MaPSS: Maths Postgraduate Seminar Series: Mitchell Curran -- Eigenvalues of the linearised Nonlinear Schrodinger Equation on a compact interval

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! 

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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! 

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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