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Zbl 0782.65099
Bassom, A.P.; Clarkson, P.A.; Hicks, A.C.
Numerical studies of the fourth Painlevé equation. (English)
[J] IMA J. Appl. Math. 50, No.2, 167-193 (1993). ISSN 0272-4960; ISSN 1464-3634/e

The authors investigate numerically solutions of a special case of the fourth Painlevé equation $d\sp 2\eta/d\xi\sp 2 = 3\eta\sp 5 + 2\xi\eta\sp 3 + ((1/4)\xi\sp 2 -\nu-1/2)\eta$ with $\nu$ a parameter, satisfying the boundary condition $\eta(\xi)\to 0$ as $\xi\to + \infty$. The equation arises as a symmetric reduction of the derivative Schrödinger equation, which is a completely integrable soliton equation solvable by inverse scattering techniques.\par A numerical approach to describe the solution of the equation for noninteger $\nu$ is adopted, and information is obtained characterizing connection formulae which describe how the asymptotic behaviour of solutions as $\xi \to +\infty$ relates to that as $\xi \to -\infty$. A typical result shows the solution blows up whenever $\nu < -1$.
[T.Mitsui (Chikusa-ku / Nagoya)]

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
34B30 Special ODE
34B15 Nonlinear boundary value problems of ODE

Keywords: numerical investigation; fourth Painlevé equation; Schrödinger equation; soliton equation; inverse scattering; asymptotic behaviour

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