In this talk we are going to construct a family of conserved energies for the one-dimensional cubic nonlinear defocusing Schrödinger equation, but with nonzero boundary conditions at infinity. This system is completely integrable by means of inverse scattering methods and admits Lax-pair structure. We will introduce the associated (generalized) energy spaces, and define the renormalized transmission coefficient (associated to the Lax operator). The conserved energies will be constructed in terms of this time-invariant transmission coefficient. This is joint work with Herbert Koch (Bonn, Germany).

 

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