Numerical studies of the fourth Painlevé equation

In this paper the authors investigate numerically solutions of a special case of the fourth Painlevé equation given by

d²η/dξ² = 3η⁵ + 2ξη³ + (1/4ξ² - ν - 1/2)η

with ν a parameter, satisfying the boundary condition

η(ξ)→0 as ξ→ + ∞.

Equation (1a) arises as a symmetry reduction of the derivative nonlinear Schrödinger (DNLS) equation, which is a completely integrable soliton equation solvable by inverse scattering techniques. Previous results concerned with solutions of (1) are largely restricted to the case when ν is an integer and very little has been proved when ν is a noninteger. Here a numerical approach to describing solutions of (1) for noninteger ν is adopted, and information is obtained characterizing connection formulae which describe how the asymptotic behaviours of solutions as §→+∞ relate to those as §→−∞.