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International Journal of Differential Equations
Volume 2011, Article ID 738509, 13 pages
http://dx.doi.org/10.1155/2011/738509
Research Article

Geometric Integrability of Some Generalizations of the Camassa-Holm Equation

Faculty of Mathematics and Informatics, Sofia University, 5 J. Bouchier boulevard., 1164 Sofia, Bulgaria

Received 27 May 2011; Accepted 17 July 2011

Academic Editor: V. A. Yurko

Copyright © 2011 Ognyan Christov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the Camassa-Holm (CH) equation and recently introduced μCH equation from the geometric point of view. We show that Kupershmidt deformations of these equations describe pseudospherical surfaces and hence are geometrically integrable.

1. Introduction

The modern theory of integrable nonlinear partial differential equations arose as a result of the inverse scattering method (ISM) discovered by Gardner et al. [1] for Korteweg de Vries (KdV) equation. Soon after, it was realized that this method can be applied to several important nonlinear equations like nonlinear Shrödinger equation, sine-Gordon, and so forth.

Sasaki [2] gave a natural geometric interpretation for ISM in terms of pseudospherical surfaces. Motivated by Sasaki, Chern and Tenenblat [3] introduce the notion of a scalar equation of pseudospherical type and study systematically the evolution equations that describe pseudospherical surfaces. It appears that almost all important equations and systems in mathematical physics enjoy this property [37]. The advantage of this geometric treatment is that most of the ingredients connected with the integrable equations such as Lax pair, zero curvature representation, conservation laws, and symmetries come naturally.

Let us recall some facts about the equations we study in this paper.

The Camassa-Holm equation (CH)𝑢𝑡𝑢𝑥𝑥𝑡+2𝜔𝑢𝑥+3𝑢𝑢𝑥2𝑢𝑥𝑢𝑥𝑥𝑢𝑢𝑥𝑥𝑥=0,(1.1) where 𝜔, has appeared in [8] as an equation with bi-Hamiltonian structure. Later, in [9], it was considered as a model, describing the unidirectional propagation of shallow water waves over a flat bottom. CH is a completely integrable equation; see, for example, [1012]. Furthermore, CH is geometrically integrable [6]. Here, we consider the case 𝜔=0. Then, (1.1) is𝑚𝑡+𝑢𝑚𝑥+2𝑚𝑢𝑥,𝑚=𝑢𝑥𝑥𝑢.(1.2)

The bi-Hamiltonian form of (1.2) is [8, 9]𝑚𝑡=1𝛿𝐻2[𝑚]𝛿𝑚=2𝛿𝐻1[𝑚],𝛿𝑚(1.3) where 1=𝜕𝜕3,2=𝑚𝜕+𝜕𝑚 are the two compatible Hamiltonian operators (𝜕 stands for 𝜕/𝜕𝑥), and the corresponding Hamiltonians are𝐻1[𝑚]=12𝑚𝑢𝑑𝑥,𝐻2[𝑚]=12𝑢3+𝑢𝑢2𝑥𝑑𝑥.(1.4)

There exists an infinite sequence of conservation laws (multi-Hamiltonian structure) 𝐻𝑛[𝑚],𝑛=0,±1,±2, including (1.4) such that 1𝛿𝐻𝑛[𝑚]𝛿𝑚=2𝛿𝐻𝑛1[𝑚]𝛿𝑚.(1.5) Very recently a “modified” CH equation (mCH) was introduced by analogy of “Miura transform” from the theory of KdV equation in [5].

The 𝜇CH equation was derived recently in [13, 14] as𝑢𝑥𝑥𝑡=2𝜇(𝑢)𝑢𝑥+2𝑢𝑥𝑢𝑥𝑥+𝑢𝑢𝑥𝑥𝑥,(1.6) where 𝜇(𝑢)=10𝑢𝑑𝑥 and 𝑢(𝑡,𝑥) is a spatially periodic real-valued function of time variable 𝑡 and space variable 𝑥𝑆1=[0,1). In order to keep a certain symmetry and analogy with CH, one can write the above equation in the form (see also [15])𝜇𝑢𝑡𝑢𝑥𝑥𝑡=2𝜇(𝑢)𝑢𝑥+2𝑢𝑥𝑢𝑥𝑥+𝑢𝑢𝑥𝑥𝑥.(1.7) Note that 𝜇(𝑢𝑡)=0 in the periodic case. Introducing 𝑚=𝐴𝑢=𝜇(𝑢)𝑢𝑥𝑥 (1.7) becomes𝑚𝑡+𝑢𝑚𝑥+2𝑚𝑢𝑥=0,𝑚=𝜇(𝑢)𝑢𝑥𝑥.(1.8) The 𝜇CH is an integrable equation and arises as an asymptotic rotator equation in a liquid crystal with a preferred direction if one takes into account the reciprocal action of dipoles on themselves [13, 14].

The bi-Hamiltonian form of (1.8) is𝑚𝑡=1𝛿𝐻2[𝑚]𝛿𝑚=2𝛿𝐻1[𝑚],𝛿𝑚(1.9) where 1=𝜕𝐴=𝜕3,2=𝑚𝜕+𝜕𝑚 are the two compatible Hamiltonian operators, and the corresponding Hamiltonians are𝐻1[𝑚]=12𝑚𝑢𝑑𝑥,𝐻2[𝑚]=𝜇(𝑢)𝑢2+12𝑢𝑢2𝑥𝑑𝑥.(1.10) Also the 𝜇CH equation is geometrically integrable [15] (see there for other geometric descriptions of this equation).

Recently in [16], a new 6th-order wave equation, named KdV6, was derived. After some rescaling, this equation can be presented as the following system; 𝑢𝑡=6𝑢𝑢𝑥+𝑢𝑥𝑥𝑥𝑤𝑥,𝑤𝑥𝑥𝑥+4𝑢𝑤𝑥+2𝑢𝑥𝑤=0.(1.11) This system gives a perturbation to KdV equation (𝑤=0), and since the constrain on 𝑤 is differential, this is a nonholonomic deformation.

Kupershmidt [17] suggested a general construction applicable to any bi-Hamiltonian systems providing a nonholonomic perturbation on it. This perturbation is conjectured to preserve integrability. In the case of KdV6, the system (1.11) can be converted into𝑢𝑡=1𝛿𝐻𝑛+1𝛿𝑢1(𝑤)=2𝛿𝐻𝑛𝛿𝑢1(𝑤),2(𝑤)=0,(1.12) where 1=𝜕,2=𝜕3+2(𝑚𝜕+𝜕𝑚) are the two standard Hamiltonian operators of the KdV hierarchy and𝐻1=𝑢𝑑𝑥,𝐻2=12𝑢2𝑑𝑥,.(1.13) In the same paper, Kupershmidt verifies integrability of KdV6, as well as integrability of such nonholonomic deformations for some representative cases: the classical long-wave equation, the Toda lattice (both continuous and discrete), and the Euler top.

In fact, Kersten et al. [18] prove that the Kupershmidt deformation of every bi-Hamiltonian equation is again bi-Hamiltonian system and every hierarchy of conservation laws of the original bi-Hamiltonian system, gives rise to a hierarchy of conservation laws of the Kupershmidt deformation.

The aim of this paper is to show that the CH and the 𝜇CH equations have geometrically integrable Kupershmidt deformations. We also show that the KdV6 equation and two-component CH system [19] are also geometrically integrable.

As a matter of fact, Yao and Zeng [20] propose a generalized Kupershmidt deformation and verify that this generalized deformation also preserves integrability in few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation, and CH equation.

Kundu et al. [21] consider slightly generalized form of deformation for the KdV equation and extend this approach to mKdV equation and to AKNS system.

Guha [22] uses Kirillov's theory of coadjoint representation of Virasoro algebra to obtain a large class of KdV6-type equations, equivalent to the original one. Also, applying the Adler-Konstant-Symes scheme, he constructs a new nonholonomic deformation of the coupled KdV equation.

The paper is organized as follows. In Section 2, we recall some facts about equations that describe pseudospherical surfaces. The main results are in Section 3. There we show that the Kupershmidt deformations for the CH equation and for the 𝜇CH equation are of pseudospherical type and hence are geometrically integrable. We also derive the corresponding quadratic pseudopotentials which turn out to be very useful in obtaining conservation laws and symmetries. At the end, we make some speculations about “modified” 𝜇CH equation.

2. Equations of Pseudospherical Type and Pseudopotentials

In this section, we recall some definitions and facts. One can consult, for example, [37] for more details.

Definition 2.1. A scalar differential equation Ξ(𝑥,𝑡,𝑢,𝑢𝑥,,𝑢𝑥𝑛𝑡𝑚)=0 in two independent variables 𝑥,𝑡 is of pseudospherical type (or, it describes pseudospherical surfaces) if there exist one-forms 𝜔𝛼0𝜔𝛼=𝑓𝛼1𝑥,𝑡,𝑢,,𝑢𝑥𝑟𝑡𝑝𝑑𝑥+𝑓𝛼2𝑥,𝑡,𝑢,,𝑢𝑥𝑠𝑡𝑞𝑑𝑡,𝛼=1,2,3,(2.1) whose coefficients 𝑓𝛼𝛽 are smooth functions which depend on 𝑥,𝑡 and finite number of derivatives of 𝑢, such that the 1-forms 𝜔𝛼=𝜔𝛼(𝑢(𝑥,𝑡)) satisfy the structure equations 𝑑𝜔1=𝜔3𝜔2,𝑑𝜔2=𝜔1𝜔3,𝑑𝜔3=𝜔1𝜔2,(2.2) whenever 𝑢=𝑢(𝑥,𝑡) is a solution of Ξ=0.
Equations (2.2) can be interpreted as follows. The graphs of local solutions of equations of pseudospherical type can be equipped with structure of pseudospherical surface (see [3, 6, 7]): if 𝜔1𝜔20, the tensor 𝜔1𝜔1+𝜔2𝜔2 defines a Riemannian metric of constant Gaussian curvature −1 on the graph of solution 𝑢(𝑥,𝑡), and 𝜔3 is the corresponding metric connection one-form.
An equation of pseudospherical type is the integrability condition for a sl(2,)-valued problem 𝑑𝜓=Ω𝜓,(2.3) where Ω is the matrix-valued one-form 1Ω=𝑋𝑑𝑥+𝑇𝑑𝑡=2𝜔2𝜔1𝜔3𝜔1+𝜔3𝜔2.(2.4)

Definition 2.2. An equation Ξ=0 is geometrically integrable if it describes a nontrivial one-parameter family of pseudospherical surfaces.

Hence, if Ξ=0 is geometrically integrable, it is the integrability condition of one-parameter family of linear problems 𝜓𝑥=𝑋𝜓,𝜓𝑡=𝑇𝜓. In fact, this is equivalent to the zero curvature equation𝑋𝑡𝑇𝑥+[]𝑋,𝑇=0,(2.5) which is an essential ingredient of integrable equations.

Another important property of equations of pseudospherical type is that they admit quadratic pseudopotentials. Pseudopotentials are a generalization of conservation laws.

Proposition 2.3 (see [6]). Let Ξ=0 be a differential equation describing pseudospherical surfaces with associated one-forms 𝜔𝛼. The following two Pfaffian systems are completely integrable whenever 𝑢(𝑥,𝑡) is a solution of Ξ=0: 2𝑑Γ=𝜔3+𝜔22Γ𝜔1+Γ2𝜔3𝜔2,(2.6)2𝑑𝛾=𝜔3𝜔22𝛾𝜔1+𝛾2𝜔3+𝜔2.(2.7) Moreover, the one-forms Θ=𝜔1𝜔Γ3𝜔2,Θ=𝜔1𝜔+𝛾3+𝜔2(2.8) are closed whenever 𝑢(𝑥,𝑡) is a solution of Ξ=0 and Γ (resp. 𝛾) is a solution of (2.6) (resp. (2.7)).

Geometrically, Pfaffian systems (2.6) and (2.7) determine geodesic coordinates on the pseudospherical surfaces associated with the equation Ξ=0 [3, 6].

3. Results

In this section, we consider the nonholonomic deformation of CH equation and 𝜇CH equation. We show that they are geometrically integrable and consider their quadratic pseudopotentials. The nonlocal symmetries will be studied elsewhere.

3.1. CH Equation

Recall from Introduction the CH equation (1.2) and its bi-Hamiltonian form (1.3) with the corresponding Hamiltonian operators 1 and 2.

Following Kupershmidt's construction, we introduce the nonholonomic deformation of the CH equation𝑚𝑡=𝑢𝑚𝑥2𝑚𝑢𝑥1(𝑤),2(𝑤)=0.(3.1)

Proposition 3.1. The system (3.1) describes pseudospherical surface and hence is geometrically integrable.

Let us give the corresponding 1-forms𝜔1=(𝑚+1)𝑑𝑥+𝑢𝑚+𝜂+1𝜂𝑢𝑥+1𝑢𝜂𝑤𝑥𝑥+(𝜂+1)𝑤𝑥𝜔𝜂𝑤(𝑚+1)𝑑𝑡,2=(𝜂+1)𝑑𝑥+𝜂+1𝜂(𝜂+1)𝑢+𝑢𝑥𝜂(𝜂+1)𝑤+𝜂𝑤𝑥𝜔𝑑𝑡,3=+(𝑚+1+𝜂)𝑑𝑥𝑢𝑚+𝜂(𝜂+1)2𝜂𝑢+𝜂+1𝜂𝑢𝑥+𝜂+1𝜂𝑤𝑥𝑥+(𝜂+1)𝑤𝑥𝜂𝑤(𝑚+1+𝜂)𝑑𝑡.(3.2)

For the proof of Proposition 3.1, we need only to verify the structure equations (2.2) and to check that the parameter 𝜂 is intrinsic.

For the matrices 𝑋 and 𝑇 in, we get1𝑋=21𝜂+1𝜂2(𝑚+1)+𝜂(𝜂+1),𝑇=2𝑇11𝑇12𝑇21𝑇11,(3.3) where𝑇11=𝜂+1𝜂(𝜂+1)𝑢+𝑢𝑥𝜂(𝜂+1)𝑤+𝜂𝑤𝑥,𝑇12=𝜂𝑢+𝜂2𝑇𝑤1,21=2𝑢𝑚(𝜂+1)2+1𝜂𝑢+2(𝜂+1)𝜂𝑢𝑥+𝜂+2𝜂2𝑤𝑥𝑥+2(𝜂+1)𝑤𝑥𝜂(𝜂+2)𝑤2𝜂𝑤𝑚.(3.4) Hence, we have a zero curvature representation 𝑋𝑡𝑇𝑥+[𝑋,𝑇]=0 for the system (3.1). From (3.3), it is straightforward to obtain the corresponding scalar linear problem𝜓𝑥𝑥=14𝜂2𝑚𝜓𝜓,𝑡=1𝑢𝜂𝑤+𝜂𝜓𝑥+𝑢𝑥+𝜂𝑤𝑥2𝜓.(3.5)

In order to apply Proposition 2.3 to nonholonomic deformation of the CH equation, we consider new 1-forms 𝜔𝛼new𝜔1new=𝜔2,𝜔2new=𝜔1,𝜔3new=𝜔3.(3.6) With these forms the Pfaffian system (2.7) becomes2𝛾𝑥=𝜂𝛾22(𝜂+1)𝛾+2(𝑚+1)+𝜂,2𝛾𝑡=𝛾22𝑤𝑥𝑥[]𝑤2𝜂𝛾(𝜂+1)𝑥2𝑤𝜂𝛾𝑥2𝛾𝜂+1𝜂𝑢𝑥𝛾2𝑢𝑥+1𝜂+𝜂+2𝜂2𝛾𝜂+1𝜂.(3.7) Applying the transform 𝛾𝛾+𝜂+1𝜂,(3.8) after some algebraic manipulations and setting 𝜆=1/𝜂, we obtain the following result.

Proposition 3.2. The nonholonomic deformation of the CH equation (3.1) admits a quadratic pseudopotential 𝛾, defined by the equations 𝛾𝑚=22𝜆+𝛾𝑥𝜆2,𝛾(3.9)𝑡=𝛾2211+𝜆𝑤𝑢𝜆𝑤𝛾𝑢𝜆𝑥𝑤𝑢𝜆𝜆𝑚+2𝜆+𝜆𝑢22,(3.10) where 𝜆0,𝑚=𝑢𝑥𝑥𝑢. Moreover, (3.1) possesses the parameter-dependent conservation law 𝛾𝑡=𝜆(𝑢+𝑤)𝑥1𝛾𝜆𝑤𝑢𝜆𝛾𝑥.(3.11)

Conservation densities can be obtained by expanding (3.9) and (3.11) in powers of 𝜆. Note that the left hand side of (3.11) and (3.9) does not depend on 𝑤 as it should be. The corresponding expansions are performed in [6].

3.2. 𝜇CH Equation

Consider now the 𝜇CH equation (1.8), its bi-Hamiltonian form (1.9) with the corresponding Hamiltonian operators 1,2. Applying Kupershmidt's procedure to (1.8), we obtain the nonholonomic deformation of the 𝜇CH equation𝑚𝑡=𝑢𝑚𝑥2𝑚𝑢𝑥1(𝑤),2(𝑤)=0,(3.12) or𝑚𝑡=𝑢𝑚𝑥2𝑚𝑢𝑥+𝑤𝑥𝑥𝑥,2𝑚𝑤𝑥+𝑤𝑚𝑥=0,𝑚=𝜇(𝑢)𝑢𝑥𝑥.(3.13)

Proposition 3.3. The nonholonomic deformation of the 𝜇CH equation (3.13) describes pseudospherical surfaces and, hence, is geometrically integrable.

For validation of Proposition 3.3, we give the 1-forms associated with (3.13) 𝜔1=12𝜂𝜂𝑚22+1+2𝑑𝑥2𝜂22𝑢𝑢𝜂𝑥1+𝑢𝑚+22+𝜇(𝑢)2𝑢+𝜂+𝜂322𝜂𝑤𝜂2𝑤𝑥+𝜂𝑤𝑥𝑥𝜂2𝜔𝑚𝑤𝑑𝑡,2=𝜂𝑑𝑥+1𝜂𝑢+𝑢𝑥𝜂2𝑤+𝜂𝑤𝑥𝜔𝑑𝑡,3=12𝜂𝜂𝑚22+12𝑑𝑥2𝜂22𝑢𝑢𝜂𝑥1+𝑢𝑚+22+𝜇(𝑢)+2𝑢𝜂+𝜂32+2𝜂𝑤𝜂2𝑤𝑥+𝜂𝑤𝑥𝑥𝜂2𝑚𝑤𝑑𝑡.(3.14)

For the matrices 𝑋 and 𝑇 in, we get1𝑋=2𝜂𝜂2𝜂𝑚221𝜂,𝑇=2𝑇11𝑇12𝑇21𝑇11,(3.15) where𝑇11=1𝜂𝑢+𝑢𝑥𝜂2𝑤+𝜂𝑤𝑥,𝑇121=2𝑢+𝜂,𝑇𝜂𝑤21=𝜂22𝑢𝑢𝜂𝑥1+𝑢𝑚+2+𝜇(𝑢)𝜂2𝑤𝑥+𝜂𝑤𝑥𝑥𝜂2𝜂𝑚𝑤+32𝑤.(3.16) Hence, we have a zero curvature representation 𝑋𝑡𝑇𝑥+[𝑋,𝑇]=0 for the system (3.13). From (3.3), it is straightforward to obtain the corresponding scalar linear problem𝜓𝑥𝑥=𝜂2𝑚𝜓𝜓,𝑡=1𝑢𝜂𝑤+𝜂𝜓𝑥+𝑢𝑥+𝜂𝑤𝑥2𝜓,(3.17) which coincides with those in [13] upon setting 𝑤=0 and 𝜆=𝜂/2.

In order to find pseudopotentials for the nonholonomic deformation of the 𝜇CH equation, we proceed as before denoting 𝜔1new=𝜔2,𝜔2new=𝜔1,𝜔3new=𝜔3.(3.18) With these forms, the Pfaffian system (2.7) becomes2𝛾𝑥=2𝛾2𝜂2𝜂𝛾+𝜂𝑚22,(3.19)2𝛾𝑡=2𝛾2𝜂+2𝛾2(𝑢+𝜂𝑤)2𝛾1𝜂𝑢+𝑢𝑥𝜂2𝑤+𝜂𝑤𝑥+𝜂22𝑢𝑢𝜂𝑥1+𝑚+2𝜂+𝜇(𝑢)+32𝑤𝜂2𝑤𝑥+𝜂𝑤𝑥𝑥𝜂2.𝑚𝑤(3.20) After some manipulations, the above system obtains the form 2𝛾𝑥=2𝛾2𝜂2𝜂𝛾+𝜂𝑚22,2𝛾𝑡2=𝜂𝛾2+𝜂𝑤𝑥𝑥[](2𝛾+𝜂)(𝑢+𝜂𝑤)𝑥𝜂+𝜇(𝑢)2𝛾2.(3.21) Applying the transform 𝛾𝛾𝜂/2, we get𝛾𝑥=𝛾2+𝜂2𝛾𝑚,(3.22)𝑡𝛾=2𝜂+𝜂2𝑤𝑥𝑥[]𝛾(𝑢+𝜂𝑤)𝑥+𝜇(𝑢)2.(3.23) Multiplying the first equation (3.22) by 1/𝜂 and then adding the result to the second equation (3.23), we get the following result denoting 𝜆=𝜂/2.

Proposition 3.4. The nonholonomic deformation of the 𝜇CH equation (3.13) admits a quadratic pseudopotential 𝛾, defined by the equations 𝛾𝑚=2𝜆+𝛾𝑥𝜆,𝛾(3.24)𝑡=2𝛾2𝜆𝜆𝑤𝑥𝑥[]𝛾(𝑢+2𝜆𝑤)𝑥+𝜇(𝑢)2,(3.25) where 𝜆0,𝑚=𝜇(𝑢)𝑢𝑥𝑥. Moreover, (3.13) possesses the parameter-dependent conservation law 𝛾𝑡=12𝜆𝛾+𝜆(𝑢+2𝜆𝑤)𝑥2𝜆(𝑢+2𝜆𝑤)𝛾𝑥.(3.26)

As the conserved densities for the nonholonomic deformation are the same as for the original bi-Hamiltonian system, we make use of the pseudopotentials to obtain them for the 𝜇CH equation. One possible expansion of 𝛾 is𝛾=𝜆1/2𝛾1+𝛾0+𝑗=1𝜆𝑗/2𝛾𝑗.(3.27) Substituting this into (3.24) yields 𝛾1=𝑚,𝛾0𝑚=𝑥4𝑚,𝛾1=1𝑚322𝑥𝑚5/2+18𝑚𝑥𝑚3/2𝑥,andsoforth.(3.28) In this way, we can obtain local functionals; see [13].

We finish this section with the geometric integrability of one of the most popular two-component generalization of CH equation and of KdV6 equation.

Another generalization of the Camassa-Holm equation is the following integrable two-component CH system [19]:𝑢𝑡𝑢𝑥𝑥𝑡=3𝑢𝑢𝑥+2𝑢𝑥𝑢𝑥𝑥+𝑢𝑢𝑥𝑥𝑥+𝜎𝜌𝜌𝑥,𝜌𝑡+(𝑢𝜌)𝑥=0,(3.29) where 𝜎=±1. Introducing a new variable 𝑣=𝜌2/2, the above system becomes𝑢𝑡𝑢𝑥𝑥𝑡=3𝑢𝑢𝑥+2𝑢𝑥𝑢𝑥𝑥+𝑢𝑢𝑥𝑥𝑥+𝜎𝑣𝑥,𝑣𝑡+2𝑣𝑢𝑥+𝑢𝑣𝑥=0.(3.30) The system (3.30) is geometrically integrable. The corresponding 1-forms, satisfying the structure equations (2.2), are the following: 𝜔1=𝑢𝑥𝑥𝑢𝑢𝜎𝜂𝑣+1𝑑𝑥+2𝑢𝑢𝑥𝑥+𝜂+1𝜂𝑢𝑥+1𝑢𝜂𝜔𝜎𝑣+𝜂𝜎𝑢𝑣𝑑𝑡,2=(𝜂+1)𝑑𝑥+𝜂+1𝜂(𝜂+1)𝑢+𝑢𝑥𝜔𝑑𝑡,3=𝑢𝑥𝑥+𝑢𝑢𝜎𝜂𝑣+𝜂+1𝑑𝑥2𝑢𝑢𝑥𝑥+𝜂(𝜂+1)2𝜂𝑢+𝜂+1𝜂𝑢𝑥+1𝜎𝑣+𝜂𝜎𝑢𝑣𝑑𝑡.(3.31) We could easily include two-component Hunter-Saxon system [19] into this picture (see also [7]).

Finally, we note that nonholonomic perturbation of KdV equation, known as KdV6 equation, is also of pseudospherical type, that is, KdV6 equation is geometrically integrable. We just give the corresponding 1-forms𝜔1=(1𝑢)𝑑𝑥+𝑢𝑥𝑥+𝜂𝑢𝑥2𝑢2+1𝜂𝑤𝑥1𝜂2𝑤𝑥𝑥++2𝑢𝑤2𝜂22𝑢+𝜂2𝑤+𝜂2𝜔𝑑𝑡,2𝜂=𝜂𝑑𝑥+32+2𝜂𝑢+𝜂𝑤2𝑢𝑥2𝜂2𝑤𝑥𝜔𝑑𝑡,3=(1+𝑢)𝑑𝑥+𝑢𝑥𝑥+𝜂𝑢𝑥2𝑢2+1𝜂𝑤𝑥1𝜂2𝑤𝑥𝑥+2𝑢𝑤2+𝜂22𝑢𝜂2𝑤𝜂2𝑑𝑡,(3.32) which coincide with those for KdV equation [3] when 𝑤0.

4. Concluding Remarks

In this paper, we study the CH equation and some of its generalizations from the geometric point of view. We show that Kupershmidt deformations for CH and 𝜇CH equations preserve integrability and derive some important objects like quadratic pseudopotentials which turn out to be useful for obtaining conservation laws and nonlocal symmetries. It is also shown that the KdV6 equation and two-component CH system are also geometrically integrable.

Having at hand these examples of geometrically integrable Kupershmidt deformations, it is natural to think that maybe there exists a general link in this sense: a Kupershmidt deformation of geometrically integrable system is again geometrically integrable. We have not succeeded in establishing such a link up to now, but we believe that this is true at least for the systems with local Hamiltonian pair of operators as in the above examples.

Let us return, however, to the 𝜇CH equation (𝑤=0). It is obvious that pseudopotentials for the 𝜇CH equation (1.8) and parameter-dependent conservation law are obtained from𝛾𝑚=2𝜆+𝛾𝑥𝜆,𝛾(4.1)𝑡=12𝜆𝛾+𝜆𝑢𝑥2𝜆𝑢𝛾𝑥=𝛾𝑥+𝑢2𝜆𝑥𝑥2𝜕(𝛾𝑢).(4.2) Equation (4.1) is an analogue of the Miura transformation of KdV theory. We can repeat, purely formally, the procedure for obtaining the “modified” CH (mCH) equation [5] in this case. However, it is clear that since 𝜇CH contains nonlocal term, one can expect that the “modified” equation also will have nonlocal terms.

Denote by 𝐴 the operator 𝐴=𝜇𝜕2,𝐴(𝑢)=𝑚=𝜇(𝑢)𝑢𝑥𝑥. The operators 𝐴1 and 𝜕 commute and 𝜇(𝑢)=𝜇(𝐴𝑢).

We have𝑢=𝐴1𝑚,𝑢𝑥=𝐴1𝑚𝑥,𝑢𝑥𝑥=𝐴1𝑚𝑥𝑥,(4.3) in which 𝑚 is determined by (4.1). Then, the second equation (4.2) takes the form 𝛾𝑡=𝛾𝑥+𝐴2𝜆1𝑚𝑥𝑥2𝛾𝐴1𝑚𝑥𝛾𝑥𝐴1𝑚,(4.4) or 𝐴𝛾𝑡=𝐴𝛾𝑥+𝑚2𝜆𝑥𝑥2𝐴𝜕𝛾𝐴1𝑚.(4.5) Formally, this equation can be named as a “modified” 𝜇CH equation. One can simplify further this equation using (4.1) or even to present it as a system as in [5], it remains nonlocal and, hence, it is of no immediate advantage.

Acknowledgments

This work is partially supported by Grant no. 169/2010 of Sofia University and by Grant no. DD VU 02/90 with NSF of Bulgaria.

References

  1. C. Gardner, J. Green, M. Kruskal, and R. Miura, “Method for solving the Korteweg-de Vriez equation,” Physical Review Letters, vol. 19, pp. 1095–1097, 1967. View at Google Scholar
  2. R. Sasaki, “Soliton equations and pseudospherical surfaces,” Nuclear Physics. B, vol. 154, no. 2, pp. 343–357, 1979. View at Publisher · View at Google Scholar
  3. S. S. Chern and K. Tenenblat, “Pseudospherical surfaces and evolution equations,” Studies in Applied Mathematics, vol. 74, no. 1, pp. 55–83, 1986. View at Google Scholar · View at Zentralblatt MATH
  4. Q. Ding and K. Tenenblat, “On differential systems describing surfaces of constant curvature,” Journal of Differential Equations, vol. 184, no. 1, pp. 185–214, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. P. Górka and E. G. Reyes, “The modified Camassa-Holm equation,” IMRN, vol. 2011, no. 12, pp. 2617–2649, 2011. View at Publisher · View at Google Scholar
  6. E. Reyes, “Geometric integrability of the Camassa-Holm equation,” Letters in Mathematical Physics, vol. 59, no. 2, pp. 117–131, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. E. Reyes, “Pseudo-potentials, nonlocal symmetries and integrability of some shallow water equations,” Selecta Mathematica. New Series, vol. 12, no. 2, pp. 241–270, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. S. Fokas and B. Fuchssteiner, “Symplectic structures, their Bäcklund transformations and hereditary symmetries,” Physica D. Nonlinear Phenomena, vol. 4, no. 1, pp. 47–66, 1981. View at Publisher · View at Google Scholar
  9. R. Camassa and D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. A. Constantin, “On the inverse spectral problem for the Camassa-Holm equation,” Journal of Functional Analysis, vol. 155, no. 2, pp. 352–363, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. A. Constantin and H. McKean, “A shallow water equation on the circle,” Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949–982, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. A. Constantin and R. Ivanov, “Poisson structure and action-angle variables for the Camassa-Holm equation,” Letters in Mathematical Physics, vol. 76, no. 1, pp. 93–108, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. B. Khesin, J. Lenells, and G. Misiołek, “Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms,” Mathematische Annalen, vol. 342, no. 3, pp. 617–656, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Lenells, G. Misiołek, and F. Tiğlay, “Integrable evolution equations on spaces of tensor densities and their peakon solutions,” Communications in Mathematical Physics, vol. 299, no. 1, pp. 129–161, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. F. Ying, L. Yue, and O. Chougzheng, “On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,” arXiv: 1009.2466v2.
  16. A. Karasu-Kalkanlı, A. Karasu, A. Sakovich, S. Sakovich, and R. Turhan, “A new integrable generalization of the Korteweg-de Vries equation,” Journal of Mathematical Physics, vol. 49, no. 7, article 073516, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. B. Kupershmidt, “KdV6: an integrable system,” Physics Letters. A, vol. 372, no. 15, pp. 2634–2639, 2008. View at Publisher · View at Google Scholar
  18. P. Kersten, I. Krasil'shchik, A. Verbovetsky, and R. Vitolo, “Integrability of Kupershmidt deformations,” Acta Applicandae Mathematicae, vol. 109, no. 1, pp. 75–86, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. A. Constantin and R. Ivanov, “On an integrable two-component Camassa-Holm shallow water system,” Physics Letters. A, vol. 372, no. 48, pp. 7129–7132, 2008. View at Publisher · View at Google Scholar
  20. Y. Yao and Y. Zeng, “The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems,” Journal of Mathematical Physics, vol. 51, no. 6, 2010. View at Publisher · View at Google Scholar
  21. A. Kundu, R. Sahadevan, and R. Nalinidevi, “Nonholonomic deformation of KdV and mKdV equations and their symmetries, hierarchies and integrability,” Journal of Physics. A: Mathematical and Theoretical, vol. 42, no. 11, article 115213, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. P. Guha, “Nonholonomic deformation of generalized KdV-type equations,” Journal of Physics. A: Mathematical and Theoretical, vol. 42, no. 34, article 345201, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH