and regularity of autonomous differential systems****Author(s)**:**Xiang Zhang**(Shanghai Jiao Tong University)

**Abstract**: For finite dimensional smooth autonomous differential systems, which have a singularity with one zero eigenvalue and the others nonresonant, we present our results on existence of local first integrals at the singularity, with emphasis on the regularity of the local first integrals. We first explore the existence of $C^\infty$ local first integrals for analytic differential systems under the Poincar\'e's non-resonant condition. We then show that for the Gevrey class of vector fields, their local first integrals have the same regularity as that of the vector fields provided that the real parts of the nonresonant eigenvalues are all positive or all negative. Lastly, a sharper expression of the loss of the regularity is presented by the lowest order of the resonant terms together with the indices of Gevrey smoothness and the diophantine condition for the case that the Jacobian matrix of the vector field at the singularity is in the diagonal form. The main tools are the homological method, the KAM theory, and the Gevrey normalization theory.

**[04369] Singular solitary waves in the KdV equation****Author(s)**:**Kazuyuki Yagasaki**(Kyoto University)

**Abstract**: In this talk we consider the KdV equation and discuss its singular solitons which are called rational solitons, positons or negatons. We are especially interested in the solvability by quadrature for Schrodinger equations appearing as one of the related Lax pair. Some formulas for scattering coefficients of positons or negatons are also given. This is joint work with Katsuki Kobayashi.

**[04393] Obstructions to integrability of nearly integrable dynamical systems****Author(s)**:**Shoya Motonaga**(Ritsumeikan University)

**Abstract**: We study necessary conditions for the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems including non-Hamiltonian ones in the sense of Bogoyavlenskij. Moreover, we compare our results with the classical results of Poincar\'{e} and Kozlov for systems written in action and angle coordinates and discuss their relationships with the Melnikov methods for periodic perturbations of single-degree-of-freedom Hamiltonian systems. This is joint work with Kazuyuki Yagasaki at Kyoto University.

**[04649] Korteweg-de Vries traveling waves and Differential Galois Theory****Author(s)**:**Maria-Angeles Zurro**(Autonomous University of Madrid)

**Abstract**: It was conjectured that the abelianity of the identity component of the Galois group of the variational equation is a necessary condition for the integrability of the non-linear PDE itself. In my lecture I will present an algebraic and spectral study of the variational equation around a KdV solitonic potential from the point of view of Galois differential theory. This is part of an ongoing joint work with J. J. Morales-Ruiz and J.-P. Ramis.

**[04873] A Tale of Two Polytopes related to geodesic flows on spheres****Author(s)**:**Holger Rainer Dullin**(University of Sydney)- Diana Nguyen (University of Sydney)
- Sean Dawson (University of Sydney)

**Abstract**: Separation of variables for the geodesic flows on round spheres leads to a large family of integrable systems whose integrals are defined through the separation constants. Reduction by the periodic geodesic flow leads to integrable systems on Grassmanians. Specifically for the geodesic flow on the round $S^3$ the reduced system defines a family of integrable systems on $S^2\times S^2$. We show that the image of these systems under a continuous momentum map defined through the action variables has a triangle as its image. The image is rigid and does not change when the integrable system is changed within the family. Each member of the family can be identified with a point inside a Stasheff polytope. Corners of the polytope correspond to toric systems (possibly with degenerations), edges correspond to semi-toric systems (in various meanings of the word), and the face corresponds to ``generic'' integrable systems. A fundamental difference of this momentum map to that of a toric or semi-toric system is that the number of tori in the preimage of a non-critical point may be 1, 2, or 4. The momentum map is continuous but not smooth along the images of hyperbolic singularities. The corresponding quantum problem and generalisations to higher dimensional spheres will be discussed.

**[05314] The geodesic deviation equation for null geodesics in the Schwarzschild black-hole****Author(s)**:**Juan José Morales-Ruiz**(Universidad Politécnica de Madrid)- Alvaro Pérez-Raposo (Universidad Politécnica de Madrid)

**Abstract**: The Schwarzschild black-hole is an integrable Hamiltonian system with four degrees of freedom. The geodesic deviation equations are the variational equations for this Hamiltonian system. By a joint theorem with Ramis, these equations can be solved in closed form in the framework of the differential Galois theory. This talk will be devoted to give the solutions of these equations around some null geodesics. This a joint work with Álvaro Pérez-Raposo.