Long-time asymptotics of the defocusing mKdV equation with step initial data
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911023915597824 |
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| author | Wang, Deng-Shan Wen, Ding |
| author_facet | Wang, Deng-Shan Wen, Ding |
| contents | This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by applying Deift-Zhou nonlinear steepest descent method. In this process, the construction of odd-symmetry g-function is generalized and the method of genus reduction on the Riemann-theta function is proposed via conformal transformation and symmetries. It is revealed that for sufficiently large time, the solution manifests a tripartite spatiotemporal structure, i.e., in the left plane-wave region, the solution decays to a modulated plane wave with oscillatory correction; in the central dispersive shock wave region, the solution is governed by a modulated elliptic periodic wave; in the right plane wave region, the solution converges exponentially to a constant. The results from the long-time asymptotic analysis have been shown to match remarkably well with that obtained by direct numerical simulations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_01570 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long-time asymptotics of the defocusing mKdV equation with step initial data Wang, Deng-Shan Wen, Ding Analysis of PDEs Mathematical Physics This work investigates the long-time asymptotics of solution to defocusing modified Korteweg-de Vries equation with a class of step initial data. A rigorous asymptotic analysis is conducted on the associated Riemann-Hilbert problem by applying Deift-Zhou nonlinear steepest descent method. In this process, the construction of odd-symmetry g-function is generalized and the method of genus reduction on the Riemann-theta function is proposed via conformal transformation and symmetries. It is revealed that for sufficiently large time, the solution manifests a tripartite spatiotemporal structure, i.e., in the left plane-wave region, the solution decays to a modulated plane wave with oscillatory correction; in the central dispersive shock wave region, the solution is governed by a modulated elliptic periodic wave; in the right plane wave region, the solution converges exponentially to a constant. The results from the long-time asymptotic analysis have been shown to match remarkably well with that obtained by direct numerical simulations. |
| title | Long-time asymptotics of the defocusing mKdV equation with step initial data |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2506.01570 |