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On the application of solutions of the fourth Painlevé equation to various physically motivated nonlinear partial differential equations
journal contribution
posted on 2023-05-18, 18:30 authored by Andrew BassomAndrew Bassom, Clarkson, PA, Hicks, ACRecently significant developments have been made in the understanding of the theory and solutions of the fourth Painlevé equation given by
ww" = 12 (w')2 + 32 w4 + 4zw3 + 2(z2 − α)w2 + β,
where α and β are arbitrary constants. All exact solutions of (1) are thought to belong to one of three solution hierarchies. In two of these hierarchies solutions may be determined in terms of the complementary error and parabolic cylinder functions whilst the third hierarchy contains rational solutions which may be expressed as ratios of polynomials in z. It is known that the fourth Painlevé equation (1) can arise as a symmetry reduction of a number of significant partial differential equations and here we consider the application of solutions of (1) to the nonlinear Schrödinger, potential nonlinear Schrödinger, spherical Boussinesq and 2+1-dimensional dispersive long wave equations. Consequently we generate a number of new exact solutions of these equations. The fourth Painlevé equation (1) is also related to problems that arise in the consideration of two-dimensional quantum gravity and this aspect is also discussed.History
Publication title
Advances in Differential EquationsPagination
175-198ISSN
1079-9389Department/School
School of Natural SciencesPublisher
Khayyam Publishing Company, IncPlace of publication
United States of AmericaRepository Status
- Restricted