A Computational Exploration of the Second Painlevé Equation

A Computational Exploration of the Second Painlevé Equation

The pole field solver developed recently by the authors (J. Comput. Phys. 230:5957–5973, 2011 ) is used to survey the space of solutions of the second Painlevé equation that are real on the real axis. This includes well-known solutions such as the Hastings–McLeod and Ablowitz–Segur type of solutions...

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Veröffentlicht in:Foundations of computational mathematics 2014-10, Vol.14 (5), p.985-1016
Hauptverfasser: Fornberg, Bengt, Weideman, J. A. C.
Format: Artikel
Sprache:eng
Schlagworte:
Analysis
Applications of Mathematics
Computation
Computer Science
Differential equations
Differential equations, Nonlinear
Economics
Exploration
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
Poles
Robustness
Solvers
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Beschreibung
Zusammenfassung:The pole field solver developed recently by the authors (J. Comput. Phys. 230:5957–5973, 2011 ) is used to survey the space of solutions of the second Painlevé equation that are real on the real axis. This includes well-known solutions such as the Hastings–McLeod and Ablowitz–Segur type of solutions, as well as some novel solutions. The speed and robustness of this pole field solver enable the exploration of pole dynamics in the complex plane as the parameter and initial conditions of the differential equation are varied. Theoretical connection formulas are also verified numerically.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-013-9156-x