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.
Dispersive shock wave, generalized Laguerre polynomials, and asymptotic solitons of the focusing nonlinear Schrödinger equation[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.