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.
Long-time asymptotics of the Camassa-Holm equation on the line[C]//Integrable Systems and Random Matrices
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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.