Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866918415408562176 |
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| author | Wang, Deng-Shan Yan, Peng |
| author_facet | Wang, Deng-Shan Yan, Peng |
| contents | The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger 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 Schrödinger 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_12968 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics Wang, Deng-Shan Yan, Peng Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems The rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger 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 Schrödinger 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. |
| title | Rigorous asymptotic analysis for the Riemann problem of the defocusing nonlinear Schrödinger hydrodynamics |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2305.12968 |