We study certain 2 degree-of-freedom Hamiltonian systems arising from an
\napproximation scheme for solutions of the 2d nonlinear Schrodinger equation with cubic-quintic or
\nsaturated nonlinearities (possibly in a grade-indexed medium). The solutions of these systems
\ncan be of various different types, depending on the values of the parameters and the initial
\nconditions, with transitions – which we call \\breathing-beating” transitions – associated with
\nsolutions displaying extremely long time periodic behavior. We show that this behavior is typically
\nassociated with a 1-1 Hamiltonian resonance, and use multiscale analysis to successfully
\npredict some, but certainly not all, of the transitions.
\nFor those who maybe did not quite manage to understand all of the above: It is remarkable
\nfact that seemingly simple looking differential equations can have solutions that behave in
\nqualitatively \u000bdifferent ways over substantially different scales. Multiscale analysis is a set of
\ntools for explaining such phenomena. I will try to explain one of the key tricks of multiscale
\nanalysis (not assuming any previous familiarity) and its application to a problem of interest in
\nthe \ffield of nonlinear optics.
\nJoint work with David Ianetz.<\/p>\n","protected":false},"excerpt":{"rendered":"
We study certain 2 degree-of-freedom Hamiltonian systems arising from an approximation scheme for solutions of the 2d nonlinear Schrodinger equation with cubic-quintic or saturated nonlinearities (possibly in a grade-indexed medium). The solutions of these systems can be of various different…
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