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Absorption/emission of solitons by an integrable boundary in the nonlinear Schrödinger model

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HY2W02 - Analysis of dispersive systems

I will review the history of the main ideas involved in the problem of identifying integrable boundary conditions for nonlinear PDEs amenable to the IST and of constructing solutions for the associated initial-boundary value problems. Special emphasis will be put on the current understanding of the interrelation between three influential ideas associated to Sklyanin, Habibullin and Fokas. The last two deal with solution methods while the first one deals with the identification of boundary conditions explicitly preserving integrability. I will then explain, on the example of the nonlinear Schrödinger equation, how these connections naturally lead to the investigations of so-called time-dependent integrable boundary conditions. The new feature of these boundary conditions is that they allow for solutions where a soliton can be absorbed or emitted by the boundary. This was originally unexpected in a classical model. In hindsight, it is consistent with the intuition developed in the quantum case for the analog of these boundary conditions and it can also be nicely explained classically by analysing the conservation laws of the model including the boundary. Time permitting, I will take advantage of this programme to present some speculative ideas related to the theme of soliton gases which are suggested by the spectral structure of general time-dependent boundary conditions.  

This talk is part of the Isaac Newton Institute Seminar Series series.

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