Abstract
odinger operators. 2000 Mathematics Subject Classification Primary 35J10, 45P05; Secondary 35P05, 47B10. The Evans function is a Wronskian type determinant used to detect point spectrum of differential operators obtained by linearizing PDEs about special solutions such as traveling waves, etc. This work is a sequel to the paper “Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves”, published by F. Gesztesy, K. Zumbrun and the second author in J. Math. Pures Appl. 90, 160‐200 (2008), where the Evans and Jost functions for the Schr¨ odinger equations have been considered. In the current work, we study the Evans function for the general case of linear ODE systems, and choose it to agree with the modified Fredholm determinant of the respective Birman-Schwinger type integral operator. The Evans function is thus the determinant of the matrix composed of the so-called generalized Jost solutions. These are the solutions of the homogeneous perturbed differential equation which are asymptotic to some reference solutions of the unperturbed equation. One of the main results of the current paper is a formula for the derivative of the Evans function for the first order systems. Its proof uses a matrix composed of the newly introduced modified Jost solutions. These are the solutions of an inhomogeneous perturbed differential equation with the inhomogeneous term constructed by means of the above-mentioned generalized Jost solutions. Copyright line will be provided by the publisher
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.