2-Modified Characteristic Fredholm Determinants, Hill’s Method, and the Periodic Evans Function of Gardner

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.

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Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.123 ◽

2020 ◽

pp. 1-14

Author(s):

SHOTA OSADA

Keyword(s):

Point Processes ◽

Duke Math ◽

Random Point ◽

Determinantal Point Processes ◽

Fredholm Determinants ◽

Poisson Point Processes ◽

Translation Invariant ◽

Ergodic Properties ◽

Tree Representations ◽

Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

Advances in Mathematical Physics ◽

10.1155/2018/1642139 ◽

2018 ◽

Vol 2018 ◽

pp. 1-18

Author(s):

Pierre Gaillard

Keyword(s):

Rational Solutions ◽

Fredholm Determinants ◽

Infinite Hierarchy

We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.

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