The Toda Flow as a Porous Medium Equation
Journal article, 2023

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

Author

Boris Khesin

University of Toronto

Klas Modin

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Communications in Mathematical Physics

0010-3616 (ISSN) 1432-0916 (eISSN)

Vol. 401 2 1879-1898

Long-time 2D hydrodynamics via quantization

Swedish Research Council (VR) (2022-03453), 2023-01-01 -- 2026-12-31.

Subject Categories

Computational Mathematics

Fluid Mechanics and Acoustics

Mathematical Analysis

DOI

10.1007/s00220-023-04680-2

More information

Latest update

3/7/2024 9