70 research outputs found
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
Partial differential systems with nonlocal nonlinearities: Generation and solutions
We develop a method for generating solutions to large classes of evolutionary
partial differential systems with nonlocal nonlinearities. For arbitrary
initial data, the solutions are generated from the corresponding linearized
equations. The key is a Fredholm integral equation relating the linearized flow
to an auxiliary linear flow. It is analogous to the Marchenko integral equation
in integrable systems. We show explicitly how this can be achieved through
several examples including reaction-diffusion systems with nonlocal quadratic
nonlinearities and the nonlinear Schrodinger equation with a nonlocal cubic
nonlinearity. In each case we demonstrate our approach with numerical
simulations. We discuss the effectiveness of our approach and how it might be
extended.Comment: 4 figure
Stochastic expansions and Hopf algebras
We study solutions to nonlinear stochastic differential systems driven by a
multi-dimensional Wiener process. A useful algorithm for strongly simulating
such stochastic systems is the Castell--Gaines method, which is based on the
exponential Lie series. When the diffusion vector fields commute, it has been
proved that at low orders this method is more accurate in the mean-square error
than corresponding stochastic Taylor methods. However it has also been shown
that when the diffusion vector fields do not commute, this is not true for
strong order one methods. Here we prove that when there is no drift, and the
diffusion vector fields do not commute, the exponential Lie series is usurped
by the sinh-log series. In other words, the mean-square error associated with a
numerical method based on the sinh-log series, is always smaller than the
corresponding stochastic Taylor error, in fact to all orders. Our proof
utilizes the underlying Hopf algebra structure of these series, and a
two-alphabet associative algebra of shuffle and concatenation operations. We
illustrate the benefits of the proposed series in numerical studies.Comment: 23 pages, 4 figure
Levy Processes and Quasi-Shuffle Algebras
We investigate the algebra of repeated integrals of semimartingales. We prove
that a minimal family of semimartingales generates a quasi-shuffle algebra. In
essence, to fulfill the minimality criterion, first, the family must be a
minimal generator of the algebra of repeated integrals generated by its
elements and by quadratic covariation processes recursively constructed from
the elements of the family. Second, recursively constructed quadratic
covariation processes may lie in the linear span of previously constructed ones
and of the family, but may not lie in the linear span of repeated integrals of
these. We prove that a finite family of independent Levy processes that have
finite moments generates a minimal family. Key to the proof are the Teugels
martingales and a strong orthogonalization of them. We conclude that a finite
family of independent Levy processes form a quasi-shuffle algebra. We discuss
important potential applications to constructing efficient numerical methods
for the strong approximation of stochastic differential equations driven by
Levy processes.Comment: 10 page
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