Integral Representations for Solutions of the Linearized Fourth Painleve Equation

Abstract

We formulate the linearization of the fourth Painlevé equation as a linearized Hamiltonian system and transform the scalar variational equation to normal form, with the first-derivative term eliminated. From the canonical fundamental solutions of the Jimbo--Miwa Lax pair, we construct quadratic expressions $Q_{IV}$ and $R_{IV}$ that satisfy the identity $\partial_x^2Q_{IV}-U_{IV}Q_{IV}=\partial_λR_{IV}$. Here $U_{IV}$ is the coefficient in the normal-form equation. A rapid-decay cycle with a vanishing boundary term yields an integral solution of the linearized equation. Every nontrivial such cycle has asymptotic branches in at least two distinct Stokes sectors; otherwise the integral vanishes by Cauchy's theorem. A second rapid-decay homology class gives another integral solution. The Wronskian of the two integral solutions depends holomorphically on the monodromy data, so a single nonzero value implies generic linear independence. For one set of complex data, direct quadrature gives numerical evidence of a nonzero Wronskian. We also derive integral formulas for variations of the Stokes multipliers, the connection matrix, and the local monodromy, and verify the contour representation by substitution into the linearized equation.