Premium
The Painlevé Property and Hirota's Method
Author(s) -
Gibbon J. D.,
Radmore P.,
Tabor M.,
Wood D.
Publication year - 1985
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm198572139
Subject(s) - eigenfunction , mathematics , connection (principal bundle) , property (philosophy) , partial differential equation , nonlinear system , mathematical analysis , simple (philosophy) , soliton , variety (cybernetics) , inverse scattering transform , inverse , inverse scattering problem , pure mathematics , mathematical physics , eigenvalues and eigenvectors , inverse problem , geometry , physics , quantum mechanics , philosophy , statistics , epistemology
The connection between the Painlevé property for partial differential equations, proposed by Weiss, Tabor, and Carnevale, and Hirota's method for calculating N ‐soliton solutions is investigated for a variety of equations including the nonlinear Schrödinger and mKdV equations. Those equations which do not possess the Painlevé property are easily seen not to have self‐truncating Hirota expansions. The Bäcklund transformations derived from the Painlevé analysis and those determined by Hirota's method are shown to be directly related. This provides a simple route for demonstrating the connection between the singular manifolds used in the Painlevé analysis and the eigenfunctions of the AKNS inverse scattering transform.