For focusing NLS with finite-genus algebro-geometric background, long-time asymptotics are Painlevé II for odd genus and parabolic cylinder functions for even genus, with explicit error bounds uniform in x.
Inverse scattering transform for the focusing nonlinear Schrödinger equation with nonzero boundary conditions[J]
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Painlev\'e Asymptotics of the Focusing Nonlinear Schr\"odinger Equation with a Finite-Genus Algebro-Geometric Background
For focusing NLS with finite-genus algebro-geometric background, long-time asymptotics are Painlevé II for odd genus and parabolic cylinder functions for even genus, with explicit error bounds uniform in x.