文档介绍：中国科学技术大学
博士学位论文
几类经典可积系列的对称与递归算子*
姓名:程纪鹏
申请学位级别:博士
专业:数学系数学物理专业
指导教师:胡森;贺劲松
2011-03
Á
© Ì ï Ä
A a ² ; È X é ¡ 4 8 f : 1)
Ñ
B. Kadomtsev-PetviashviliX (BKPX ) Ì L « ± 9 B.
Squared Eigenfunction Potential (BSEP), 3 d Ä : þ X Ú ¨ E
BKPX “ghost”é ¡ , ¿ ? Ø
Ù 3 í å BKPX V
5 ª ± 9 BKPX Adler-Shiota-van-Moerbeke (ASvM)ú ª
¡ A^ ; 2) ¨ E
B. Ú C. Toda : X (BTL & CTLX )
N \ é ¡ Ñ
Ù ê ( ¨ © O B B ± 9 C C Ó
, w∞× w∞ w∞× w∞,
? Ø
N \ é ¡ ) ¤ ¼ ê ; 3) l KP, modified KP (mKP)
& Harry-Dym (HD)X Lax f Ñ u , Ñ
ù n È X
3 n z ^ e , Ú 4 8 f ¨ E ª ; 4) Ñ
HDX
6 § L ª .
'
c : KP, mKP & HD X , TLX , BTL&CTLX , BKPX
, “ghost”é ¡ , N \ é ¡ , 4 8 f , 6 § .
i
Abstract
The symmetries and the recursion operators for several kinds of classi-
cal integrable hierarchies are studied: 1) the spectral representation of the
Kadomtsev-Petviashvili hierarchy of B type (BKP hierarchy) and the squared
eigenfunction potential of B type (BSEP) are given, then based upon these
results, the “ghost” symmetry for the BKP hierarchy is systematically con-
structed, and some applications in the derivations of the bilinear identity for
the constrained BKP hierarchy and the Adler-Shiota-van-Moerbeke (ASvM)
formula for the BKP hierarchy are considered; 2) the additional symmetries
for the Toda lattice hierarchies of B type and C type (BTL and CTL hierar-
B B
chies) are constructed, then its algebraic structures are showed as w∞× w∞
C C
and w∞× w∞ respectively, and also the generating functions of the corre-
ponding additional symmetries are discussed; 3) a unified construction of
the recursion operators from the Lax representation for three integrable hi-
erarchies: KP, modified KP (mKP) and Harry-Dym (HD) hierarchies under
n-reduction is given; 4) a general formula of the flow equtions for the HD
hierarchy is derived.
Keywords: KP, mKP&HD