Recursion operators for KP, mKP and Harry-Dym Hierarchies
In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym under $n$-reduction. This shows a …
Authors: Jipeng Cheng, Lihong Wang, Jingsong He
RECURSION OPERA TORS FOR KP , MKP AND HARR Y-D YM HIE RAR CHI ES JIPENG CHENG 1 , LIHONG W AN G 2 , JINGSONG HE 2 ∗ 1 Dep artment of Mathematics, USTC, Hef ei, 230026 Anhui, P. R. China 2 Dep artment of Mathematics, NBU, Ni ngb o, 315211 Zhejiang, P. R. China Abstra ct. In this paper, w e give a unified construction of the recursion operators from the Lax representa tion f or three integ rable hierarc hies: Kadomtse v-Petviash vili (KP), mo dified Kadom tsev- P etviashvili (mKP) and H arry-Dym u nder n - reduction. This shows a new inherent relationship b etw een them. T o illustrate o ur construction, th e recursion operator are calculated ex plicitly for 2- red uction and 3-reduction. AMS class ification(2010): 35Q53, 37K10, 37K40. Keywords : K P , mKP and Harry-Dym hierarchie s, recursion op erator. 1. Intr o duction The recursion o p erator Φ, firstly presen ted by P .J. Olv er [1], pla y s a k ey role (see [2–4] and references therein) in th e study of the integrable system. F or single integrable evo lution equation, it alwa ys o wn s infin itely man y comm uting symmetries and bi-Hamiltonian s tr uctures [2 –4 ] which th e recursion op erator can link. As for an integrable hierarch y , th e h igher fl o ws can b e generated from the lo wer flow with the help of th e recurs ion op erator, whic h offers a natural wa y to construct the whole integrable hierarc hy from a single seed system (see [2–4] and r eferences therein). By no w, m u c h work has b een done on the recur sion op erator. F or example, th e construction of the recursion for a giv en in tegrable system [5–17], and the p r op erties of the recursion op erator [18–23]. In general, the recurs ion op erator has non-local term. So it is a highly n on-trivial problem to und erstand the lo calit y of higher ord er symmetries and higher order flo ws generated b y recursion op erator [22, 24]. In this pap er, we shall fo cus on t he construction of th e recur sion op erato r and explain the lo cali t y of their higher fl o ws although the recursion op erator asso ciated is n on-lo cal. The m ain ob ject that we will inv estigate is three in teresting in tegrable hierarc hies, i.e. Kadomtsev- P etviashvili (KP ), mo difi ed Kadomtsev- P etviashvili (mKP) and Harry-Dym h ierarc hies [25, 26], which are corresp ond ing to the d ecomp ositions of the algebra g of p seudo-differen tial op erators g := { X i ≪∞ u i ∂ i } = { X i ≥ k u i ∂ i } ⊕ { X i n − k in terms of ( u 2 − k , u 3 − k , · · · , u n − k ). Thus only n − 1 co ordinates ( u 2 − k , u 3 − k , · · · , u n − k ) are in d ep endent, whic h are in one-to-one corresp ond with ( a k ( n ) , a k +1 ( n ) , · · · , a k + n − 2 ( n )). F or example, u n der the 2- reduction, only u 2 − k is indep end en t, then th e flo w equ ation (15) imp lies the follo wing 1 + 1 dimensional equations, for k = 0 u 2 t 3 = 1 4 u 2 xxx + 3 u 2 u 2 x , (19) u 2 t 5 = 15 2 u 2 2 u 2 x + 5 4 u 2 u 2 xxx + 5 2 u 2 xx u 2 x + 1 16 u 2 xxxxx , (20) for k = 1 u 1 t 3 = 1 4 u 1 xxx − 3 2 u 2 1 u 1 x , (21) u 1 t 5 = 15 8 u 4 1 u 1 x − 5 8 u 1 xxx u 2 1 + 1 16 u 1 xxxxx − 5 8 u 3 1 x − 5 2 u 1 u 1 xx u 1 x , (22) for k = 2 u 0 t 3 = 1 4 u 3 0 u 0 xxx , (23) u 0 t 5 = 1 32 u 3 0 (10 u 0 u 0 xx u 0 xxx + 5 u 0 xxx u 2 0 x + 10 u 0 u 0 xxxx u 0 x + 2 u 0 u 0 xxxxx ) . (24) After the preparation ab ov e, under n -r eduction we can at last rewrite the Lax equations (9) in to matrix forms in terms of a j ( m ). F or this, w e denote U ( n ) = ( u 2 − k , u 3 − k , · · · , u n − k ) t , A ( n, m ) = ( a − 1+ k ( m ) , a − 2+ k ( m ) , · · · , a − n +1+ k ( m )) t , O ( n ) = O 2 − k , 1 − k 0 · · · 0 O 3 − k , 1 − k O 3 − k , 2 − k · · · 0 . . . . . . . . . . . . O n − k , 1 − k O n − k , 2 − k · · · O n − k ,n − 1 − k , (25) 5 where t denotes the transp ose of the matrix, then we can rewrite (15 ) into U ( n ) t m = O ( n ) A ( n, m ) . (26) It is trivial to kno w that all of the flo w equations in U ( n ) t m are lo cal, includin g those in U ( n ) t m + j n . 3. Rec ursion Formulas In th is section, we will construct the recursion op erator. T o do this, we h a ve to first obtain a recursion formula r elating A ( n, m ) and A ( n, m + n ) under n -reduction constrain t, that is, w e try to seek an op erator R ( n ), s.t. A ( n, m + n ) = R ( n ) A ( n, m ). F or th is, we consid er th e relation L m + n = L m L n = L n L m . Assumin g n -r eduction, we fi nd ( L m L n )
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