Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2020
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| _version_ | 1866916493225099264 |
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| author | Cafasso, Mattia Claeys, Tom Ruzza, Giulio |
| author_facet | Cafasso, Mattia Claeys, Tom Ruzza, Giulio |
| contents | We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_07723 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations Cafasso, Mattia Claeys, Tom Ruzza, Giulio Mathematical Physics Classical Analysis and ODEs Probability We study a family of unbounded solutions to the Korteweg-de Vries equation which can be constructed as log-derivatives of deformed Airy kernel Fredholm determinants, and which are connected to an integro-differential version of the second Painlevé equation. The initial data of the Korteweg-de Vries solutions are well-defined for $x>0$, but not for $x<0$, where the solutions behave like $\frac{x}{2t}$ as $t\to 0$, and hence would be well-defined as solutions of the cylindrical Korteweg-de Vries equation. We provide uniform asymptotics in $x$ as $t\to 0$; for $x>0$ they involve an integro-differential analogue of the Painlevé V equation. A special case of our results yields improved estimates for the {tails} of the narrow wedge solution to the Kardar-Parisi-Zhang equation. |
| title | Airy kernel determinant solutions to the KdV equation and integro-differential Painlevé equations |
| topic | Mathematical Physics Classical Analysis and ODEs Probability |
| url | https://arxiv.org/abs/2010.07723 |