Equations of Lax type, height functions and deformations

Lunedi' 27 Giugno 2011 - Alessandro Arsie


Lunedi' 27 giugno 2011 alle 12:15 in aula 1C150 il Prof. Alessandro Arsie (Toledo, Ohio) terra' una conferenza di Fisica Matematica dal titolo "Equations of Lax type, height functions and deformations".

In this talk I will discuss some examples of systems of nonlinear ODEs of Lax type.
These are systems of the form $frac{d}{dt}A(t)=[U(A), A],$ where $A$ is an $ntimes n$ matrix, $U(A)$ is a matrix possibly depending on $A$ and the bracket denotes their commutator. These systems have the remarkable property that the spectrum of $A$ is preserved under evolution (isospectral flows).
I will focus on their applications to the study of direct and inverse problems in linear algebra and on their geometric meaning. In particular I will discuss in detail the dynamics of a system of Lax type that generalizes the celebrated Brockett's double bracket equation and which can be used to compute the eigenvalues of a Hamiltonian matrix while preserving the structure of the matrix itself.
I will also introduce the concept of height function and show how this can be used to effectively study the asymptotic behavior of some of these systems, providing also a strengthening of LaSalle's invariance principle.
Finally I will discuss some ideas on how to deform an equation of Lax type in such a way to selectively dissipate any of its first integrals, while preserving the others. Numerical simulations will be provided to illustrate some of the key points of the talk.
Most of these results have been obtained in collaboration with Christian Ebenbauer (University of Stuttgart, Germany).

Rif. int. F. Cardin