Journal of Nonlinear Mathematical Physics

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Volume 26, Issue 1, December 2018, Pages 91 - 97

A 3-Lie algebra and the dKP Hierarchy

Authors
Min-Ru Chen
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
Ying Chen
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
Zhao-Wen Yan
School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Jian-Qin Mei
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
Xiao-Li Wang
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China,wxlspu@qlu.edu.cn
Received 18 May 2018, Accepted 15 July 2018, Available Online 6 January 2021.
DOI
10.1080/14029251.2019.1544791How to use a DOI?
Keywords
3-Lie algebra; dKP hierarchy; Integrable systems
Abstract

In terms of a 3-Lie algebra and the classical Poisson bracket {Bn,L} of the dKP hierarchy, a special 3-bracket {Bm,Bn,L} is proposed. When m = 0 or m = 1, the 3-lax equation ∂L∂t={Bm,Bn,L} is the dKP hierarchy and the corresponding proof is given. Meanwhile, for the generalized case (m,n), the generalized dKP hierarchy is also investigated.

Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

Nambu mechanics [14, 15] is a generalization of classical Hamiltonian mechanics. In the context of integrable systems, the integrable hydrodynamical systems have been investigated via Nambu mechanics [9, 10]. Moreover various super-integrable systems, such as Calogero-Moser system, Kepler problem, three Hamiltonian structures of Landau problem, have been analyzed in the framework of Nambu mechanics [1, 16], where a super-integrable system means that it is not only an integrable system in the Liouville-Arnold sense, but also possesses more constants of motion than degrees of freedom. With the development of infinite-dimensional 3-algebras [2, 3, 6, 8], recently more attempts have been made to understand the connection between the infinite-dimensional 3-algebras and the integrable systems in the framework of Nambu mechanics. Chen et al. [4] investigated the classical Heisenberg and w∞ 3-algebras, and established the relations 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. The W1+∞ 3-algebra was constructed in [5] and its connection with the integrable system has also been investigated.

The dispersionless Kadomtsev-Petviashvili (dKP) hierarchy is a paradigm of the integrable systems, which arises as the quasi classical limit of the KP hierarchy. It consists of an infinite number of nonlinear differential equations. This kind of integrable system was introduced by Lebedev, Manin and Zakharov [13]. Many special solutions were obtained by Kodama and Gibbons [11]. Krichever [12] studied the dKP hierarchy and introduced the analogue of the tau function to integrate the consistency conditions for the free energy of the topological minimal models. It is well-known that the dKP hierarchy can be represented in terms of a Lax equation ∂L∂tn={Bn,L}. In this paper, we reinvestigate the property of the Lax equation Bn and L of the dKP hierarchy in the framework of 3-Lie algebra. In terms of a 3-bracket and Lax equation of the dKP hierarchy, we present a 3-Lax equation with respect to the Lax triple {Bm, Bn, L}, and derive the corresponding (non)integrable nonlinear evolution equations for the cases of the different Lax triples {Bm, Bn, L}.

This paper is organized as follows. In section 2, we briefly review the definition of the dKP hierarchy and define a 3-Lax equation ∂L∂tm,n={Bm,Bn,L}. Meanwhile, We prove that when (m,n) takes the special values (0,n + 1) and (1,n), the dKP hierarchy is derived. In addition, the generalized case of the 3-Lax equation ∂L∂tm,n={Bm,Bn,L} is discussed. We end this paper with the concluding remarks in section 3.

2. The dKP Hierarchy

The Lax equation with respect to a series of independent time variables (t1, t2,...) for the dKP hierarchy is

∂L∂tn={Bn,L},Bn=(Ln)+,(2.1)
where L is a Laurent series of λ as the form
L=λ+∑i=2+∞uiλ−i+1,(2.2)
(Ln)+ is the nonnegative powers of λ in the Laurent series Ln, ui = ui(t1, t2,...), i ⩾ 2, the bracket { , } is the Poisson bracket in 2D phase space (λ, x = t1)
{f,g}=∂f∂λ∂g∂x−∂g∂λ∂f∂x.(2.3)

The dKP hierarchy (2.1) is a collection of nonlinear differential equations for un(x, t2,...) with respect to (x, t2,...). This system is obtained by replacing microdifferential operators ∂ (in x) and their commutators of the KP hierarchy by Laurent series (in λ) and Poisson brackets. A number of characteristics of the KP hierarchy, indeed, persist in this hierarchy. For example, one can prove that the Lax equations (2.1) are equivalent to the zero-curvature equations

∂Bn∂tm−∂Bm∂tn+{Bn,Bm}=0,
with purely algebraic manipulation as done for the ordinary KP hierarchy [7].

Let us define a 3-bracket

{f,g,h}=f{g,h}+g{h,f}+h{f,g},(2.4)
where f, g, h are smooth functions on 2D phase space (λ, x) and the bracket { , } is the Poisson bracket (2.3). The 3-bracket (2.4) satisfies the following properties:
  1. 1.

    skew-symmetry: { fσ(1), fσ(2), fσ(3)} = (−1)ε(σ){ f1, f2, f3},

  2. 2.

    Fundamental identity:

    {f1,f2,{f3,f4,f5}}={{f1,f2,f3},f4,f5}+{f3,{f1,f2,f4},f5}+{f3,f4,{f1,f2,f5}},
    where ε(σ) equals to 0 or 1 depending on the parity of the permutation σ. Therefore, with the 3-bracket (2.4), the set of all the smooth functions on the 2D phase space (λ, x) over a field of characteristic zero forms a 3-Lie algebra.

By means of the 3-bracket (2.4), we introduce the 3-Lax equation

∂L∂tm,n={Bm,Bn,L}.(2.5)

When m = 0, it is easy to see that (2.5) becomes (2.1).

Suppose | · | express the weight of ·. Let us assign the weights as follows,

|ui|=i(i⩾2),   |x|=−1,   |1|=0,   |tm,n|=−(m+n),
then we have the following lemma.

Lemma 2.1.

For the multivariate polynomial

(n−1)(1+u2+⋯+un)n+1(u2+2u3⋯+(n−1)un)−(n+1)(1+u2+⋯+un)n(2u2+⋯+nun)+3(1+u2+⋯+un)n+1,(2.6)
the terms with weight n + 2 (n ⩾ 2) disappear.

Proof.

The terms with weight n + 2 in (2.6) can be written as the form Cu2i2u3i3⋯unin, where 0 ⩽ i1,i2,...,in ⩽ n and 2i2 + 3i3 + ··· + nin = n + 2. The coefficient C is

C=(n−1)[Cn+1i2Cn+1−i2i3⋯Cn+1−i2−⋯−in−2in−1Cn+1−i2−⋯−in−1in−1⋅(n−1)   +Cn+1i2Cn+1−i2i3⋯Cn+1−i2−⋯−in−3in−2Cn+1−i2−⋯−in−2in−1−1Cn+1−i2−⋯−in−1+1in⋅(n−2)   +⋯+Cn+1i2−1Cn+1−i2+1i3⋯Cn+1−i2−⋯in−1+1in⋅1]   −(n+1)[Cni2Cn−i2i3⋯Cn−i2−⋯−in−2in−1Cn−i2−⋯−in−1in−1⋅n   +Cni2Cn−i2i3⋯Cn−i2−⋯−in−3in−2Cn−i2−⋯−in−2in−1−1Cn−i2−⋯−in−1+1in⋅(n−1)   +⋯+Cni2−1Cn−i2+1i3⋯Cn−i2−⋯−in−1+1in⋅2]+3Cn+1i2Cn+1−i2i3⋯Cn+1−i2−⋯−in−1in   =(n−1)(n+1)!i2!i3!⋯in!(n+2−i2−⋯−in)!⋅[in(n−1)+in−1(n−2)+⋯+i2⋅1]   −(n+1)n!i2!i3!⋯in!(n+1−i2−⋯−in)![inn+in−1(n−1)+⋯+i2⋅2]   +3(n+1)!i2!i3!⋯in!(n+1−i2−⋯−in)!.(2.7)

Using the condition 2i2 + 3i3 + ··· + nin = n + 2, (2.7) equals to

(n+1)!i2!i3!⋯in![(n−1)(n+2−i2−⋯−in)(n+2−i2−⋯−in)!+−n−2+3(n+1−i2−⋯−in)!]=(n+1)!i2!i3!⋯in![(n−1)(n+1−i2−⋯−in)!+−n+1(n+1−i2−⋯−in)!]=0.

Theorem 2.1.

For the 3-brackets n−1n+1{B0,Bn+1,L} and {B1,Bn,L}, the corresponding coefficients of λ−1are the same, i.e. n−1n+1∂u2∂t0,n+1=∂u2∂t1,n by(2.5). Thus when m = 1 the 3-Laxequation (2.5)is in fact the dKP hierarchy as the case m = 0.

Proof.

By direct calculations, we obtain

Bn=∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin⩽nCni1Cn−i1i2⋯Cn−i1−i2−⋯−in−1inu2i2u3i3⋯uninλn−(2i2+3i3+⋯+nin),(2.8)
and
{B0,Bn+1,L}=∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin⩽n∑k=2∞Cn+1i1+1Cn−i1i2⋯Cn−i1−i2−⋯−in−1inu2i2u3i3⋯uninuk,x   [(n+1)−(2i2+3i3+⋯+nin)]λn+1−k−(2i2+3i3+⋯+nin)   −∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin⩽n+1Cn+1i1+1Cn−i1i2⋯Cn−i1−i2−⋯−in−1in∂(u2i2u3i3⋯unin)∂x   λ(n+1)−(2i2+3i3+⋯+nin)−(n+1)un+1,x   +∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin⩽n+1∑k=2∞Cn+1i1+1Cn−i1i2⋯Cn−i1−i2−⋯−in−1in∂(u2i2u3i3⋯unin)∂xuk(k−1)   λn+1−k−(2i2+3i3+⋯+nin)+(n+1)un+1,x∑k=2∞(k−1)ukλ−k.(2.9)

From the right side of (2.9), the coefficient of λ−1 in n−1n+1{B0,Bn+1,L} is

(n−1)un+2,x+∑k=2n∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin=n+2−kn−1i1+1Cni1Cn−11i2⋯Cn−i1−i2−⋯−in−1in(k−1)∂(u2i2u3i3⋯uninuk)∂x.(2.10)

Similarly, the coefficient of λ−1 in {B1,Bn,L} is

(n−1)un+2,x+∑k=2n∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin=n+2−kCni1Cn−i1i2⋯Cn−i1−i2−⋯−in−1ink∂(uku2i2u3i3⋯unin)∂x   −3∑k=2n∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin=n+2−kCni1Cn−i1i2⋯Cn−i1−i2−⋯−in−1inuk,xu2i2u3i3⋯unin.(2.11)

To prove the theorem, we need prove the following equality

∑k=2n∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin=n+2−k[(n−1)(k−1)i1+1]Cni1Cn−i1i2⋯Cn−i1−i2−⋯−in−1in∂(u2i2u3i3⋯uninuk)∂x   +3∑k=2n∑0⩽i1,i2,…,in⩽ni1+i2+⋯+in=n2i2+3i3+⋯+nin=n+2−kCni1Cn−i1i2⋯Cn−i1−i2−⋯−in−1inuk,xu2i2u3i3⋯unin=0.(2.12)

On the left side of (2.12), the first term is the terms with weight n + 3 among

∂∂x[∑k=2n∑0⩽i1⩽n+1n−1n+1Cn+1i1(u2+⋯+un)n+1−i1(k−1)uk−(1+u2+⋯+un)n(2u2+⋯+nun)]=∂∂x[n−1n+1(1+u2+⋯+un)n+1(u2+2u3⋯+(n−1)un)−(1+u2+⋯+un)n(2u2+⋯+nun)].(2.13)

Meanwhile the second term is the terms with weight n + 3 among

3(1+u2+⋯+un)n(u2,x+⋯+un,x)=3n+1∂∂x[(1+u2+⋯+un)n+1].(2.14)

So the proof of (2.12) is equivalent to the proof of the terms with weight n + 2 among

(n−1)(1+u2+⋯+un)n+1(u2+2u3⋯+(n−1)un)−(n+1)(1+u2+⋯+un)n(2u2+⋯+nun)+3(1+u2+⋯un)n+1
disappear, which follows from Lemma 2.1.

We further calculate the more general case of the 3-Lax equation (2.5). For convenience, let I represent the condition 0 ⩽ i1,...,im ⩽ m; i1 + i2 + ··· + im = m; 2i2 + 3i3 + ··· + mim ⩽ m, J represent 0 ⩽ j1,..., jn ⩽ n; j1 + j2 + ··· + jn = n; 2 j2 + 3 j3 + ··· + n jn ⩽ n and K represent the condition (2i2 + ··· + mim) + (2 j2 + ··· + n jn) = m + n − k + 1.

Substituting (2.8) into (2.5) and comparing the coefficient of λ−1 on both sides, we obtain

∂u2∂tm,n=(n−m)um+n+1,x+∑k=2m+n∑I,J,KCmi1Cm−i1i2⋯Cm−i1−⋯−im−1imCnj1Cn−j1j2   ⋯Cn−j1−⋯−jn−1jn{[uk(u2i2⋯umim)(u2j2⋯unjn)]x[−n−k+1+(2j2+⋯njn)]+3(k−1)   uk(u2i2⋯umim)(u2j2⋯unjn)x+3[n−(2j2+⋯+njn)]uk,x(u2i2⋯umim)(u2j2⋯unjn)}.(2.15)

Similarly, we obtain

∂u2∂t0,m+n=(m+n)um+n+1,x+∑k=2m+n∑0⩽i1,…,im+n⩽m+ni1+i2+⋯+im+n=m+n2i2+⋯+(m+n)im+n=m+n+1−kCm+ni1Cm+n−i1i2⋯   Cm+n−i1−⋯−im+n−1im+n(k−1)(uku2i2⋯um+nim+n)x.(2.16)

When m = 1, as the result in the Theorem 2.1.(2.15) and (2.16) support n−1n+1∂u2∂t0,n+1=∂u2∂t1,n. But for the general case of m, n ⩾ 2 and m ≠ n, there is not such kind of the equivalent relation between ∂u2∂t0,m+n and ∂u2∂tm,n(2.17)

Substituting (2.8) into (2.5), when m = 0, n = 2, we obtain

∑i=1∞∂ui+1∂t02λ−i=2u3,xλ−1+∑i=2∞2[ui+2,x+(i−1)uiu2,x]λ−i.(2.17)

Comparing the coefficient of λ−i, i ⩾ 1 on both sides of (2.17), we obtain

∂u2∂t0,2=2u3,x,   ∂ui+1∂t0,2=2ui+2,x+2(i−1)uiu2,x,   i≥2.(2.18)

Let t02 = y, (2.18) gives the recursion relations

u3=12∂x−1u2,y,ui+2=12∂x−1ui+1,y−(i−1)∂−1(uiu2,x),   i⩾2.(2.19)

Substituting (2.19) for the case m = 0, n ⩾ 3 and m = 1, n ⩾ 2 into (2.15), we get the dKP hierarchy. Substituting (2.19) for the case m, n ⩾ 1 and m ≠ n into (2.15), we get a general hierarchy. The properties of the general hierarchy deserve our further study.

3. Summary

By combining the Lax equation of the dKP hierarchy and the 3-bracket, we have defined a Lax triple {Bm,Bn,L} and investigated the relations between different pairs of (m,n). Here the 3-bracket is not a Nambu-Poisson bracket, but it satisfies the conditions of 3-Lie algebra. We have proved that when m = 0 and m = 1, the 3-Lax equation ∂L∂tm,n={Bm,Bn,L} is the dKP hierarchy. For the generalized case of (m,n), the 3-Lax equation means a generalized dKP hierarchy including the dKP hierarchy. Whether the hierarchy of the general case constitute a new integrable hierarchy remains to be studied.

Our analyses suggest that there exists much deeper connections between the infinite-dimensional 3-algebra and the integrable system. More properties with respect to their relations still deserve further study. We believe that 3-algebras may shed new light on the integrable systems.

The authors are grateful to Professor Wei-Zhong Zhao for his valuable discussions and suggestions. We also thank Doctor Chun-Hong Zhang and Rui Wang for their discussions during our calculations. This work is partially supported by National Science Foundation (NSF) projects (Grant Nos. 11505046, 11605096 and 11801292).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
26 - 1
Pages
91 - 97
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2019.1544791How to use a DOI?
Copyright
© 2019 The Authors. Published by Atlantis and Taylor & Francis
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This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

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TY  - JOUR
AU  - Min-Ru Chen
AU  - Ying Chen
AU  - Zhao-Wen Yan
AU  - Jian-Qin Mei
AU  - Xiao-Li Wang
PY  - 2021
DA  - 2021/01/06
TI  - A 3-Lie algebra and the dKP Hierarchy
JO  - Journal of Nonlinear Mathematical Physics
SP  - 91
EP  - 97
VL  - 26
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1544791
DO  - 10.1080/14029251.2019.1544791
ID  - Chen2021
ER  -