The Painlevé-type asymptotics of defocusing complex mKdV equation with finite density initial data
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915198687772672 |
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| author | Wen, Lili Fan, Engui |
| author_facet | Wen, Lili Fan, Engui |
| contents | We consider the Cauchy problem for the defocusing
complex mKdV equation with finite density initial data
\begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*}
which can be formulated into a Riemann-Hilbert (RH) problem. With
$\bar\partial$-generation of the nonlinear steepest descent approach and a double scaling limit technique,
in the transition region $$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C< \left(x/(2t)+3/2\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$
we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem
is associated with the Painlevé-II transcendents. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02740 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Painlevé-type asymptotics of defocusing complex mKdV equation with finite density initial data Wen, Lili Fan, Engui Mathematical Physics We consider the Cauchy problem for the defocusing complex mKdV equation with finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-3|q|^2q_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be formulated into a Riemann-Hilbert (RH) problem. With $\bar\partial$-generation of the nonlinear steepest descent approach and a double scaling limit technique, in the transition region $$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C< \left(x/(2t)+3/2\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$ we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem is associated with the Painlevé-II transcendents. |
| title | The Painlevé-type asymptotics of defocusing complex mKdV equation with finite density initial data |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2308.02740 |