TY - JOUR
T1 - A unified approach to Painlevé expansions
AU - Newell, A. C.
AU - Tabor, M.
AU - Zeng, Y. B.
N1 - Funding Information:
This work was supported by DMS-8403187 NSF, DAAG-29-85-K-0091 Army, N0014-84-K-0420 ONR Physics, F4962086C0130 AFOSR-Center, U.S. Department of Energy, DE-FG02-84ER 13190, The Alfred P. Sloan Foundation (M.T.) and the Science Fund of the Chinese Academy of Sciences (Y.B.Z.). The authors would like to thank N. Ercolani, H. Flaschka, J.D. Gibbon, and E. Siggia for many valuable discussions.
PY - 1987
Y1 - 1987
N2 - The Painlevé test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Bäcklund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy.
AB - The Painlevé test for partial differential equations developed by Weiss, Tabor and Carnevale (WTC) is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations. A simple modification of the WTC procedure used for partial differential equations enables us to determine the Lax pairs, Hirota equations and auto-Bäcklund transformations for ordinary differential equations, including a new Lax pair for an integrable case of the Henon-Heiles system. A detailed study of the KdV hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained. The role of the singular branches is examined in detail and important new insights obtained. In particular we find that each singular branch is simply a re-expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce. A parallel analysis is carried out for the AKNS hierarchy.
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U2 - 10.1016/0167-2789(87)90046-7
DO - 10.1016/0167-2789(87)90046-7
M3 - Article
AN - SCOPUS:0001952488
VL - 29
SP - 1
EP - 68
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
SN - 0167-2789
IS - 1-2
ER -