Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions
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| Natura: | Preprint |
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2023
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| _version_ | 1866916153515835392 |
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| author | Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei |
| author_facet | Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei |
| contents | The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm d}x}+\frac{αu^2+β}{x}+4u^3-\frac{4}{u}, \qquad α,β\in \mathbb C.$$ Starting from a generic initial solution $u_0(x)$ corresponding to parameters $α$, $β$, denoted as the triple $(u_0(x),α,β)$, we apply an explicit Bäcklund transformation to generate a family of solutions $(u_n(x),α+4n,β+4n)$ indexed by $n \in \mathbb N$. We study the large $n$ behavior of the solutions $(u_n(x),α+4n,β+4n)$ under the scaling $x=z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z}\frac{{\rm d}U}{{\rm d}z}+\frac{4U^2+4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m \in \mathbb C \setminus \big(\mathbb Z +\frac{1}{2}\big)$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z=0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $D_6$ and $D_8$ at $z=0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11217 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions Barhoumi, Ahmad Lisovyy, Oleg Miller, Peter D. Prokhorov, Andrei Classical Analysis and ODEs Mathematical Physics Complex Variables Exactly Solvable and Integrable Systems Primary 34M55, Secondary 34E05, 34M50, 34M56, 33E17 The third Painlevé equation in its generic form, often referred to as Painlevé-III($D_6$), is given by $$ \frac{{\rm d}^2u}{{\rm d}x^2} =\frac{1}{u}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2-\frac{1}{x}\frac{{\rm d}u}{{\rm d}x}+\frac{αu^2+β}{x}+4u^3-\frac{4}{u}, \qquad α,β\in \mathbb C.$$ Starting from a generic initial solution $u_0(x)$ corresponding to parameters $α$, $β$, denoted as the triple $(u_0(x),α,β)$, we apply an explicit Bäcklund transformation to generate a family of solutions $(u_n(x),α+4n,β+4n)$ indexed by $n \in \mathbb N$. We study the large $n$ behavior of the solutions $(u_n(x),α+4n,β+4n)$ under the scaling $x=z/n$ in two different ways: (a) analyzing the convergence properties of series solutions to the equation, and (b) using a Riemann-Hilbert representation of the solution $u_n(z/n)$. Our main result is a proof that the limit of solutions $u_n(z/n)$ exists and is given by a solution of the degenerate Painlevé-III equation, known as Painlevé-III($D_8$), $$ \frac{{\rm d}^2U}{{\rm d}z^2} =\frac{1}{U}\left(\frac{{\rm d}U}{{\rm d}z}\right)^2-\frac{1}{z}\frac{{\rm d}U}{{\rm d}z}+\frac{4U^2+4}{z}.$$ A notable application of our result is to rational solutions of Painlevé-III($D_6$), which are constructed using the seed solution $(1,4m,-4m)$ where $m \in \mathbb C \setminus \big(\mathbb Z +\frac{1}{2}\big)$ and can be written as a particular ratio of Umemura polynomials. We identify the limiting solution in terms of both its initial condition at $z=0$ when it is well defined, and by its monodromy data in the general case. Furthermore, as a consequence of our analysis, we deduce the asymptotic behavior of generic solutions of Painlevé-III, both $D_6$ and $D_8$ at $z=0$. We also deduce the large $n$ behavior of the Umemura polynomials in a neighborhood of $z=0$. |
| title | Painlevé-III Monodromy Maps Under the $D_6\to D_8$ Confluence and Applications to the Large-Parameter Asymptotics of Rational Solutions |
| topic | Classical Analysis and ODEs Mathematical Physics Complex Variables Exactly Solvable and Integrable Systems Primary 34M55, Secondary 34E05, 34M50, 34M56, 33E17 |
| url | https://arxiv.org/abs/2307.11217 |