Recursion operators for KP, mKP and Harry-Dym Hierarchies

The recursion operator Φ, firstly presented by P.J. Olver [1], plays a key role (see [2][3][4] and references therein) in the study of the integrable system. For single integrable evolution equation, it always owns infinitely many commuting symmetries and bi-Hamiltonian structures [2][3][4] which the recursion operator can link. As for an integrable hierarchy, the higher flows can be generated from the lower flow with the help of the recursion operator, which offers a natural way to construct the whole integrable hierarchy from a single seed system (see [2][3][4] and references therein). By now, much work has been done on the recursion operator. For example, the construction of the recursion for a given integrable system [5][6][7][8][9][10][11][12][13][14][15][16][17], and the properties of the recursion operator [18][19][20][21][22][23]. In general, the recursion operator has non-local term. So it is a highly non-trivial problem to understand the locality of higher order symmetries and higher order flows generated by recursion operator [22,24]. In this paper, we shall focus on the construction of the recursion operator and explain the locality of their higher flows although the recursion operator associated is non-local.

The main object that we will investigate is three interesting integrable hierarchies, i.e. Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym hierarchies [25,26], which are corresponding to the decompositions of the algebra g of pseudo-differential operators

for k = 0, 1, 2 respectively, where u i are the functions of t = (t 1 = x, t 2 , • • • ) and ∂ = ∂ x . The algebraic multiplication of ∂ i with the multiplication operator u are defined by

where u (j) = ∂ j u ∂x j , with

In fact, g ≥k and g <k are the sub-Lie algebra of g: [g ≥k , g ≥k ] ⊂ g ≥k and [g <k , g <k ] ⊂ g <k when k = 0, 1, 2. The projections of L = i u i ∂ i ∈ g to g ≥k and g <k are

Then according to the famous Adler-Kostant-Symes scheme [27], the following commuting Lax equations [25,26] on g can be constructed

where k = 0, 1, 2 are corresponding to KP, mKP and Harry-Dym hierarchies respectively, with the Lax operator L given by

(

For simplicity, we rewrite (5) in a unified form [25,26]

i.e.

and let

Then (4) becomes into

These three integrable hierarchies have been studied intensively in literatures [26,[28][29][30], which contain the following well-known 2 + 1 dimensional equations

where we have set t 2 = y, t 3 = t.

There are some inherent relationships discovered among these three integrable hierarchies. For example, their flow equations are defined by a unified Lax equation (9) although their B m are different, their Hamiltonian structure is given by a general way, i.e. r-matrix method [28], and there exists an interesting link among them in the flow equations and gauge transformations [26]. So it is very natural to ask whether there exists a unified way to deal with their recursion operators, which is just our central aim of this paper. For the KP hierarchy, W.Strampp & W.Oevel [9] and V.V.Sokolov et al [11] separately developed a general method to construct the recursion operator by the Lax representation (9). V.V.Sokolov et al [11][12][13][14] used an important ansatz B = PB n + R that relates B n operator for different n, where P is some operator that commutes with the L operator and R is the remainder.

While, W.Strampp & W.Oevel derived a general expression (see eq.(47) of reference [9]) for the recursion operators of the KP hierarchy under n-reduction starting from Lax equations. In this paper, we will use W.Strampp & W.Oevel’s method. However, their method is not applicable to get a similar and compact formula for the mKP and Harry-Dym hierarchies due to following two observations:

2. It is not affirmative to get a compact form of the flow equations of mKP hierarchy and Harry-Dym hierarchy as eq.( 6) and eq.( 17) of reference [9] for the KP hierarchy because of the second summation terms in the last two cases of 1).

In this paper, to further find inherent relations between above three hierarchies, we shall improve W.Strampp & W.Oevel method (use a j (m) only ) and give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: KP, mKP and Harry-Dym under n-reduction (see eq.( 18)). There are two advantages in our construction: 1) it is easy to explain why nonlocal recursion operators produce local flows, since the L.H.S. of (9) only produces the differential polynomials of u i , thus the flow equations of (9) are naturally local; 2) a formula of the recursion operator for arbitrary n-reduction are derived, which shows the existence of recursion operators for the three kinds of integrable hierarchies, and provides a constructive way to get recursion operators for higher order reductions although the calculation is not an easy task.

This paper is organized as follows. In Section 2, we rewrite the unified Lax equati

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