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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.16237 |
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| _version_ | 1866917969386274816 |
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| author | Liu, Xuliang Wang, Deng-Shan |
| author_facet | Liu, Xuliang Wang, Deng-Shan |
| contents | This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for $x>0$, while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with $L$-spikes is also studied and it is proved that one to $L$ solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16237 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics of the KdV equation with delta function initial profile Liu, Xuliang Wang, Deng-Shan Analysis of PDEs This work investigates the long-time asymptotic behaviors of the solution to the KdV equation with delta function initial profiles in different regions, employing the Riemann-Hilbert formulation and Deift-Zhou nonlinear steepest descent method. When the initial value is a delta potential well, the asymptotic solution is predominantly dominated by a single soliton in certain region for $x>0$, while in other regions, the dispersive tails including self-similar region, collisionless shock region and dispersive wave region, play a more significant role. Conversely, when the initial value is a delta potential barrier, the soliton region is absent, although the dispersive tails still persist. Moreover, the general delta function initial profile with $L$-spikes is also studied and it is proved that one to $L$ solitons will be generated in soliton region, which depends on the sizes of the distance and height of the spikes. The leading-order terms of the solution in each region are derived, highlighting the efficacy of the Riemann-Hilbert formulation in elucidating the long-time behaviors of integrable systems. |
| title | Long-time asymptotics of the KdV equation with delta function initial profile |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.16237 |