Seminar GDEq. Bi-Hamiltonian systems from homogeneous operators
18.03.26 Speaker: Raffaele Vitolo Title: Bi-Hamiltonian systems from homogeneous operators Abstract: Many "famous" integrable systems (KdV, AKNS, dispersive water waves, etc.) have a bi-Hamiltonian pair of the following form: A_1 = P_1 + R_k and A_2 = P_2, where P_1, P_2 are homogeneous first-order Hamiltonian operators and Rk is a homogeneous Hamiltonian operator of degree (order) k. The Hamiltonian property of P_1, P_2 and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of R_k was studied in the past (Doyle, Potemkin; k = 2, 3) and recently revisited with interesting results. In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between P_i and R_k, k = 2, 3.] See the recent papers arXiv:2602.14739 (https://arxiv.org/abs/2602.14739), arXiv:2407.17189 (https://arxiv.org/abs/2407.17189), arXiv:2311.13932 (https://arxiv.org/abs/2311.13932). Joint work with P. Lorenzozi and S. Opanasenko. Geometry of differential equations Страница семинара - https://gdeq.org/Seminar Плейлист на YouTube - https://www.youtube.com/playlist?list=PLp9ABVh6_x4HN6Cf6w0mau4KXaaoVmX3w Плейлист на RuTube - https://rutube.ru/plst/893946 The seminar works at the Independent University of Moscow - https://mccme.ru/ru/nmu/
18.03.26 Speaker: Raffaele Vitolo Title: Bi-Hamiltonian systems from homogeneous operators Abstract: Many "famous" integrable systems (KdV, AKNS, dispersive water waves, etc.) have a bi-Hamiltonian pair of the following form: A_1 = P_1 + R_k and A_2 = P_2, where P_1, P_2 are homogeneous first-order Hamiltonian operators and Rk is a homogeneous Hamiltonian operator of degree (order) k. The Hamiltonian property of P_1, P_2 and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of R_k was studied in the past (Doyle, Potemkin; k = 2, 3) and recently revisited with interesting results. In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between P_i and R_k, k = 2, 3.] See the recent papers arXiv:2602.14739 (https://arxiv.org/abs/2602.14739), arXiv:2407.17189 (https://arxiv.org/abs/2407.17189), arXiv:2311.13932 (https://arxiv.org/abs/2311.13932). Joint work with P. Lorenzozi and S. Opanasenko. Geometry of differential equations Страница семинара - https://gdeq.org/Seminar Плейлист на YouTube - https://www.youtube.com/playlist?list=PLp9ABVh6_x4HN6Cf6w0mau4KXaaoVmX3w Плейлист на RuTube - https://rutube.ru/plst/893946 The seminar works at the Independent University of Moscow - https://mccme.ru/ru/nmu/