Numerical studies of the fourth Painlev equation

In this paper the authors investigate numerically solutionsof a special case of the fourth Painlev equation given by with a parameter, satisfying theboundary condition Equation (1a) arises asa symmetry reduction of the derivative nonlinear Schrdinger(DNLS) equation, which is a completely integrable soliton equationsolvable by inverse scattering techniques. Previous resultsconcerned with solutions of (1) are largely restricted to thecase when is an integer and very little has been proved when is a noninteger. Here a numerical approach to describing solutionsof (1) for noninteger is adopted, and information is obtainedcharacterizing connection formulae which describe how the asymptoticbehaviours of solutions as + relate to those as –.