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

^{1}, Beibei Hu

^{1}

^{, 2}

^{, *}

^{*}Corresponding authors.

- 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

*μ*HS equation [16] which is closely related to the HS equation

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

*μ*HS equation when the values of

*c*and

*k*are given respectively

*P*

_{1}and

*P*

_{2}

*μ*HS 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

*H*

^{1}metric, see [1–5,11,12] and references therein.

The Kuper-CH equation [6] and Kuper-*μ*HS equation [23] we firstly proposed and further researched in [23, 24] as Euler equation related to the Neveu-Schwarz superalgebra, especially, when taking *H*^{1}-metric and *μ*HS system with Lax pair and local super-biHamiltonian structures, which are fermionic versions of the CH equation and *μ*HS 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- *μ*HS equation can also be rewritten as a unified form, which is called Kuper-CH-HS euqation here.

For an arbitrary integrable equation

*P*

_{1}and

*P*

_{2}, Kupershmidt [15] proposed a nonholonomic

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

*m*=

*u*−

_{xx}*cμ*(

*u*)−4

*ku*is a bosonic function and

*α*=

*η*−

_{xx}*kη*is a fermionic function.

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

*K*and

*J*

*H*

_{1},

*H*

_{2}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

*m*=

*u*−

_{xx}*cμ*(

*u*)−4

*ku*and

*f*are bosonic functions and

*α*=

*η*−

_{xx}*kη*and

*ϕ*are fermionic functions.

Corresponding we can get

### 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]

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

*a*,

*b*,

*c*

_{1},

*c*

_{2}are bosonic fields,

*ξ*,

*β*are fermionic fields. From the compatibility condition

*b*= −

*f*and

## 4. Geodesic Flow

The CH equation can be described as the geodesic flow on the Bott-Virasoro group for the right-invariant *H*^{1}-metric on the group of diffeomorphisms [3, 11, 13]. The HS equation can be regarded as geodesic equations on the quotient space Diff(*S*^{1}) / *S*^{1} of the group Diff(*S*^{1}) of orientation-preserving diffeomorphisms of the unit circle *S*^{1} 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-*μ*HS 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

Let us denote

Observe that the stabilizer space of the coadjoint action of the Neveu-Schwarz superalgebra

### 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*

*with*

*where*

## 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.

## References

### 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 -