32 Painlevé TranscendentsProperties32.4 Isomonodromy Problems32.6 Hamiltonian Structure

§32.5 Integral Equations

ⓘ

Keywords:

Painlevé transcendents, applications, integral equations

Notes:

See Ablowitz and Segur (1977, 1981); also Ablowitz and Clarkson (1991).

Permalink:

http://dlmf.nist.gov/32.5

See also:

Annotations for Ch.32

Let $K(z,\zeta )$ be the solution of

32.5.1		$$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$, $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution Permalink: http://dlmf.nist.gov/32.5.E1 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

where $k$ is a real constant, and $\mathrm{Ai}\left(z\right)$ is defined in §9.2. Then

32.5.2		$$w(z)=K(z,z),$$
ⓘ Symbols: $z$: real and $K(z,\zeta )$: solution Permalink: http://dlmf.nist.gov/32.5.E2 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and the boundary condition

32.5.3		$$w(z)\sim k\mathrm{Ai}\left(z\right),$$
$z\to +\mathrm{\infty }$.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real Permalink: http://dlmf.nist.gov/32.5.E3 Encodings: TeX, pMML, png See also: Annotations for §32.5 and Ch.32

32.4 Isomonodromy Problems32.6 Hamiltonian Structure

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