| Abstract: |
| In this talk I am going cover various topics regarding the validation of the NLS approximation in dispersive systems. In the first half of the talk we will study a family of solutions to the NLS called the Peregrine Soliton family, which are not covered by previously known approximation theory, as they do not approach zero for $|x|\to\infty$. Either the envelope spatially decays to zero over a temporally oscillating background, or it is fully spatially periodic, and special care is necessary when it comes to choice of function spaces to work in.
The second half of the talk will instead focus on the NLS approximation in the presence of a fractional Laplacian. Still, a fully local NLS can be derived from the original, non-local system. The fact that the approximation is strongly localized around the wave number $k_0$ is used to, in lowest order, be able to use similar methods as in the regular case. Something of note is that the character of the derived NLS (focusing/defocusing) may change depending on choice of basic wave number, which is a stark difference to the classic case.
This talk is based on joint work with Guido Schneider, Anna Logioti and Theo Belin. |
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