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