NUMERICAL-STUDIES OF THE 4TH PAINLEVE EQUATION

In this paper the authors investigate numerically solutions of a speci al case of the fourth Painleve equation given by d2eta/dxi2 = 3eta5 2xieta3 + (1/4xi2 - nu - 1/2)eta, (1a) with v a parameter, satisfying the boundary condition eta(xi) --> 0 as xi --> + infinity. (1b) Equati on (1a) arises as a symmetry reduction of the derivative nonlinear Sch rodinger (DNLS) equation, which is a completely integrable soliton equ ation solvable by inverse scattering techniques. Previous results conc erned with solutions of (1) are largely restricted to the case when nu is an integer and very little has been proved when nu is a noninteger . Here a numerical approach to describing solutions of (1) for noninte ger nu is adopted, and information is obtained characterizing connecti on formulae which describe how the asymptotic behaviours of solutions as xi --> + infinity relate to those as xi --> - infinity.