Seminar talk, 18 March 2020
Speaker: Raffaele Vitolo
Title: Jacobi structures of evolutionary partial differential equations
I will report on the paper by Si-Qi Liu and Youjin Zhang published in Adv. Math. 227 (2011) 73-130, https://doi.org/10.1016/j.aim.2011.01.015. In this paper the authors introduce the notion of Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems. They prove that Jacobi structures are invariant under the action of reciprocal transformations that only change the spatial variable.
Then the authors compute the Lichnerowicz-Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. The notion of bi-Jacobi structures is introduced to the purpose of proving integrability of PDEs.