Nonlinear differential equations associated with the first Painlevé hierarchy
An integrable hierarchy associated with the first Painleve equation was introduced in the author's work [Phys. Lett., A 224, No. 6, 353--360 (1996; Zbl 0962.35504)]. It takes the form \[ L_n[w] = z, \,n = 1, 2, . . ., \] where \(z\) is the independent variable, \(w(z)\) is the dependent variable and the operator \(L_n[w]\) is determined by formula \[ \frac{d L_n[w]}{dz} =\Bigg(\frac{d^3}{dz^3}-4w \frac{d}{d^z}-2 \frac{dw}{dz}\Bigg) L_{n-1}[w],\,\, L_0[w]=-1/2. \] The case \(n=2\) corresponds to the original first Painleve equation. This hierarchy can be obtained as a compatibility condition for the following linear system \[ \Psi_{zz}=U(z,\lambda)\Psi,\,\,\, \omega(\lambda)\Psi_{\lambda}=2A(z,\lambda)\Psi_{z}-A_z(z,\lambda)\Psi. \] In the paper under review, the Cauchy problem for the equations of the hierarchy are solved by means of the inverse isomonodromic transform for the associated linear system.
scientific article; zbMATH DE number 7026125