Derivation of a matrix Painlevé equation germane to wave scattering by a broken corner (Q1076910)

The author considers the problem of wave scattering by a broken corner in two dimensions, modelled by the equation \(\nabla^ 2\psi =\lambda \psi\) subject to certain boundary conditions. This partial differential equation is reduced to a small number of linear ordinary differential equations whose coefficients involve the solution of a nonlinear ordinary differential equation resembling Painlevé's equation but with the dependent variable being a \(2\times 2\) symmetric matrix.