Type: Seminar

Distribution: World

Expiry: 17 Mar 2020

CalTitle1: Eigenvalues of the linearised Nonlinear SchrÃ¶dinger Equation on a compact interval

Auth: wenqi@dora.maths.usyd.edu.au

Hello all, The next MaPSS talk of this semester will be at 17:00 on Mon 16th of September in Carslaw 535. It’s a great opportunity to meet fellow postgrads, listen to an interesting talk, and of course get some free pizza! ************************************************************************************** Speaker: Mitchell Curran Title: Eigenvalues of the linearised Nonlinear Schrodinger Equation on a compact interval Abstract: In 1988 Jones proved a theorem regarding the existence of a positive eigenvalue for the linearised operator associated with the nonlinear Schrodinger equation with spatial domain the real line. Specifically, one linearises this complex-valued second order partial differential equation about a standing wave and splits the system into real and complex parts. The resulting operator N is not self-adjoint, and much of its spectrum lies on the imaginary axis; however, it can be written in terms of 2 self-adjoint operators (L_+, L_-) whose spectra are real. With P being the number of positive eigenvalues of L_+ and Q the number of positive eigenvalues of L_- (both well-defined quantities), we arrive at the neat relationship: P - Q = the number of positive real eigenvalues of N (well, almost). I am looking at this statement for the case when the spatial domain is a compact interval - we will see some pretty plots which shows the relationship holds true in this case. What remains is to rigorously prove the statement! ************************************************************************************** See you there! Details can also be found on the school’s Postgraduate Society website: http://www.maths.usyd.edu.au/u/MaPS/mapss.2019.html