Geometric Integrability of Some Generalizations of the Camassa-Holm Equation

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 [3–7]. 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)𝑢_𝑡−𝑢_𝑥𝑥𝑡+2𝜔𝑢_𝑥+3𝑢𝑢_𝑥−2𝑢_𝑥𝑢_𝑥𝑥−𝑢𝑢_𝑥𝑥𝑥=0,(1.1) where 𝜔∈ℝ, 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, [10–12]. Furthermore, CH is geometrically integrable [6]. Here, we consider the case 𝜔=0. Then, (1.1) is𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥,𝑚=𝑢_𝑥𝑥−𝑢.(1.2)

The bi-Hamiltonian form of (1.2) is [8, 9]𝑚_𝑡=−ℬ¹𝛿𝐻₂[𝑚]𝛿𝑚=−ℬ²𝛿𝐻₁[𝑚]𝛿𝑚,(1.3) where ℬ¹=𝜕−𝜕³,ℬ²=𝑚𝜕+𝜕𝑚 are the two compatible Hamiltonian operators (𝜕 stands for 𝜕/𝜕𝑥), and the corresponding Hamiltonians are𝐻₁[𝑚]=12𝑚𝑢𝑑𝑥,𝐻²[𝑚]=12𝑢³+𝑢𝑢^2𝑥𝑑𝑥.(1.4)

There exists an infinite sequence of conservation laws (multi-Hamiltonian structure) 𝐻_𝑛[𝑚],𝑛=0,±1,±2,… including (1.4) such that ℬ¹𝛿𝐻_𝑛[𝑚]𝛿𝑚=ℬ²𝛿𝐻_𝑛−1[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] as−𝑢_𝑥𝑥𝑡=−2𝜇(𝑢)𝑢_𝑥+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥,(1.6) where 𝜇(𝑢)=∫¹⁰𝑢𝑑𝑥 and 𝑢(𝑡,𝑥) is a spatially periodic real-valued function of time variable 𝑡 and space variable 𝑥∈𝑆¹=[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])𝜇𝑢_𝑡−𝑢_𝑥𝑥𝑡=−2𝜇(𝑢)𝑢_𝑥+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥.(1.7) Note that 𝜇(𝑢_𝑡)=0 in the periodic case. Introducing 𝑚=𝐴𝑢=𝜇(𝑢)−𝑢_𝑥𝑥 (1.7) becomes𝑚_𝑡+𝑢𝑚_𝑥+2𝑚𝑢_𝑥=0,𝑚=𝜇(𝑢)−𝑢_𝑥𝑥.(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) is𝑚_𝑡=−ℬ¹𝛿𝐻₂[𝑚]𝛿𝑚=−ℬ²𝛿𝐻₁[𝑚]𝛿𝑚,(1.9) where ℬ¹=𝜕𝐴=−𝜕³,ℬ²=𝑚𝜕+𝜕𝑚 are the two compatible Hamiltonian operators, and the corresponding Hamiltonians are𝐻₁[𝑚]=12𝑚𝑢𝑑𝑥,𝐻²[𝑚]=𝜇(𝑢)𝑢²+12𝑢𝑢^2𝑥𝑑𝑥.(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; 𝑤𝑢_{𝑡𝑥𝑥𝑥}=6𝑢𝑢_𝑥+𝑢_𝑥𝑥𝑥−𝑤_𝑥,+4𝑢𝑤_𝑥+2𝑢_𝑥𝑤=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 into𝑢_𝑡=ℬ¹𝛿𝐻_𝑛+1𝛿𝑢−ℬ¹(𝑤)=ℬ²𝛿𝐻_𝑛𝛿𝑢−ℬ¹ℬ(𝑤),₂(𝑤)=0,(1.12) where ℬ¹=𝜕,ℬ²=𝜕³+2(𝑚𝜕+𝜕𝑚) are the two standard Hamiltonian operators of the KdV hierarchy and𝐻₁=𝑢𝑑𝑥,𝐻₂=12𝑢²𝑑𝑥,….(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, [3–7] for more details.

Definition 2.1. A scalar differential equation Ξ(𝑥,𝑡,𝑢,𝑢_𝑥,…,𝑢_{𝑥𝑛𝑡𝑚})=0 in two independent variables 𝑥,𝑡 is of pseudospherical type (or, it describes pseudospherical surfaces) if there exist one-forms 𝜔^𝛼≠0𝜔^𝛼=𝑓_𝛼1𝑥,𝑡,𝑢,…,𝑢_𝑥^𝑟_𝑡^𝑝𝑑𝑥+𝑓_𝛼2𝑥,𝑡,𝑢,…,𝑢_𝑥^𝑠_𝑡^𝑞𝑑𝑡,𝛼=1,2,3,(2.1) whose coefficients 𝑓_𝛼𝛽 are smooth functions which depend on 𝑥,𝑡 and finite number of derivatives of 𝑢, such that the 1-forms 𝜔^𝛼=𝜔^𝛼(𝑢(𝑥,𝑡)) satisfy the structure equations 𝑑𝜔¹=𝜔³∧𝜔²,𝑑𝜔²=𝜔¹∧𝜔³,𝑑𝜔³=𝜔¹∧𝜔²,(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 𝜔¹∧𝜔²≠0, the tensor 𝜔¹⊗𝜔¹+𝜔²⊗𝜔² 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,ℝ)-valued problem 𝑑𝜓=Ω𝜓,(2.3) where Ω is the matrix-valued one-form Ω=𝑋𝑑𝑥+𝑇𝑑𝑡=12⎛⎜⎜⎝𝜔²𝜔¹−𝜔³𝜔¹+𝜔³−𝜔²⎞⎟⎟⎠.(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 𝜓_𝑥=𝑋𝜓,𝜓_𝑡=𝑇𝜓. In fact, this is equivalent to the zero curvature equation𝑋_𝑡−𝑇_𝑥+[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 𝜔^𝛼. The following two Pfaffian systems are completely integrable whenever 𝑢(𝑥,𝑡) is a solution of Ξ=0: −22𝑑𝑑𝛾Γ==𝜔𝜔³₃+𝜔²−2Γ𝜔¹+Γ²𝜔³−𝜔²,(2.6)−𝜔²−2𝛾𝜔¹+𝛾²𝜔³+𝜔².(2.7) Moreover, the one-forms ΘΘ==−𝜔𝜔¹₁−Γ𝜔³−𝜔²,+𝛾𝜔³+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 ℬ¹ and ℬ².

Following Kupershmidt's construction, we introduce the nonholonomic deformation of the CH equation𝑚_𝑡=−𝑢𝑚ℬ_𝑥−2𝑚𝑢_𝑥−ℬ¹(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+1𝜂𝑢^𝑥−𝑢+1𝜂−𝑤^𝑥𝑥𝜔+(𝜂+1)𝑤_𝑥−𝜂𝑤(𝑚+1)𝑑𝑡,₂=(𝜂+1)𝑑𝑥+𝜂+1𝜂−(𝜂+1)𝑢+𝑢^𝑥−𝜂(𝜂+1)𝑤+𝜂𝑤_𝑥𝜔𝑑𝑡,₃=+(𝑚−+𝑢1𝑚++𝜂𝜂)𝑑−𝑥(𝜂+1)₂𝜂𝑢+𝜂+𝜂1𝑢^𝑥+𝜂+1𝜂−𝑤^𝑥𝑥+(𝜂+1)𝑤_𝑥−𝜂𝑤(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 get𝑋=12⎛⎜⎜⎝2(𝑚𝜂++11)+𝜂−(𝜂−+𝜂1)⎞⎟⎟⎠,𝑇=12⎛⎜⎜⎝𝑇¹¹𝑇₁₂𝑇₂₁−𝑇₁₁⎞⎟⎟⎠,(3.3) where𝑇₁₁=𝜂+1𝜂−(𝜂+1)𝑢+𝑢^𝑥𝑇₁₂−𝜂(𝜂+1)𝑤+𝜂𝑤_𝑥,𝑇=𝜂𝑢+𝜂²𝑤−1,₂₁=−2𝑢𝑚−(𝜂+1)²+1𝜂𝑢+2(𝜂𝜂+1)𝑢^𝑥+𝜂+2𝜂−2𝑤^𝑥𝑥+2(𝜂+1)𝑤_𝑥−𝜂(𝜂+2)𝑤−2𝜂𝑤𝑚.(3.4) Hence, we have a zero curvature representation 𝑋_𝑡−𝑇_𝑥+[𝑋,𝑇]=0 for the system (3.1). From (3.3), it is straightforward to obtain the corresponding scalar linear problem𝜓_𝑥𝑥=1𝜓4−𝜂2𝑚𝜓,_𝑡=−𝑢−𝜂𝑤+1𝜂𝜓^𝑥+𝑢^𝑥+𝜂𝑤_𝑥2𝜓.(3.5)

In order to apply Proposition 2.3 to nonholonomic deformation of the CH equation, we consider new 1-forms 𝜔^𝛼_new𝜔¹_new=𝜔²,𝜔²_new=−𝜔¹,𝜔³_new=𝜔³.(3.6) With these forms the Pfaffian system (2.7) becomes22𝛾𝛾_𝑥𝑡=𝜂𝛾²−2(𝜂+1)𝛾+2(𝑚+1)+𝜂,=𝛾²−2𝑤_𝑥𝑥−2[𝜂𝛾−(𝜂+1)]𝑤_𝑥−2𝑤𝜂𝛾_𝑥−2𝛾−𝜂+1𝜂𝑢^𝑥−2𝑢𝛾_𝑥+1𝜂+𝜂+𝜂2−2𝛾𝜂+𝜂1.(3.7) Applying the transform 𝛾⟼𝛾+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 𝑚=𝛾²2𝜆+𝛾^𝑥−𝜆𝛾2,(3.9)_𝑡=𝛾²21+𝜆1𝑢−𝑤𝜆−𝛾𝑢−𝑤𝜆_𝑥−𝑢−𝑤𝜆𝑚+𝜆2+𝜆𝑢−𝜆2²,(3.10) where 𝜆≠0,𝑚=𝑢_𝑥𝑥−𝑢. Moreover, (3.1) possesses the parameter-dependent conservation law 𝛾_𝑡=𝜆(𝑢+𝑤)_𝑥−𝛾−1𝜆𝑢−𝑤𝜆𝛾_𝑥.(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 ℬ¹,ℬ². Applying Kupershmidt's procedure to (1.8), we obtain the nonholonomic deformation of the 𝜇CH equation𝑚_𝑡=−𝑢𝑚ℬ_𝑥−2𝑚𝑢_𝑥−ℬ¹(𝑤),₂(𝑤)=0,(3.12) or2𝑚𝑤𝑚_𝑡𝑥=−𝑢𝑚_𝑥−2𝑚𝑢_𝑥+𝑤_𝑥𝑥𝑥,+𝑤𝑚_𝑥=0,𝑚=𝜇(𝑢)−𝑢_𝑥𝑥.(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=1+21𝜂𝑚−𝜂2²+2𝑑𝑥2𝜂2²𝑢−𝜂𝑢^𝑥+𝑢𝑚+12+𝜇(𝑢)−2𝑢+2𝜂+𝜂2³−2𝜂𝑤−𝜂²𝜔𝑤_𝑥+𝜂𝑤_𝑥𝑥−𝜂²𝑚𝑤𝑑𝑡,₂𝜔=𝜂𝑑𝑥+1−𝜂𝑢+𝑢_𝑥−𝜂²𝑤+𝜂𝑤_𝑥𝑑𝑡,₃=12+1𝜂𝑚−𝜂2²−2𝑑𝑥2𝜂2²𝑢−𝜂𝑢^𝑥+𝑢𝑚+12+𝜇(𝑢)+2𝑢−2𝜂+𝜂2³+2𝜂𝑤−𝜂²𝑤_𝑥+𝜂𝑤_𝑥𝑥−𝜂²𝑚𝑤𝑑𝑡.(3.14)

For the matrices 𝑋 and 𝑇 in, we get𝑋=12⎛⎜⎜⎝𝜂𝑚−𝜂𝜂²22−𝜂⎞⎟⎟⎠,𝑇=12⎛⎜⎜⎝𝑇¹¹𝑇₁₂𝑇₂₁−𝑇₁₁⎞⎟⎟⎠,(3.15) where𝑇𝑇₁₁₁₂=1−𝜂𝑢+𝑢_𝑥−𝜂²𝑤+𝜂𝑤_𝑥,=2−𝑢+1𝑇𝜂−𝜂𝑤₂₁,=𝜂²2𝑢−𝜂𝑢^𝑥+𝑢𝑚+12+𝜇(𝑢)−𝜂²𝑤_𝑥+𝜂𝑤_𝑥𝑥−𝜂²𝑚𝑤+𝜂³2𝑤.(3.16) Hence, we have a zero curvature representation 𝑋_𝑡−𝑇_𝑥+[𝑋,𝑇]=0 for the system (3.13). From (3.3), it is straightforward to obtain the corresponding scalar linear problem𝜓_𝑥𝑥=𝜂𝜓2𝑚𝜓,_𝑡=−𝑢−𝜂𝑤+1𝜂𝜓^𝑥+𝑢^𝑥+𝜂𝑤_𝑥2𝜓,(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 𝜔¹_new=𝜔²,𝜔²_new=−𝜔¹,𝜔³_new=𝜔³.(3.18) With these forms, the Pfaffian system (2.7) becomes2𝛾_𝑥=−2𝛾²−2𝜂𝛾+𝜂𝑚−𝜂²2𝛾_𝑡2,(3.19)=−2𝛾²𝜂+2𝛾²+𝜂₂(𝑢+𝜂𝑤)−2𝛾1−𝜂𝑢+𝑢_𝑥−𝜂²𝑤+𝜂𝑤_𝑥2𝑢−𝜂𝑢^𝑥+𝑚+12+𝜇(𝑢)+𝜂2³𝑤−𝜂²𝑤_𝑥+𝜂𝑤_𝑥𝑥−𝜂²𝑚𝑤.(3.20) After some manipulations, the above system obtains the form 2𝛾_𝑥=−2𝛾²−2𝜂𝛾+𝜂𝑚−𝜂²2𝛾_𝑡2,=−2𝜂𝛾²+𝜂𝑤_𝑥𝑥−[(2𝛾+𝜂)(𝑢+𝜂𝑤)]_𝑥+𝜇(𝑢)−2𝛾−𝜂2.(3.21) Applying the transform 𝛾↦𝛾−𝜂/2, we get𝛾_𝑥=−𝛾²+𝜂𝛾2𝑚,(3.22)_𝑡=−𝛾²𝜂+𝜂2𝑤^𝑥𝑥−[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 𝑚=𝛾²𝛾𝜆+𝛾𝜆^𝑥,(3.24)_𝑡=−2𝛾²𝜆𝜆𝑤^𝑥𝑥−[𝛾(𝑢+2𝜆𝑤)]_𝑥+𝜇(𝑢)2,(3.25) where 𝜆≠0,𝑚=𝜇(𝑢)−𝑢_𝑥𝑥. Moreover, (3.13) possesses the parameter-dependent conservation law 𝛾_𝑡=12𝜆𝛾+𝜆(𝑢+2𝜆𝑤)^𝑥−2𝜆(𝑢+2𝜆𝑤)𝛾_𝑥.(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=𝜆^1/2𝛾₁+𝛾₀+^∞_𝑗=1𝜆^−𝑗/2𝛾_−𝑗.(3.27) Substituting this into (3.24) yields 𝛾₁=√𝑚,𝛾₀=−𝑚^𝑥4𝑚,𝛾⁻¹=132𝑚𝑚^2𝑥5/2+18𝑚𝑚^𝑥3/2_𝑥,…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]:𝑢_𝑡−𝑢_𝑥𝑥𝑡=−3𝑢𝜌𝑢_𝑥𝑡+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝜎𝜌𝜌_𝑥,+(𝑢𝜌)_𝑥=0,(3.29) where 𝜎=±1. Introducing a new variable 𝑣=𝜌²/2, the above system becomes𝑢_𝑡−𝑢_𝑥𝑥𝑡=𝑣−_𝑡3𝑢𝑢_𝑥+2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥+𝜎𝑣_𝑥,+2𝑣𝑢_𝑥+𝑢𝑣_𝑥=0.(3.30) The system (3.30) is geometrically integrable. The corresponding 1-forms, satisfying the structure equations (2.2), are the following: 𝜔¹=𝑢_𝑥𝑥−𝑢−𝜎𝜂𝑣+1𝑑𝑥+𝑢²−𝑢𝑢_𝑥𝑥+𝜂+1𝜂𝑢^𝑥−𝑢+1𝜔𝜂−𝜎𝑣+𝜂𝜎𝑢𝑣𝑑𝑡,₂=(𝜂+1)𝑑𝑥+𝜂+1𝜔𝜂−(𝜂+1)𝑢+𝑢^𝑥𝑑𝑡,₃=+𝑢𝑢_𝑥𝑥−𝑢−𝜎𝜂𝑣+𝜂+1𝑑𝑥₂−𝑢𝑢_𝑥𝑥+𝜂−(𝜂+1)²𝜂𝑢+𝜂+𝜂1𝑢^𝑥+1−𝜎𝑣+𝜂𝜎𝑢𝑣𝑑𝑡.(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=(1−𝑢)𝑑𝑥+−𝑢_𝑥𝑥+𝜂𝑢_𝑥−2𝑢²+1𝜂𝑤^𝑥−1𝜂²𝑤_𝑥𝑥+2𝑢𝑤+2−𝜂²𝑢+2𝜂²𝑤+𝜂²𝜔𝑑𝑡,₂=𝜂𝑑𝑥+𝜂³+2𝜂𝑢+2𝜂𝑤−2𝑢^𝑥−2𝜂²𝑤_𝑥𝜔𝑑𝑡,₃=−(1+𝑢)𝑑𝑥+−𝑢_𝑥𝑥+𝜂𝑢_𝑥−2𝑢²+1𝜂𝑤^𝑥−1𝜂²𝑤_𝑥𝑥+2𝑢𝑤−2+𝜂²𝑢−2𝜂²𝑤−𝜂²𝑑𝑡,(3.32) which coincide with those for KdV equation [3] when 𝑤→0.

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 from𝑚=𝛾²𝛾𝜆+𝛾𝜆^𝑥,(4.1)_𝑡=12𝜆𝛾+𝜆𝑢^𝑥−2𝜆𝑢𝛾_𝑥=𝛾^𝑥2𝜆+𝑢2^𝑥𝑥−𝜕(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)−𝑢_𝑥𝑥. The operators 𝐴⁻¹ and 𝜕 commute and 𝜇(𝑢)=𝜇(𝐴𝑢).

We have𝑢=𝐴⁻¹𝑚,𝑢_𝑥=𝐴⁻¹𝑚_𝑥,𝑢_𝑥𝑥=𝐴⁻¹𝑚_𝑥𝑥,(4.3) in which 𝑚 is determined by (4.1). Then, the second equation (4.2) takes the form 𝛾_𝑡=𝛾^𝑥2𝜆+𝐴⁻¹𝑚_𝑥𝑥2−𝛾𝐴⁻¹𝑚_𝑥−𝛾_𝑥𝐴⁻¹𝑚,(4.4) or 𝐴𝛾_𝑡=𝐴𝛾^𝑥2𝜆+𝑚2^𝑥𝑥−𝐴𝜕𝛾𝐴⁻¹𝑚.(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.