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Nonlocal Symmetries for Bilinear Equations and Their Applications
Author(s) -
Hu XingBiao,
Lou SenYue,
Qian XianMin
Publication year - 2009
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 - 10.1111/j.1467-9590.2009.00435.x
Subject(s) - bilinear interpolation , homogeneous space , hierarchy , integrable system , mathematics , bilinear form , flow (mathematics) , pure mathematics , mathematical physics , mathematical analysis , geometry , law , statistics , political science
In this paper, nonlocal symmetries for the bilinear KP and bilinear BKP equations are re‐studied. Two arbitrary parameters are introduced in these nonlocal symmetries by considering gauge invariance of the bilinear KP and bilinear BKP equations under the transformation . By expanding these nonlocal symmetries in power series of each of two parameters, we have derived two types of bilinear NKP hierarchies and two types of bilinear NBKP hierarchies. An impressive observation is that bilinear positive and negative KP (NKP) and BKP hierarchies may be derived from the same nonlocal symmetries for the KP and BKP equations. Besides, as two concrete examples, we have derived bilinear Bäcklund transformations for t −2 ‐flow of the NKP hierarchy and t −1 ‐flow of the NBKP hierarchy. All these results have made it clear that more nice integrable properties would be found for these obtained NKP hierarchies and NBKP hierarchies. Because KP and BKP hierarchies have played an essential role in soliton theory, we believe that the bilinear NKP and NBKP hierarchies will have their right place in this field.