On $W_{1+\infty}$ 3-algebra and integrable system

We construct the $W_{1+\infty}$ 3-algebra and investigate the relation between this infinite-dimensional 3-algebra and the integrable systems. Since the $W_{1+\infty}$ 3-algebra with a fixed generator $W^0_0$ in the operator Nambu 3-bracket recovers the $W_{1+\infty}$ algebra, it is natural to derive the KP hierarchy from the Nambu-Poisson evolution equation. For the general case of the $W_{1+\infty}$ 3-algebra, we directly derive the KP and KdV equations from the Nambu-Poisson evolution equation with the different Hamiltonian pairs. We also discuss the connection between the $W_{1+\infty}$ 3-algebra and the dispersionless KdV equations. Due to the Nambu-Poisson evolution equation involves two Hamiltonians, the deep relationship between the Hamiltonian pairs of KP hierarchy is revealed. Furthermore we give a realization of $W_{1+\infty}$ 3-algebra in terms of a complex bosonic field. Based on the Nambu 3-brackets of the complex bosonic field, we derive the (generalized) nonlinear Schr\"{o}dinger equation and give an application in optical soliton.


Introduction
The infinite-dimensional algebras play a very important role in the study of integrable systems. This is due to the fact that they have a deep intrinsic connection to the integrable systems.
The Virasoro algebra is an important infinite-dimensional algebra which has been turned out to be a universal symmetry algebra in two-dimensional conformal field theory. Gervais and Neveu [1,2] found that this infinite-dimensional algebra is intrinsically related to the Kortewegde Vries (KdV) equation where the Poisson bracket of the KdV equation is indeed the classical Virasoro algebra. The lattice Virasoro algebra appears in the study of the Toda field theory and Toda integrable systems [3,4]. Furthermore the intrinsic connection between the super Virasoro algebra and super KdV equation has been studied in [5]- [10]. As a higher-spin extension of the Virasoro algebra, the W N algebra has been found to be associated to the generalized KdV hierarchies [11,12].
The KP hierarchy is an integrable system consisting of an infinite number of non-linear differential equations. This hierarchy has proven to be bi-Hamiltonian and to possess an infinite number of conserved quantities. All the (generalized) KdV hierarchies can be incorporated into the KP hierarchy. The W 1+∞ algebra [13] can be regards as a linear deformation of the W N algebras in the large N limit [14]. It has very important applications in physics, such as quantum Hall effect [15], topological string [16], Alday-Gaiotto-Tachikawa (AGT) relation [17], 2-dimensional quantum gravity [18] and crystal melting [19]. In the context of integrable systems, it was shown that the W 1+∞ algebra is intrinsically related to the first Hamiltonian structure of the KP hierarchy [20,21]. The second Hamiltonian structure of the KP hierarchy was then shown to be isomorphic toŴ ∞ [22], a centerless deformation of W ∞ .
Nambu mechanics [23] is a generalization of classical Hamiltonian mechanics in which the usual binary Poisson bracket is replaced by the Nambu bracket. Based on the Nambu bracket, a notion of a Nambu 3-algebra was introduced in [24] as a natural generalization of a Lie algebra for higher-order algebraic operations. Recently 3-algebras have been paid great attention due to a world-volume description of multiple M2-branes proposed by Bagger and Lambert [25], and Gustavsson [26] (BLG). There has been an increasing interest in the applications of 3algebras to the string/M-theory [27]- [34]. The properties of various 3-algebras especially the infinite-dimensional cases have been widely investigated, such as Virasoro-Witt 3-algebra [35]- [37], Kac-Moody 3-algebra [38] and w ∞ 3-algebra [39,40]. Furthermore the q-deformation of Virasoro-Witt 3-algebra has also been investigated. The nontrivial q-deformed Virasoro-Witt 3-algebra constructed in [41] is the so-called sh-3-Lie algebra.
More recently, the relation between the infinite-dimensional 3-algebras and the integrable systems has been investigated. Chen er al. [42] investigated the classical Heisenberg and w ∞ 3-algebras and established the relation between the dispersionless KdV hierarchy and these two infinite-dimensional 3-algebras. They found that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. It should be pointed out that the dispersionless KdV hierarchy is a very simple integrable system. An interesting open question is whether we can derive other integrable nonlinear evolution equations from the infinitedimensional 3-algebras. We have mentioned that the first Hamiltonian structure of the KP hierarchy is indeed corresponding to the W 1+∞ algebra. In this paper, we shall construct the W 1+∞ 3-algebra and discuss the relation between this infinite-dimensional 3-algebra and the integrable systems. This paper is organized as follows. In section 2, we construct the W 1+∞ 3-algebra. In section 3, we establish the relation between the W 1+∞ 3-algebra and the KP hierarchy in the framework of Nambu mechanics. In section 4, By means of the W 1+∞ 3-algebra, We derive the KP and KdV equations from the Nambu-Poisson (N-P) evolution equation with the different Hamiltonian pairs. In section 5, we establish the connections between the W 1+∞ 3-algebra and the dispersionless KdV equations. A conjecture is presented. In section 6, we give a realization of W 1+∞ 3-algebra and investigate the (generalized) nonlinear Schrödinger equation. Moreover we give an application of the generalized nonlinear Schrödinger equation derived from the N-P evolution equation. We end this paper with the concluding remarks in section 7.
. The first few brackets read The first three commutators can be recognized as the semidirect product of the Witt algebra with an abelian current algebra.
Taking the Fourier transformation field (FTF) and defining {, } = −i[, ], we may rewrite (1) as the following Poisson bracket algebra derived in [20]: The W 1+∞ algebra is an infinite-dimensional higher-spin algebra with generators having all integer conformal spins s ≥ 1 [13]. It is worth noting that the Poisson bracket algebra (4) has been proved to be isomorphic to the W 1+∞ algebra [20]. Thus we call (1) the W 1+∞ algebra.
Another important infinite-dimensional algebra which should be mentioned is the higherorder Virasoro algebra (HOVA). By investigating the higher-order differential neighbourhoods in holomorphic mappings of S 1 a vector space, Zha [43] first constructed this infinite-dimensional algebra. Its generators are {L r m = (−1) r z m+r d r dz r |m, r ∈ Z, r ≥ 0}. Taking z = e ix , we find that the generators of HOVA and W 1+∞ algebra can be linearly expressed reciprocally. It implies that the W 1+∞ algebra (1) is also indeed isomorphic to the HOVA.

2.2
W 1+∞ 3-algebra The operator Nambu 3-bracket was defined to be a sum of commutators of two operators multiplying the remaining one [23], i.e., Based on the operator Nambu 3-bracket (5) and the commutation relation (1), after a straightforward calculation we obtain the W 1+∞ 3-algebra Taking k = h = 0 in (6) It is known that all three types of 3-brackets do satisfy a 3-on-3-on-3 multiple bracket relation, i.e., so-called BI condition [44,36] where i 1 , · · · , i 6 are implicitly summed from 1 to 6. Thus the BI may be confirmed to hold for the ternary algebra (6).
It is worthwhile to mention another condition, i.e., Filippov condition or fundamental identity (FI) condition [45] [ The FI condition (9) is not an operator identity. It holds only in special circumstances. For the W 1+∞ 3-algebra (6), we find that it does not satisfy the FI condition (9).
In order to achieve a better understanding of the W 1+∞ 3-algebra (6), we list the first few brackets as follows: We observe that these brackets do not close except for the generators W i m , i = 0, 1. The Nambu 3-bracket (5) with respect to the generators W 0 m and W 1 n indeed give the so-called the Virasoro-Witt 3-algebra [35,37]. However, this 3-algebra does not satisfy the FI condition (9). The general Virasoro-Witt 3-algebra is given by Curtright et al. [35] [R p , R q , R k ] = 0, where z is a parameter. They showed that the Virasoro-Witt 3-algebra (11) does not satisfy the FI condition (9), except when z = ±2i.
In order to derive the classical w 1+∞ 3-algebra, we take the generators to be {W r m = (−i ) r e imx d r dx r |m, r ∈ Z, r ≥ 0}, where is introduced into the generators. By means of (6) and classical limit relation {, , } = lim →0 1 i [, , ], we have the classical w 1+∞ 3-algebra as follows: Not as the case of W 1+∞ 3-algebra, we find that the classical w 1+∞ 3-algebra (12) satisfies the FI condition.

W 1+∞ 3-algebra and KP hierarchy
The KP hierarchy is a paradigm of the hierarchies of integrable systems. It is defined as an infinite system of equations given in Lax form by under the compatible condition, (13) is equivalent to the Zakharov-Shabat (ZS) equation where the pseudo-differential operator L is given by and B n = (L) n + is the differential part of L n . The Hamiltonians of the KP Hierarchy are given by the general expression The first few members are where the subscript x denotes the derivative with respect to the variable x.
It is known that the Poisson bracket (4) leads to the Hamiltonian description of the KP flow equations [20,21]. Substituting the Hamiltonians (16) into the Poisson evolution equation and using the Poisson bracket (4), we may derive the KP hierarchy.
In order to establish the relation between the W 1+∞ 3-algebra and integrable systems, let us introduce the FTF as follows: and define the classical Nambu bracket { , , } = −i[ , , ]. Then we can rewrite the W 1+∞ 3-algebra (6) as the following Nambu 3-bracket relation: Not as the Poisson evolution equation, the N-P evolution equation involves two Hamiltonians.
It is given by In terms of the Nambu 3-bracket relation (20), we have Comparing (22) with (4), it gives the equivalent relation where the FTFs in Nambu 3-bracket and Poisson bracket are given by (19) and (3), respectively.
This equivalent relation is indeed corresponding to the relation (7)  means of (20), we note that they are not in involution with the Nambu 3-bracket structure, i.e., For the Hamiltonian pairs (H 0 , H n ), by means of (23), it is not difficult to show that Thus based on (20) and (24), it is natural to derive the KP hierarchy from the N-P evolution equation (21) with H m = H 0 . For later convenience, we list the first few members of the hierarchy here.
• Let us consider the following equations: From (25b) and (25c), we obtain Substituting (26) into (25a), we obtain the usual KP equation where y = t 02 and t = t 03 . The KP hierarchy can also be obtained from (14). It is easy to derive the KP equation (27) from (14) by taking m = 2 and n = 3.
• To give the integrable equation corresponding to the case of m = 2 and n = 4 in (14), we consider the following equations: From (28b), (28c) and (28d) we may give the expressions of v 1 , v 2 and v 3 with respect to v 0 , respectively. Then substituting them into (28a), we obtain where y = t 02 and t = t 04 .
• The integrable equation corresponding to the case of m = 2 and n = 5 in (14) can be derived from the following equations: The integrable equation is where y = t 02 and t = t 05 .
• The integrable equation corresponding to the case of m = 2 and n = 6 in (14) can be derived from the following equations: The integrable equation is where y = t 02 and t = t 06 .

KP equation, KdV equation and W 1+∞ 3-algebra
One has already known that the W 1+∞ algebra is related to the KP hierarchy. Since the W 1+∞ 3-algebra with a fixed generator W 0 0 in the operator Nambu 3-bracket (6) recovers the W 1+∞ algebra, it is natural to derive the KP hierarchy from the W 1+∞ 3-algebra. To explore the deep connection between the W 1+∞ 3-algebra and the integrable equations, we need to deal with new case studies. Hereafter we focus on the general case of the W 1+∞ 3-algebra. Due to the N-P evolution equation involving two Hamiltonians, this may hide some still unknown relations between these Hamiltonians. Thus it would be quite interesting to understand these Hamiltonians from the N-P evolution equation.
To start with let us assign the weights as follows: Theorem For any Hamiltonian pair (H m , H n ) of the KP hierarchy, 0 ≤ m < n, from the N-P evolution equations where i = 0, 1, · · · , s − 2 and s = m + n ≥ 1, it gives the evolution equation with respect to v 0 .
The weight of equation is s + 2.
Proof By weight counting, it is easy to see that the pseudo-differential operator L (15)  finite v i , i = 0, 1, 2, · · · , n being permitted to emerge in H n . Note that for these finite v i , v n has the highest weight n + 2. Thus to determine how many v i appearing in (35a), let us consider In terms of (36), we find that there are only finite v i , i = 0, 1, 2, · · · , s − 1 being permitted to emerge in the evolution equation of v 0 (35a).
After a straightforward calculation, (35b) gives its final form Thus we have a recursion relation from (37) By means of the recursion relation (38), we may express v i , i = 1, 2, · · · , s − 1 as the function of v 0 . Then substituting these v i into (35a), we obtain the evolution equation with respect to v 0 .
Since the Nambu 3-bracket relation (20) and the Hamiltonian (16) Substituting (26) into (39), we obtain where y = t 02 and t = t 12 . Under the scaling transformations y → 1 √ −3 y and v 0 → 3 7 v 0 , (40) becomes the usual KP equation (27). What is surprising here is that we derive the KP equation Comparing (39) with (25a), we note that the coefficients of fields v 1 and v 2 in (25a) are three times that of (39). To explore more relationship between the Hamiltonian pairs (H 0 , H 3 ) and (H 1 , H 2 ), we consider the following evolution equation: Substituting the expressions of (25a) and (39) into (41), we obtain where t = t 03 = t 12 . Under the scaling transformations t → −2t and v 0 → 1 4 v 0 , (42) becomes the well-known KdV equation The connection between the KdV equation and the Virasoro algebra was first pointed out by Gervais then from (44) and the last three equations in (28), we obtain where y = t 02 and t = t 13 . Its single soliton solution is where in which ω = 1 4 (−k 2 + 1), k and c are the constants. Although (45) has the single soliton solution, by performing a Painlevé analysis of this equation, we note that it does not pass the test. Hence (45) does not have the Painlevé integrable property. In spite of this negative result it is instructive to pursue the analysis of (45). Taking the reduction x = y, it is interesting to note that (45) may reduce to the KdV equation (43) with the appropriate scaling transformations.
Under the low dimensional reduction, we may also derive the KdV equation from the following N-P evolution equations with respect to v 0 : Comparing (48a) with (28a), we observe that there are not the fields v 2 and v 3 in (48a). Thus not as the case of (28), we do not need the evolution equations of v 1 and v 2 here. Eliminating v 1 via the evolution equations of v 0 (48b), we can rewrite (48a) as where t = t 04 = t 13 and y = t 02 . Its single soliton solution is where ξ = k(−k 2 t + x + y) + c, k and c are the constants. By performing a Painlevé analysis of this equation, we note that it does not pass the test. Taking the reduction x = y and the scaling transformations t → −t and v 0 → 1 4 v 0 , (49) reduces to the KdV equation (43).

W 1+∞ 3-algebra and dispersionless KdV hierarchy
In the previous section, we derived the (2+1)-dimensional equation (45) then from (51) and the last four equations in (30), we obtain where y = t 02 and t = t 14 . Its single soliton solution is where ξ is given by (47) we can derive the following equation: where y = t 02 and t = t 23 . By applying the Painlevé analysis to (55), we find that it does not pass the test. As the case of (52), (55) has the single soliton solution as follows: where ξ is given by (47) with ω = 3 28 k 4 + 67 700 . Under the reduction y = 0, (31), (52) and (55) reduce to the following equations: and respectively. In the dispersionless limit ∂ t → ǫ∂ t and ∂ x → ǫ∂ x with ǫ → 0, (57), (58) and (59) reduce to the following dispersionless KdV equation with the appropriate scaling transformations: Note that (57) is the completely integrable Lax's fifth order KdV equation [46], but (58) and (59) are not integrable. The fifth order KdV equations describe motions of long waves in shallow water under gravity and in a one-dimensional nonlinear lattice and has wide applications in quantum mechanics and nonlinear optics. Thus a great deal of research work has been invested for the study of the fifth order KdV equations. Recently the following general fifth order KdV equation has been investigated [47]: where α, β, γ and ω are the arbitrary real parameters. By using the exp function method, the solutions of (61) have also been presented. It can be very easily seen that (58) and (59) are the special cases of (61).
The integrable equation corresponding to the case of m = 2 and n = 7 in (14) can be derived from the following equations: Due to too many terms in this integrable equation, we do not exhibit its explicit expression here.
Let us replace the first equation in (62) by respectively. We may also derive three non-integrable equations. Taking the reduction y = 0 and the dispersionless limit, these four equations reduce to the following dispersionless KdV equation with the appropriate scaling transformations: The dispersionless KdV hierarchy is a well studied integrable system [48,49]. It is given by where A n is given by We have derived the third and fourth order dispersionless KdV equations from the N-P evolution equation. For the second order dispersionless KdV equation in (65), it is easy to obtain from (27) and (40) under the low dimensional reduction with the dispersionless limit, respectively.
From above several examples, it is of interest to note that for a higher-order dispersionless KdV equation, we may derive it from the N-P evolution equation with the different Hamiltonian pairs.
It indicates that there is the intrinsic equivalent relation between the Hamiltonian pairs. We call this intrinsic relation between the Hamiltonian pairs "degeneration". Based on above analysis, we present a conjecture as follows: Conjecture For any Hamiltonian pair (H m , H n ) of the KP hierarchy, 0 ≤ m < n, when s = m + n is odd, we may choose s+1 2 different Hamiltonian pairs (H m , H n ) for (35a) such that under the low dimensional reduction with the dispersionless limit, (35) leads to the dispersionless KdV hierarchy.
Let us make a comment about this conjecture. When s = 1, from (35a), it is easy to obtain This is the first equation in (65).
When s = m + n is odd and s ≥ 3, let us choose the following s+1 2 different Hamiltonian pairs (H s−1 2 −i , H s+1 2 +i ), i = 0, 1, · · · s−1 2 . Based on the theorem in the previous section, we can derive s+1 2 different evolution equations with respect to v 0 with odd weight s + 2. Under the low dimensional reduction t 02 = 0 and the dispersionless limitation, i.e., ∂ tmn → ǫ∂ tmn and ∂ x → ǫ∂ x with ǫ → 0, all higher derivation terms of v 0 in the evolution equations are vanishing. Thus we have the following equation with the weight s + 2: where α mn is a constant. When α mn = 0, by taking the appropriate scaling transformation, (68) leads to the dispersionless KdV hierarchy. However, how to prove α mn = 0 is the most challenging problem.

Non-linear Schrödinger equation
The non-linear Schrödinger equation is given by where ψ is a complex bosonic field.
The non-linear Schrödinger equation possesses an infinite set of conserved quantitiesĤ n , given by the formulaĤ where The first few such Hamiltonians arê The non-linear Schrödinger equation (69) can be obtained from where the Poisson brackets of the complex bosonic field are given by [50] {ψ It is known that there is a realization of W 1+∞ algebra in terms of a complex bosonic field with Poisson brackets (74). Taking where ψ (r) (x) = d r ψ(x) dx r , by means of (74), it is easy to verify that the generators (75) satisfy (4). Due to this kind of realization of W 1+∞ algebra, the connection between the non-linear Schrödinger equation and the KP hierarchy has been established by Freeman and West [50].
For the case of W 1+∞ 3-algebra (20), we find that its realization can be achieved in terms of a complex bosonic field with Nambu 3-brackets given by The proof is straightforward. One can confirm the W 1+∞ 3-algebra (20) by substituting v r (x) = √ 6 2 (1 + i)(−1) r ψ * (x)ψ (r) (x) and using the Nambu 3-brackets (76). As the cases of (23) and (24), in terms of the Nambu 3-bracket relations (76), we have also the similar equivalent relations Thus we may also derive the non-linear Schrödinger equation from the N-P evolution equation • Hamiltonian pair (Ĥ 0 ,Ĥ 3 ) This equation is the so-called integrable complex mKdV equation.

An application in optical soliton
It is well-known that the nonlinear Schrödinger equation plays a critical role in the study of optical solitons for optical fiber communications. To describe a large number of nonlinear effects in optical fibers, the effects of the higher-order terms on the nonlinear Schrödinger equation have been paid more attention. The propagation of optical field in a monomode optical fiber may be described by the following generalized nonlinear Schrödinger equation [52]: where Z is the normalized propagation distance, T is the normalized retarded time, i.e., time in the frame of reference moving with the wave packet, A(Z, T ) denotes the normalized slowly varying complex wave packet envelope in the electric filed, and |A| 2 represents the optical power.
(85) is an important nonlinear equation for studying higher-order nonlinear effects in optical fibers. The second and third terms on the left side of (85) describe the effects of fiber loss and chromatic dispersion, respectively. The term proportional to β 3 governs the effects of third-order dispersion and becomes important for ultrashort pulses because of their wide bandwidth. The first term on the right side of (85) describes the nonlinear effects of self-phase modulation. The terms proportional to ω −1 0 and T R are responsible for the phenomenon of self-steepening and the Raman-induced frequency shift, respectively.
Based on the results in the previous subsection, let us turn to derive generalized nonlinear Schrödinger equation in the framework of Nambu mechanics. For the Hamiltonian pairs (Ĥ 0 ,Ĥ 2 ) and (Ĥ 1 ,Ĥ 2 ), we consider the following N-P evolution equation: where the parameters β 2 and β 3 in (85) are introduced into (86).
By means of (78) and (80), (86) leads to Its single soliton solution is where k is a real constant.
Comparing (85) with (87), we observe that in absence of the fiber loss and Raman-induced frequency shift terms, i.e., α = T R = 0, (85) reduces to (87) with γ = −β 2 , and ω 0 = 3β 2 β 3 . We have already seen that the generalized nonlinear Schrödinger equation can be derived in terms of the Hamiltonian pairs (Ĥ 0 ,Ĥ 2 ) and (Ĥ 1 ,Ĥ 2 ). It should be pointed out that we may also derive the similar result just from the N-P evolution equation with the Hamiltonian pair (Ĥ 1 ,Ĥ 2 ) under the appropriate variable transformations. For the Hamiltonian pair (Ĥ 1 ,Ĥ 2 ), the corresponding N-P evolution equation is given by (80). Let us take the variable transformations as follows: It is not difficult to prove that under the variable transformations (89), (80) becomes (85) with T R = 0 and ω 0 = β 2 β 3 .

Concluding Remarks
The KP hierarchy is an important integrable system which can be regarded as the generalization of KdV hierarchies. It is well-known that there is a remarkable connection between the integrable systems and the infinite-dimensional conformal algebra and its extensions. For the KP hierarchy, one has known that its first Hamiltonian structure is related to the W 1+∞ algebra. To discuss the relation between the infinite-dimensional 3-algebra and the KP hierarchy, we constructed the W 1+∞ 3-algebra which does not satisfy the FI condition, but the BI condition holds for this ternary algebra. By introducing the FTF, we rewrote the W 1+∞ 3-algebra as the Nambu 3bracket structure of the FTF. We noted that when there is a fixed generator W 0 0 in the operator Nambu 3-bracket, the W 1+∞ 3-algebra may reduce to the W 1+∞ algebra. Thus based on this special W 1+∞ 3-algebra, we derived the KP hierarchy from the N-P evolution equation with the Hamiltonian pairs (H 0 , H i ).
To explore the relation between the general case of the W 1+∞ 3-algebra and integrable equations, we replaced the Hamiltonian pair (H 0 , H 3 ) by (H 1 , H 2 ) in the N-P evolution equation (25).
An intriguing feature is that it still leads to the KP equation. Moreover without taking the low dimensional reduction, we directly derived the KdV equation by means of the N-P evolution equation with the Hamiltonian pairs (H 0 , H 3 ) and (H 1 , H 2 ). It did provide new insight into the KP and KdV equations. Furthermore we investigated the N-P evolution equation (35) with the different Hamiltonian pairs and established the connections between the W 1+∞ 3-algebra and the dispersionless KdV equations. We pointed out that there is an intrinsic equivalent relation between the Hamiltonian pairs of KP hierarchy, i.e., "degeneration". We also presented a conjecture with respect to the "degeneration". For the Hamiltonians of KP hierarchy, one did only know that these Hamiltonians are in involution. Our investigation turned out that the intrinsic relationship between these Hamiltonians is actually more complicated than has been previously recognized.
For the W 1+∞ 3-algebra, we also presented its realization in terms of a complex bosonic field with Nambu 3-brackets (76). Based on (76) Although the W 1+∞ 3-algebra does not satisfy the so-called FI condition, our analysis suggests that there still exist the much deeper connections between the W 1+∞ 3-algebra and the integrable equations. More properties with respect to their relations still deserve further study. We believe that the infinite-dimensional 3-algebras may shed new light on the integrable systems.