Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912904341618688 |
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| author | Fan, Engui Li, Gaozhan Yang, Yiling Zhang, Lun |
| author_facet | Fan, Engui Li, Gaozhan Yang, Yiling Zhang, Lun |
| contents | In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlevé XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_13589 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background Fan, Engui Li, Gaozhan Yang, Yiling Zhang, Lun Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems In this paper, we consider the Cauchy problem for the defocusing nonlinear Schr$\ddot{\text{o}}$dinger equation with a finite genus algebro-geometric background. Long-time asymptotics of the solution are derived in four space-time regions. It comes out that the leading-order term in the expansion is, up to a constant, given by the background solution with a shift of the parameter. The subleading term, however, decays at different rates for different regions. We particularly highlight that in the two transition regions, they are of order $\mathcal{O}(t^{-1/3})$ and the coefficients involve an integral of the Painlevé XXXIV transcendent. We establish our results by applying a nonlinear steepest descent analysis to the associated Riemann-Hilbert problems. |
| title | Painlevé XXXIV asymptotics for the defocusing nonlinear Schrödinger equation with a finite-genus algebro-geometric background |
| topic | Analysis of PDEs Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2602.13589 |