6 research outputs found
Tau functions for linear systems
Let be a linear system in continuous time with input and output space and state space . The function determines a Hankel integral operator on ; if is trace class, then the Fredholm determinant defines the tau function of . Such tau functions arise in Tracy and Widom's theory of matrix models, where they describe the fundamental probability distributions of random matrix theory. Dyson considered such tau functions in the inverse spectral problem for Schr\"odinger's equation , and derived the formula for the potential in the self-adjoint scattering case {\sl Commun. Math. Phys.} {\bf 47} (1976), 171--183. This paper introduces a operator function that satisfies Lyapunov's equation and , without assumptions of self-adjointness. When is sectorial, and are Hilbert--Schmidt, there exists a non-commutative differential ring of operators in and a differential ring homomorphism such that , which extends the multiplication rules for Hankel operators considered by P\"oppe, and McKean {\sl Cent. Eur. J. Math.} {\bf 9} (2011), 205--243. \pa
The conservative Camassa-Holm flow with step-like irregular initial data
We extend the inverse spectral transform for the conservative Camassa-Holm
flow on the line to a class of initial data that requires strong decay at one
endpoint but only mild boundedness-type conditions at the other endpoint. The
latter condition appears to be close to optimal in a certain sense for the
well-posedness of the conservative Camassa-Holm flow. As a byproduct of our
approach, we also find a family of new (almost) conservation laws for the
Camassa-Holm equation, which could not be deduced from its bi-Hamiltonian
structure before and which are connected to certain Besov-type norms (however,
in a rather involved way). These results appear to be new even under positivity
assumptions on the corresponding momentum, in which case the conservative
Camassa-Holm flow coincides with the classical Camassa-Holm flow and no
blow-ups occur.Comment: 36 page