Journal of Nonlinear Mathematical Physics

Volume 25, Issue 2, March 2018, Pages 179 - 187

The Mixed Kuper-Camassa-Holm-Hunter-Saxton Equations

Authors
Ling Zhang1, Beibei Hu1, 2, *
School of Mathematics and Finance, Chuzhou University, Chuzhou, Anhui 239000, P. R. China
Department of Mathematics, Shanghai University, Shanghai 200444, China P. R. China
*Corresponding authors.
Corresponding Author
Beibei Hu
Received 24 October 2017, Accepted 5 November 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1452668How to use a DOI?
Keywords
Lax Pair; mixed Kuper-CH-HS equation; Neveu-Schwarz superalgebra; Hamiltonian structure
Abstract

In this paper, a mixed Kuper-CH-HS equation by a Kupershmidt deformation is introduced and its integrable properties are studied. Moreover, that the equation can be viewed as a constraint Hamiltonian flow on the coadjoint orbit of Neveu-Schwarz superalgebra is shown.

Copyright
© 2018 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

There are many interesting differential equations in mathematics and physics, such as the Camassa-Holm (CH in brief) equation [1] which is the model for the propagation of shallow water waves of moderate amplitude

ututxx=2uxuxx+uuxxx3uux,
the Hunter-Saxton (HS in brief) equation [9] which is used as a progressive equation of liquid crystal rotator
utxx=2uxuxx+uuxxx,
and the μHS equation [16] which is closely related to the HS equation
utxx=2μ(u)ux+2uxuxx+uuxxx,
with
μ(u)=S1udx.

It is worth mentioning that the above three equations can be expressed as

mt=2muxumx,(1.1)
where
m=uxxcμ(u)4ku.
The CH-HS equation (1.1) is just CH equation, HS equation and μHS equation when the values of c and k are given respectively
{c=0,k=14;c=0,k=0;c=1,k=0.
The CH-HS equation (1.1) admits a Lax pair, a bi-Hamiltonian structure
mt=P1δH2δm=P2δH1δm,
where the P1 and P2
P1=34k,P2=m+m,
are two compatible operators of the CH-HS equation (1.1) and
H1=12mudx,H2=(12u3+16uux2+13u2uxx)dx;
are the first two conserved quantities of the CH equation, and
H1=12mudx,H2=(16uux2+13u2uxx)dx;
are the first two conserved quantities of the HS equation, and
H1=12mudxH2=(16uux213u2uxx+23u2μ(u)u)dx
are the first two conserved quantities of the μHS equation. And CH-HS equation (1.1) is formally integrable through the inverse scattering method and can be regarded as geodesic equations on the diffeomorphism group of the circle (or of the line) for the right-invariant H1 metric, see [15,11,12] and references therein.

The Kuper-CH equation [6] and Kuper-μHS equation [23] we firstly proposed and further researched in [23, 24] as Euler equation related to the Neveu-Schwarz superalgebra, especially, when taking H1-metric and μH˙1-metric, two new super-integrable systems—Kuper-CH system and Kuper-μHS system with Lax pair and local super-biHamiltonian structures, which are fermionic versions of the CH equation and μHS equation in (1|1) superspace are given. The Super-HS equation ([17,23]) is supersymmetric extensions of HS equation, super-bi-Hamiltonian. The Kuper-CH equation, Super-HS equation and the Kuper- μHS equation can also be rewritten as a unified form, which is called Kuper-CH-HS euqation here.

For an arbitrary integrable equation

ut=P1δHnδu
with two compatible Hamiltonian operators P1 and P2, Kupershmidt [15] proposed a nonholonomic
{ut=P1δHnδuP1f,P2f=0.(1.2)
Zhou [21] proposed the concept of mixed hierarchy of soliton equations based on Lenard scheme and defined the nonholonomic deformation as Kupershmidt deformation. Naturally we want to consider the fermionic cases of the CH-HS equation (1.1) and the mixed Kuper-CH-HS equation and theirs propertities.

In this paper, motivated by the work about sKdV 6 [22], we will study the mixed kuper-CH equation and its integrable properties, and study the relation to the corresponding Neveu-Schwarz superalgebra.

2. The Kuper-CH-HS equation and the mixed Kuper-CH-HS equation

The Kuper-CH-HS equation can be rewritten as

{mt=2k1mux+k1mxu+12k1αxηx+32k1αηxx,αt=32k1uxα+k1uαx+12k1mηx.(2.1)
which is a fermionic extension of the CH-HS equation (1.1), m = uxx(u)−4ku is a bosonic function and α = ηxx is a fermionic function.
{c=0,k=14,k1=1, Kuper-CH equation; c=0,k=0,k1=1, Super-HS equation; c=1,k=0,k1=1, KuperμHS equation.

The Kuper-CH-HS equation (2.1) has the spectral problem

Φx=UΦ,U=(010k+12λm012λα12λα00).(2.2)
The Kuper-CH-HS equation (2.1) has super-bi-Hamiltonian structures, which can be rewritten as
(mα)t=K(δH2δmδH2δα)=J(δH1δmδH1δα),
the K and J
K=(34k002k),J=(m+mα+12αα+12αm2),(2.3)
are two compatible operators of Kuper-CH-HS equation and
H1=k12(mu+αηx)dx,H2=k13(6ku3+12uux2+u2uxx2cμ(u)u23k2uηηx12uηxηxx+12uηαx+32uxηα+m2uηηx)dx.
are the first two conserved quantities H1, H2 of the Kuper-CH-HS equation (2.1).

Motivated by the Kupershmidt deformation (1.2), we propose a mixed Kuper-CH-HS equation as a nonholonomic deformation of the Kuper-CH-HS equation(2.1).

Definition 2.1.

The mixed Kuper-CH-HS equation is defined as

(mα)t=K(δH2δmδH2δα)K(fϕ),J(fϕ)=0(2.4)
which is equivalent to
mt=2k1mux+k1mxu+12k1αxηx+32k1αηxx4kfx+fxxx,αt=32k1uxα+k1uαx+12k1mηxkϕ+ϕxx,2mfx+mxf+32αϕx+12αxϕ=0,32αfx+αxf+12mϕ=0.(2.5)
where m = uxx(u)−4ku and f are bosonic functions and α = ηxx and ϕ are fermionic functions.

Corresponding we can get

{c=0,k=14,k1=1, mixed Kuper-CH equation; c=0,k=0,k1=1, mixed Super-HS equation; c=1,k=0,k1=1, mixed Kuper- μ HS equation.
It's easy to prove that
dHndt=HnKH2HnK(fϕ)=Hn1J(fϕ)=0.
where functional gradient
=(δδm,δδα)T.
So we have following proposition

Proposition 2.2.

The mixed Kuper-CH-HS equation (2.5) has infinite many conserved quantities.

3. Lax Pair of mixed Kuper-CH Equation

Zero curvature representation and Lax pairs are two kinds of Commutator representations. A systematic approach for constructing zero curvature representation has been well developed by several papers, see [18,20] and references therein for details. In this section we adopt direct method to construct the Lax pair of the mixed Kuper-CH-HS equation (2.5). From the spectral problem of Kuper-CH-HS equation (2.1), We have got its the Lax pair [24]

{Φx=UΦ=(010k+12λm012λα12λα00)Φ,Φt=V0Φ=k12(ux2u+2ληx2λ2kucμ(u)(mu12ηxα)λuxαuλkηαuλ+kαηx0)Φ.(3.1)

Motivated by the (3.1), we assume that the Lax pair of the mixed Kuper-CH-HS equation has the following form

{Φx=UΦ,Φt=VΦ(3.2)
with
V=V0+λV1
and
V1=(abξc1+c2λaββξ0),
where a, b, c1, c2 are bosonic fields, ξ, β are fermionic fields. From the compatibility condition
UtVx+[U,V]=0(3.3)
considering its componentwise elements, we have
ax=c1kb+λ(c212mb12ξα),a=12bx,β=12λbα+ξx,mt=2k1mux+k1mxu+12k1αxηx+32k1αηxx+2c1x4ka+2λ(c2xmaαβ),αt=32k1uxα+k1uαx+12k1mηx+2βx2kξλ(mξ+aα).(3.4)
Furthermore, from (3.4) and the first two terms in (2.5), the following two relations are given
4kfx+fxxx2c1x+4ka=2λ(c2xmaαβ),kϕ+ϕxx2βx+2kξ=λ(mξ+aα).(3.5)
By choosing b = −f and ξ=12ϕ,we can get
a=12fx,c1=12fxxkf,c2=14ϕα12mf,β=12λfα+12ϕx.
meanwhile the system (3.5) reduces to
4mfx+2mxf+3αϕx+αxϕ=03αfx+2αxf+mϕ=0.(3.6)
which are exactly the last terms of the mixed Kuper-CH-HS equation (2.5). So we have

Proposition 3.1.

The mixed Kuper-CH-HS equation (2.5) has the Lax pair (3.2), where U and

V=V0+λV1
are defined above.

4. Geodesic Flow

The CH equation can be described as the geodesic flow on the Bott-Virasoro group for the right-invariant H1-metric on the group of diffeomorphisms [3, 11, 13]. The HS equation can be regarded as geodesic equations on the quotient space Diff(S1) / S1 of the group Diff(S1) of orientation-preserving diffeomorphisms of the unit circle S1 modulo the subgroup of rigid rotations [12]. Furthermore, the μHS equation can also be regarded as a remember of this frame. Zuo [22] described Euler equations associated to the generalized Neveu-Schwarz algebra and got many super-bi-Hamiltonian structure or supersymmetric equation, such as Kuper-CH equation, Kuper-μHS equation, super-CH equation, super-HS equation and Kuper-2CH equation etc. In this section, we want to descibe the relation between Neveu-Schwarz algebra and the mixed Kuper-CH equation (2.5).

The Neveu-Schwarz superalgebra [19,23] is an algebra

G=Vect(S1)C(S1)
with the bilinear operation
[f^,g^]=(A,B,C),
where
A=(fgfg12ϕχ)ddx,B=(fχ12fχgϕ+12gϕ)dx12,C=S1(fg4ϕχ)dx,
with
f^=(f(x,t)ddx,ϕ(x,t)dx12,a),g^=(g(x,t)ddx,χ(x,t)dx12,b),f=fx.

Let us denote

G*=C(S1)C(S1)
to be the dual space of G, under the following pair
m^,f^*=S1(mf+αϕ)dx+ςa,
where
m^=(m(x,t)dx2,α(x,t)dx32,ς)G*.
By the definition,
adf^*(m^),g^*=m^,[f^,g^]*,
using integration by parts
adf^*(m^),g^*=m^,[f^,g^]*=S1(2mf+mfςf+32αϕ+12αϕ)gdx+S1(m2ϕζϕ+32fα+fα)χdx=((2mf+mfςf+32αϕ+12αϕ)dx2,(m2ϕζϕ+32fα+fα)dx32,0),g^*.

Observe that the stabilizer space of the coadjoint action of the Neveu-Schwarz superalgebra G on the hyperplane ς=0 of G* is given by

2mf+mfςf+32αϕ+12αϕ=0,m2ϕςϕ+32fα+fα=0.
which are exactly the latter two equations in (2.5) . Thus we have

Proposition 4.1.

The mixed Kuper-CH-HS equation (2.5) is the constraint Hamiltonian flow on the NeveuSchwarz coadjoint orbit, that is to say

(mα)t=ad(δHδm,δHδα)(mα)K(fϕ),
with
adf^*(m^)=0,H=k12(mu+αηx)dx,
where
m^=(m(x,t)dx2,α(x,t)dx32,0)G*,f^=(f(x,t)ddx,ϕ(x,t)dx12,a)G.

Acknowledgments

The authors thanks the referees and the editor for their valuable and suggestions. This work is partially supported by the National Natural Science Foundation of China under Grant No.11601055, 11271345 , 11201451. Natural Science Foundation of Anhui Province under Grant No. 1408085QA06.

Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
25 - 2
Pages
179 - 187
Publication Date
2021/01/06
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
10.1080/14029251.2018.1452668How to use a DOI?
Copyright
© 2018 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/).

Cite this article

TY  - JOUR
AU  - Ling Zhang
AU  - Beibei Hu
PY  - 2021
DA  - 2021/01/06
TI  - The Mixed Kuper-Camassa-Holm-Hunter-Saxton Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 179
EP  - 187
VL  - 25
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1452668
DO  - 10.1080/14029251.2018.1452668
ID  - Zhang2021
ER  -