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.
Direct and inverse scattering transforms with arbitrary spectral singularities[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.