Seminar talk, 18 March 2026: Difference between revisions
Created page with "{{Talk | 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: <math>A_1 = P_1 + R_k</math> and <math>A_2 = P_2</math>, where <math>P_1</math>, <math>P_2</math> are homogeneous first-order Hamiltonian operators and <math>R_k</math> is a homogeneous Hamiltonian operator of degree (order) <math>k</math>. Th..." |
No edit summary |
||
| (One intermediate revision by the same user not shown) | |||
| Line 2: | Line 2: | ||
| speaker = Raffaele Vitolo | | speaker = Raffaele Vitolo | ||
| title = Bi-Hamiltonian systems from homogeneous operators | | 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: <math>A_1 = P_1 + R_k</math> and <math>A_2 = P_2</math>, where <math>P_1</math>, <math>P_2</math> are homogeneous first-order Hamiltonian operators and <math>R_k</math> is a homogeneous Hamiltonian operator of degree (order) <math>k</math>. The Hamiltonian property of <math>P_1</math>, <math>P_2</math> and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of <math>R_k</math> was studied in the past (Doyle, Potemin; <math>k=2,3</math>) and recently revisited with interesting results. | | abstract = Many "famous" integrable systems (KdV, AKNS, dispersive water waves, etc.) have a bi-Hamiltonian pair of the following form: <math>A_1 = P_1 + R_k</math> and <math>A_2 = P_2</math>, where <math>P_1</math>, <math>P_2</math> are homogeneous first-order Hamiltonian operators and <math>R_k</math> is a homogeneous Hamiltonian operator of degree (order) <math>k</math>. The Hamiltonian property of <math>P_1</math>, <math>P_2</math> and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of <math>R_k</math> was studied in the past (Doyle, Potemin; <math>k=2,3</math>) and recently revisited with interesting results. | ||
In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between <math>P_i</math> and <math>R_k</math>, <math>k=2,3</math>. | In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between <math>P_i</math> and <math>R_k</math>, <math>k=2,3</math>. | ||
See the recent papers {arXiv|2602.14739}, {arXiv|2407.17189}, {arXiv|2311.13932}. | See the recent papers {{arXiv|2602.14739}}, {{arXiv|2407.17189}}, {{arXiv|2311.13932}}. | ||
Joint work with P. Lorenzoni and S. Opanasenko. | Joint work with P. Lorenzoni and S. Opanasenko. | ||
| video = | | video = | ||
| slides = | | slides = | ||
| references = {arXiv|2602.14739}, {arXiv|2407.17189}, {arXiv|2311.13932} | | references = {{arXiv|2602.14739}}, {{arXiv|2407.17189}}, {{arXiv|2311.13932}} | ||
| 79YY-MM-DD = | | 79YY-MM-DD = | ||
}} | }} | ||
Latest revision as of 14:43, 9 March 2026
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: and , where , are homogeneous first-order Hamiltonian operators and is a homogeneous Hamiltonian operator of degree (order) . The Hamiltonian property of , and their compatibility were given an explicit analytic form and geometric interpretation long ago (Dubrovin, Novikov, Ferapontov, Mokhov). The Hamiltonian property of was studied in the past (Doyle, Potemin; ) and recently revisited with interesting results.
In this talk, we illustrate the analytic form and some preliminary geometric interpretation of the compatibility conditions between and , .
See the recent papers arXiv:2602.14739, arXiv:2407.17189, arXiv:2311.13932.
Joint work with P. Lorenzoni and S. Opanasenko.
References:
arXiv:2602.14739, arXiv:2407.17189, arXiv:2311.13932