Painlevé XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum
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| Format: | Preprint |
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2026
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| _version_ | 1866918535003897856 |
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| author | Ma, Ruihong Fan, Engui |
| author_facet | Ma, Ruihong Fan, Engui |
| contents | We investigate the Cauchy problem for the focusing modified Korteweg--de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift--Zhou to the associated Riemann--Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce with the endpoints of the finite-genus branch cuts. The collision is resolved via a local Painlevé XXXIV parametrix, and the discrete spectrum (breathers) is incorporated into the analysis. The resulting expansion is valid uniformly up to an error of order $\mathcal{O}(t^{-1/2})$. In this critical region, the leading-order term comprises the finite-genus algebro-geometric background together with breathers, whose parameters are slowly modulated by the background solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01977 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Painlevé XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum Ma, Ruihong Fan, Engui Mathematical Physics 35Q55, 35P25, 35Q15, 35C20, 35G25 We investigate the Cauchy problem for the focusing modified Korteweg--de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift--Zhou to the associated Riemann--Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce with the endpoints of the finite-genus branch cuts. The collision is resolved via a local Painlevé XXXIV parametrix, and the discrete spectrum (breathers) is incorporated into the analysis. The resulting expansion is valid uniformly up to an error of order $\mathcal{O}(t^{-1/2})$. In this critical region, the leading-order term comprises the finite-genus algebro-geometric background together with breathers, whose parameters are slowly modulated by the background solution. |
| title | Painlevé XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum |
| topic | Mathematical Physics 35Q55, 35P25, 35Q15, 35C20, 35G25 |
| url | https://arxiv.org/abs/2606.01977 |