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Long-Time Asymptotics for the Spin-1 Gross–Pitaevskii Equation

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Abstract

On the basis of the spectral analysis of the \(4\times 4\) Lax pair associated with the spin-1 Gross–Pitaevskii equation and the scattering matrix, the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is transformed into the solution to the corresponding Riemann–Hilbert problem. The Deift–Zhou nonlinear steepest descent method is extended to the Riemann–Hilbert problem, from which a model Riemann–Hilbert problem is established with the help of distinct factorizations of the jump matrix for the Riemann–Hilbert problem and a decomposition of the matrix-valued spectral function. Finally, the long-time asymptotics of the solution to the Cauchy problem of the spin-1 Gross–Pitaevskii equation is obtained.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871440).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, 450001, Henan, People’s Republic of China

    Xianguo Geng, Kedong Wang & Mingming Chen

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  1. Xianguo Geng
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  2. Kedong Wang
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  3. Mingming Chen
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Correspondence to Kedong Wang.

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Communicated by H.-T. Yau

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Geng, X., Wang, K. & Chen, M. Long-Time Asymptotics for the Spin-1 Gross–Pitaevskii Equation. Commun. Math. Phys. 382, 585–611 (2021). https://doi.org/10.1007/s00220-021-03945-y

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