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arXiv · 2305.12968

Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"odinger hydrodynamics

Deng-Shan Wang·Peng Yan
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Abstract

The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"{o}dinger hydrodynamics is a very interesting problem with many challenges. To date, the full analysis of this problem remains open. In this work, the long-time asymptotics for the defocusing nonlinear Schr\"{o}dinger equation with general step-like initial data is investigated by the Whitham modulation theory and Riemann-Hilbert formulation. The Whitham modulation theory shows that there are six cases for the initial discontinuity problem according to the orders of the Riemann invariants. The leading-order terms and the corresponding error estimates for each region of the six cases are formulated by the Deift-Zhou nonlinear steepest descent method for oscillatory Riemann-Hilbert problems. It is demonstrated that the long-time asymptotic solutions match very well with the results from Whitham modulation theory and the numerical simulations.

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Deng-Shan Wang, Peng Yan. 2023-05-22. Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schr\"odinger hydrodynamics. https://doi.org/10.1088/1361-6544/ae1dc4

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