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Numerical studies of the fourth Painlevé equation. (English) Zbl 0782.65099

The authors investigate numerically solutions of a special case of the fourth Painlevé equation d 2 η/dξ 2 =3η 5 +2ξη 3 +((1/4)ξ 2 -ν-1/2)η with ν a parameter, satisfying the boundary condition η(ξ)0 as ξ+. The equation arises as a symmetric reduction of the derivative Schrödinger equation, which is a completely integrable soliton equation solvable by inverse scattering techniques.

A numerical approach to describe the solution of the equation for noninteger ν is adopted, and information is obtained characterizing connection formulae which describe how the asymptotic behaviour of solutions as ξ+ relates to that as ξ-. A typical result shows the solution blows up whenever ν<-1.

MSC:
65L10Boundary value problems for ODE (numerical methods)
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
34B15Nonlinear boundary value problems for ODE