Painlevé functions of the third kind

Abstract

We explicitly construct the one-parameter family of solutions, η(θ;ν,λ), that remain bounded as θ→∞ along the positive real θ axis for the Painlevé equation of third kind ww″ = (w′)2-θ-1ww′+2νθ -1(w3-w) + w4-1, where, as θ→∞ , η ∼ 1-λΓ(ν+1/2)2-2νθ -ν-1/2e-2θ. We further construct a representation for ψ(t;ν,λ) = -ln[η(t/2;ν,λ)], where ψ(t;ν,λ) satisfies the differential equation ψ″ + t -1ψ′ = (1/2)sinh(2ψ)+2νt-1 sinh(ψ). The small-θ behavior of η(θ;ν,λ) is described for |λ|