Integrable Systems Seminar
Thursday, October 24, 2002, 4:00pm, 218 Physics
Greg Forest
Instability Phenomena Arising in the Nonlinear Coupling of Conservative Scalar Fields
Abstract:- This lecture explores a very simple scenario, using very simple
mathematical analysis, but with some surprising and possibly even
applicable results. In joint work with Otis Wright, we have studied
systems of coupled nonlinear Schrodinger equations as models for wave:wave
interactions on a finite spatial domain. One can think of wavetrain data
on a finite optical transmission line.
We are interested in instabilities that arise through nonlinear coupling
of scalar fields, and in particular, the scaling properties of the spatial
structures that are generated. These structures are to be understood as
one varies the background waves in each scalar field, and as one varies
the stability of each scalar wave if it were propagating alone. It will
turn out that the instability structures depend critically on the relative
amplitudes of the background data and on their wavelength.
We use the integrable coupled NLS system to study this problem, and focus
on the simplest background waves, plane waves. The mathematical problem
is then a Hamiltonian system of 4 real pdes, or 2 complex pdes, whose
"solvable" by the inverse spectral transform for both the focusing
(unstable) and defocusing (stable) sign of the cubic nonlinearity; and,
whose linearized stability admits a complete characterization for all
possible couplings (stable:stable, unstable:stable, and unstable:unstable)
and for all possible plane wave solutions. We will summarize the results
of that analysis, including earlier work with Daves McLaughlin and Muraki.
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