1,271 research outputs found
Fredholm determinants for the stability of travelling waves
This thesis investigates both theoretically and numerically the stability of travelling
wave solutions using Fredholm determinants, on the real line. We identify a class of
travelling wave problems for which the corresponding integral operators are of trace
class. Based on the geometrical interpretation of the Evans function, we give an alternative
proof connecting it to (modified) Fredholm determinants. We then extend
that connection to the case of front waves by constructing an appropriate integral
operator. In the context of numerical evaluation of Fredholm determinants, we prove
the uniform convergence associated with the modified/regularised Fredholm determinants
which generalises Bornemann's result on this topic. Unlike in Bornemann's
result, we do not assume continuity but only integrability with respect to the second
argument of the kernel functions. In support to our theory, we present some numerical
results. We show how to compute higher order determinants numerically, in particular
for integral operators belonging to classes I3 and I4 of the Schatten-von Neumann
set. Finally, we numerically compute Fredholm determinants for some travelling wave
problems e.g. the `good' Boussinesq equation and the fth-order KdV equation.UK EPSRC (Engineering and Physical Sciences Research Council) grant EP/G03613
A Fredholm Determinant for Semi-classical Quantization
We investigate a new type of approximation to quantum determinants, the
``\qFd", and test numerically the conjecture that for Axiom A hyperbolic flows
such determinants have a larger domain of analyticity and better convergence
than the \qS s derived from the \Gt. The conjecture is supported by numerical
investigations of the 3-disk repeller, a normal-form model of a flow, and a
model 2- map.Comment: Revtex, Ask for figures from [email protected]
Beyond the periodic orbit theory
The global constraints on chaotic dynamics induced by the analyticity of
smooth flows are used to dispense with individual periodic orbits and derive
infinite families of exact sum rules for several simple dynamical systems. The
associated Fredholm determinants are of particularly simple polynomial form.
The theory developed suggests an alternative to the conventional periodic orbit
theory approach to determining eigenspectra of transfer operators.Comment: 29 pages Latex2
Evans function and Fredholm determinants
We explore the relationship between the Evans function, transmission
coefficient and Fredholm determinant for systems of first order linear
differential operators on the real line. The applications we have in mind
include linear stability problems associated with travelling wave solutions to
nonlinear partial differential equations, for example reaction-diffusion or
solitary wave equations. The Evans function and transmission coefficient, which
are both finite determinants, are natural tools for both analytic and numerical
determination of eigenvalues of such linear operators. However, inverting the
eigenvalue problem by the free state operator generates a natural linear
integral eigenvalue problem whose solvability is determined through the
corresponding infinite Fredholm determinant. The relationship between all three
determinants has received a lot of recent attention. We focus on the case when
the underlying Fredholm operator is a trace class perturbation of the identity.
Our new results include: (i) clarification of the sense in which the Evans
function and transmission coefficient are equivalent; and (ii) proof of the
equivalence of the transmission coefficient and Fredholm determinant, in
particular in the case of distinct far fields.Comment: 26 page
Return probability after a quench from a domain wall initial state in the spin-1/2 XXZ chain
We study the return probability and its imaginary () time continuation
after a quench from a domain wall initial state in the XXZ spin chain, focusing
mainly on the region with anisotropy . We establish exact Fredholm
determinant formulas for those, by exploiting a connection to the six vertex
model with domain wall boundary conditions. In imaginary time, we find the
expected scaling for a partition function of a statistical mechanical model of
area proportional to , which reflects the fact that the model exhibits
the limit shape phenomenon. In real time, we observe that in the region
the decay for large times is nowhere continuous as a function
of anisotropy: it is either gaussian at root of unity or exponential otherwise.
As an aside, we also determine that the front moves as , by analytic continuation of known arctic curves in
the six vertex model. Exactly at , we find the return probability
decays as . It is argued that this
result provides an upper bound on spin transport. In particular, it suggests
that transport should be diffusive at the isotropic point for this quench.Comment: 33 pages, 8 figures. v2: typos fixed, references added. v3: minor
change
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