Large-time asymptotics for the defocusing Manakov system on nonzero background
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| Format: | Preprint |
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2025
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| _version_ | 1866912789397766144 |
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| author | Geng, Xianguo Zhang, Haibing Wei, Jiao |
| author_facet | Geng, Xianguo Zhang, Haibing Wei, Jiao |
| contents | The Manakov system is a two-component nonlinear Schrödinger equation. The long-time asymptotics for the defocusing or focusing Manakov system under nonzero background still remains open. In this paper, we derive the long-time asymptotic formula for the solution of the defocusing Manakov system on nonzero boundary conditions and provide a detailed proof. The solution of the defocusing Manakov system on such nonzero background is first transformed into the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. Then we demonstrate how to conduct the Deift-Zhou steepest descent analysis for this Riemann-Hilbert problem, thereby obtaining the long-time asymptotic behavior of the solution in the space-time soliton region. In this region, the leading order of the solution takes the form of a modulated multisoliton. Apart from the error term, we also discover that the defocusing Manakov system has a dispersive correction term of order $t^{-1/2}$, but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Large-time asymptotics for the defocusing Manakov system on nonzero background Geng, Xianguo Zhang, Haibing Wei, Jiao Exactly Solvable and Integrable Systems Analysis of PDEs The Manakov system is a two-component nonlinear Schrödinger equation. The long-time asymptotics for the defocusing or focusing Manakov system under nonzero background still remains open. In this paper, we derive the long-time asymptotic formula for the solution of the defocusing Manakov system on nonzero boundary conditions and provide a detailed proof. The solution of the defocusing Manakov system on such nonzero background is first transformed into the solution of a $3 \times 3$ matrix Riemann-Hilbert problem. Then we demonstrate how to conduct the Deift-Zhou steepest descent analysis for this Riemann-Hilbert problem, thereby obtaining the long-time asymptotic behavior of the solution in the space-time soliton region. In this region, the leading order of the solution takes the form of a modulated multisoliton. Apart from the error term, we also discover that the defocusing Manakov system has a dispersive correction term of order $t^{-1/2}$, but this term does not exist in the scalar case, and we provide the explicit expression for this dispersion term. |
| title | Large-time asymptotics for the defocusing Manakov system on nonzero background |
| topic | Exactly Solvable and Integrable Systems Analysis of PDEs |
| url | https://arxiv.org/abs/2512.21841 |