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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2407.21526 |
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| _version_ | 1866910770049056768 |
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| author | Chen, Meisen Fan, Engui Wang, Zhaoyu |
| author_facet | Chen, Meisen Fan, Engui Wang, Zhaoyu |
| contents | We investigate the soliton resolution and Painlevé asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value problem, which is further reformulated as a Riemann-Hilbert problem with higher-order poles. Using Fredholm theory, the Riemann-Hilbert problem with the jump contour consisting of three circles centered around the origin is uniquely solved. Then, by performing a $\bar\partial$-nonlinear steepest descent method to the Riemann-Hilbert problem, we obtain the asymptotic approximation to the solution of the focusing Ablowitz-Ladik system for large time in different space-time regions of the $(n,t)$-half plane. In the sectors $\{(n,t): n /(2t) <-M_0 \}$ and $\{(n,t): n /(2t) >M_0 \}$, where $M_0$ is a positive constant, the leading order asymptotics is dominated by the solitons; while in the sector $\{(n,t): |n /(2t) -1 <M_0^{-1} \}$, the long-time asymptotics is influenced by both the solitons and the oscillations; In the two transition zones $\{(n,t): |n /(2t)+1|t^{2/3} <C \}$ and $\{(n,t): |n /(2t)-1|t^{2/3} <C \}$ with $C$ being a positive constant, we find the Painlevé-type asymptotics which can be expressed in terms of the solution of the second Painlevé transcendents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21526 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons Chen, Meisen Fan, Engui Wang, Zhaoyu Analysis of PDEs Mathematical Physics We investigate the soliton resolution and Painlevé asymptotics for the focusing Ablowitz-Ladik system with the initial data in a discrete weighted $\ell^2$ space. First, we establish the global well-posedness of this initial-value problem, which is further reformulated as a Riemann-Hilbert problem with higher-order poles. Using Fredholm theory, the Riemann-Hilbert problem with the jump contour consisting of three circles centered around the origin is uniquely solved. Then, by performing a $\bar\partial$-nonlinear steepest descent method to the Riemann-Hilbert problem, we obtain the asymptotic approximation to the solution of the focusing Ablowitz-Ladik system for large time in different space-time regions of the $(n,t)$-half plane. In the sectors $\{(n,t): n /(2t) <-M_0 \}$ and $\{(n,t): n /(2t) >M_0 \}$, where $M_0$ is a positive constant, the leading order asymptotics is dominated by the solitons; while in the sector $\{(n,t): |n /(2t) -1 <M_0^{-1} \}$, the long-time asymptotics is influenced by both the solitons and the oscillations; In the two transition zones $\{(n,t): |n /(2t)+1|t^{2/3} <C \}$ and $\{(n,t): |n /(2t)-1|t^{2/3} <C \}$ with $C$ being a positive constant, we find the Painlevé-type asymptotics which can be expressed in terms of the solution of the second Painlevé transcendents. |
| title | Long-time Asymptotics for the Ablowitz-Ladik system with present of solitons |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2407.21526 |