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Exponentially small corrections to divergent asymptotic expansions of solutions of the fifth Painlevé equation
| Content Provider | Semantic Scholar |
|---|---|
| Author | Andreev, F. Kitaev, Alexander V. |
| Copyright Year | 1997 |
| Abstract | We calculate the leading term of asymptotics for the coefficients of certain divergent asymptotic expansions for the solutions of the fifth Painlevé equation (P5) by using the isomonodromy deformation method and the Borel transform. Unexpectedly, these asymptotics appear to be periodic functions of the coefficients of P5. We also show the relation of our results with some other facts already known in the theory of the Painlevé equations established by other methods: (1) a connection formula for the third Painlevé equation; (2) a condition for existence of rational solutions for P5; (3) and a numerical study of the τ function for P5. 1. Instead of introduction There are several interesting asymptotic properties of the τ -function associated with a particular one-parameter family of solutions of the third Painlevé equation (P3) with a special set of coefficients: this family can be also viewed as the fifth Painlevé transcendent for a very special choice of its parameters. Let us recall some of these properties, formulated in terms of the function σ = σ(x;λ), which is the solution of the following ordinary differential equation (ODE), (xσ′′)2 = −16(xσ′ − σ)(xσ′ − σ + 1 4 (σ′)2), (1.1) where ′ = d dx , satisfying the boundary condition σ(x;λ) = x→0 −2λx − (2λx) − . . . . (1.2) |
| Starting Page | 741 |
| Ending Page | 759 |
| Page Count | 19 |
| File Format | PDF HTM / HTML |
| DOI | 10.4310/MRL.1997.v4.n5.a12 |
| Volume Number | 4 |
| Alternate Webpage(s) | http://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1997/0004/0005/MRL-1997-0004-0005-a012.pdf |
| Alternate Webpage(s) | https://doi.org/10.4310/MRL.1997.v4.n5.a12 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
