The Toda flow as a porous medium equation. (English) Zbl 1540.37093
Summary: 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.
MSC:
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K25 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 76B03 | Existence, uniqueness, and regularity theory for incompressible inviscid fluids |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
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