Seminar talk, 6 November 2019

From Geometry of Differential Equations
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Speaker: Maxim Pavlov

Title: Bi-Hamiltonian systems of hydrodynamic type

We consider bi-Hamiltonian systems of hydrodynamic type from the viewpoint of the classical differential geometry and discuss open problems.

In Riemann invariants, this problem reduces to the integrable systems such that the coefficients of their linear Lax pairs depend on the independent variables. Thus, such problems belong to a more complicated class of integrable systems than such known systems of equations as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, and others.

In the flat coordinates, this problem reduces to the set of commuting systems of hydrodynamic type that are integrable by the inverse scattering transform. On the one hand, the coefficients of the corresponding linear Lax pairs doesn't depend explicitly on the independent variables (in the flat coordinates). On the other hand, the systems of hydrodynamic type that we get are completely not understood and offer significant prospects in their study and integrating.