Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911455598608384 |
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| author | Wang, Zhaoyu Xu, Taiyang Fan, Engui |
| author_facet | Wang, Zhaoyu Xu, Taiyang Fan, Engui |
| contents | We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align}
&q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber
&q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlevé transcendent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_07073 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions Wang, Zhaoyu Xu, Taiyang Fan, Engui Mathematical Physics 35P25, 35Q51, 35Q15, 35B40, 35C20 We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions \begin{align} &q_t(x,t)-6q^2(x,t)q_{x}(x,t)+q_{xxx}(x,t)=0, \nonumber &q(x,0)=q_{0}(x)\to \pm 1, \ \ x\rightarrow\pm\infty, \nonumber \end{align} which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the $\bar\partial$-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region $|x/t+6|t^{2/3}< C$ with $C>0$. The asymptotics can be expressed in terms of the solution of the second Painlevé transcendent. |
| title | Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions |
| topic | Mathematical Physics 35P25, 35Q51, 35Q15, 35B40, 35C20 |
| url | https://arxiv.org/abs/2306.07073 |