8,424 research outputs found
Grassmannian flows and applications to nonlinear partial differential equations
We show how solutions to a large class of partial differential equations with
nonlocal Riccati-type nonlinearities can be generated from the corresponding
linearized equations, from arbitrary initial data. It is well known that
evolutionary matrix Riccati equations can be generated by projecting linear
evolutionary flows on a Stiefel manifold onto a coordinate chart of the
underlying Grassmann manifold. Our method relies on extending this idea to the
infinite dimensional case. The key is an integral equation analogous to the
Marchenko equation in integrable systems, that represents the coodinate chart
map. We show explicitly how to generate such solutions to scalar partial
differential equations of arbitrary order with nonlocal quadratic
nonlinearities using our approach. We provide numerical simulations that
demonstrate the generation of solutions to
Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal
nonlinearities. We also indicate how the method might extend to more general
classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure
Wave relations
The wave equation (free boson) problem is studied from the viewpoint of the
relations on the symplectic manifolds associated to the boundary induced by
solutions. Unexpectedly there is still something to say on this simple,
well-studied problem. In particular, boundaries which do not allow for a
meaningful Hamiltonian evolution are not problematic from the viewpoint of
relations. In the two-dimensional Minkowski case, these relations are shown to
be Lagrangian. This result is then extended to a wide class of metrics and is
conjectured to be true also in higher dimensions for nice enough metrics. A
counterexample where the relation is not Lagrangian is provided by the Misner
space.Comment: 27 pages; minor clarifying changes added; to appear in CM
Geometry of quantum dynamics in infinite dimension
We develop a geometric approach to quantum mechanics based on the concept of
the Tulczyjew triple. Our approach is genuinely infinite-dimensional and
including a Lagrangian formalism in which self-adjoint (Schroedinger) operators
are obtained as Lagrangian submanifolds associated with the Lagrangian. As a
byproduct we obtain also results concerning coadjoint orbits of the unitary
group in infinite dimension, embedding of the Hilbert projective space of pure
states in the unitary group, and an approach to self-adjoint extensions of
symmetric relations.Comment: 32 page
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