AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 275450 10.1155/2014/275450 275450 Research Article Consecutive Rosochatius Deformations of the Garnier System and the Hénon-Heiles System Xia Baoqiang Zhou Ruguang Rui Weiguo 1 School of Mathematics and Statistics Jiangsu Normal University Xuzhou, Jiangsu 221116 China jsnu.edu.cn 2014 3132014 2014 08 01 2014 20 02 2014 31 3 2014 2014 Copyright © 2014 Baoqiang Xia and Ruguang Zhou. 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.

An algorithm of constructing infinitely many symplectic realizations of generalized sl(2) Gaudin magnet is proposed. Based on this algorithm, the consecutive Rosochatius deformations of integrable Hamiltonian systems are presented. As examples, the consecutive Rosochatius deformations of the Garnier system and the Hénon-Heiles system as well as their Lax representations, are obtained.

1. Introduction

Usually the integrability of a Hamiltonian system is destroyed even with a very small perturbation. As early as in 1877, Rosochatius first discovered that it would keep the integrability to add a potential of the sum of inverse squares of the coordinates to that of the Neumann system [1, 2]. This provides an interesting example of integrable perturbation. Nowadays, the resulting system is called the Neumann-Rosochatius system . In 1985, Wojciechowski gained an analogy system (called Garnier-Rosochatius system) for the Garnier system [8, 9]. Later in 1999, based on the Deift technique and a well-known theorem that the Gauss map transforms the Neumann system to the Jacobi system, Kubo et al. constructed the analogy system for the Jacobi system or the geodesic flow equation on the ellipsoid . In 2007, one of the authors (Zhou) generalized the Rosochatius deformations of the constrained soliton flows , and then the method has been extended to construct the integrable deformations of the symplectic maps  and the soliton equations with self-consistent sources .

There appear some important physical and mathematical applications of Rosochatius deformed integrable systems. For example, the Neumann-Rosochatius system can be used to describe the dynamics of a rotating closed string and the membranes [16, 17], the Garnier-Rosochatius system can be used to solve the multicomponent coupled nonlinear Schödinger equation [18, 19], and the Rosochatius deformation of the KdV equation with self-consistent sources can be used to establish the bi-Hamiltonian structure of the KdV6 equation .

Recently, we proposed an approach to generate integrable Rosochatius deformations of the Neumann system consecutively . The Lax matrix of the N-copies of Neumann system is of the form of classical sl(2) Gaudin magnet defined on the 2(N-1)-dimensional submanifold. In this paper, we would like to show that the approach can be applied to the integrable Hamiltonian systems whose Lax matrices are of the form of the generalized Gaudin magnet. We first present an algorithm of constructing infinitely many realizations of generalized sl(2) Gaudin magnet model. Then, we describe how to generate integrable Hamiltonian systems based on the realizations of sl(2) Gaudin magnet. The Rosochatius deformation of an integrable Hamiltonian system is explained as a special case of the realizations of generalized sl(2) Gaudin magnet model. Thus, such an algorithm enables us to construct Rosochatius deformations of the integrable Hamiltonian systems consecutively. As applications, we obtain the consecutive Rosochatius deformations of the Garnier system and the Hénon-Heiles system as well as their Lax representations.

The plan of the paper is as follows. In Section 2, we propose infinitely many symplectic realizations of sl(2) Gaudin magnet and describe how to generate the integrable Hamiltonian systems based on these realizations. In Sections 3 and 4, we pay attention to studying the integrable deformations of the Garnier system and Hénon-Heiles system, respectively. Some concluding remarks are drawn in Section 5.

2. The Generalized sl(2) Gaudin Magnet and Its Realizations 2.1. The Realizations of the Generalized sl(2) Gaudin Magnet

We consider the Lax matrix of the form of the generalized Gaudin magnet [22, 23] (1)L(λ)=L0(λ)+12j=1N1λ-λj(Sj0Sj-Sj+-Sj0), where L0(λ) is a traceless 2  ×  2 matrix whose entries are polynomials of λ or λ-1 and Sj satisfy N copies of the standard sl(2) algebra (2){Sj0,Sk+}=2δjkSk+,{Sj0,Sk-}=-2δjkSk-,{Sj+,Sk-}=4δjkSk0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhj,k=1,2,,N, with N Casimirs (3)Cj=(Sj0)2+Sj-Sj+,j=1,2,,N.

It is well known that the sl(2) algebra (2) has a symplectic realization: (4)Sj0=qjpj,Sj-=-qj2,Sj+=pj2,hhhhhhhhhhhhhhhhhhhj=1,2,,N, where qj, pj are canonical coordinates on the standard symplectic space (2N,ω2=j=1Ndpjdqj). Under the realization of (4), the Lie-Poisson brackets (2) are recovered by computing the standard Poisson bracket (5){F(q,p),G(q,p)}=j=1N(FqjGpj-FpjGqj), where q=(q1,,qN), p=(p1,,pN), F(q,p), and G(q,p) are arbitrary smooth functions about q and p.

With a direct calculation, we observe the following proposition.

Proposition 1.

If Sj0=fj(q,p), Sj-=gj(q,p), and Sj+=hj(q,p) is a realization of (2), so are (6)S~j0=fj(q,p),S~j-=gj(q,p),S~j+=hj(q,p)+γjgj-1(q,p),S^j0=fj(q,p),S^j-=gj(q,p)+βjhj-1(q,p),S^j+=hj(q,p), where γj,βj,(j=1,2,,N) are arbitrary constants.

This proposition provides us with two kinds of new realizations of sl(2) algebra (2) from a known one. Moreover, applying such two kinds of realizations in turn, we can construct an infinitely many realizations of sl(2) algebra (2). For example, from (4), we obtain the following realizations of (2):(7)S~j0=qjpj,S~j-=-qj2,S~j+=pj2+γjqj-2,(8)S^j0=qjpj,S^j-=-qj2-βj(pj2+γjqj-2)-1,S^j+=pj2+γjqj-2.

2.2. A Recipe for Generating Integrable Hamiltonian Systems Based on Realizations of sl(2) Gaudin Magnet

Now, we describe how to generate an integrable Hamiltonian system based on a symplectic realization of sl(2) Gaudin magnet. We suppose that the Lax matrix (1) satisfies an r-matrix relation  (9){L1(λ),L2(μ)}=[r12(λ,μ),L1(λ)]-[r21(λ,μ),L2(μ)], where L1(λ)=L(λ)I2, L2(μ)=I2L(μ), I2 is 2×2 identity matrix, μ is an arbitrary parameter, and [·,·] denotes the commutator of the matrices, such as [r12,L1]=r12L1-L1r12. According to the general theory of the r-matrix [24, 25], we have (10){detL(λ),detL(μ)}=0.

First, we expand detL(λ) as (11)detL(λ)=j=k0Fjλ-j,ordetL(λ)=j=k0Fjλj. From (10), we have (12){Fj,Fk}=0,j,kk0, which implies that Fk’s are in involution in pairs. Usually, we can single out N functionally independent Fk1,,FkN among {Fk}kk0. Choosing a Hamiltonian H, which is composed of some of Fk’s, we have (13){H,Fkj}=0,1jN. Functionally independent and involutive pairwise integrals, Fk1,,FkN, ensure that the Hamiltonian system H is completely integrable in the sense of Liouville .

Further, substituting a realization of (2) into the Lax matrix (1) and the corresponding Fk’s and H defined above, we finally obtain an integrable Hamiltonian system with the Hamiltonian H expressed in canonical coordinates (qj,pj):(14)qj,x=Hpj,pj,x=-Hqj,1jN.

The above recipe shows that, once having a symplectic realization of sl(2) algebra (2), we may obtain an integrable Hamiltonian system. In the next sections, we will take the Garnier system and the Hénon-Heiles system as examples to show that the Rosochatius deformations and second Rosochatius deformations of integrable systems can be generated from the realizations of (7) and (8), respectively, according to the above recipe. Thus, applying Proposition 1 in turn enables us to consecutively construct Rosochatius deformations of the integrable Hamiltonian systems.

3. Consecutive Rosochatius Deformations of the Garnier System

We take L0(λ) in (1) as (15)L0(λ)=(01-λ+12j=1NSj-0); then, the Lax matrix (1) becomes (16)L(λ)=(01-λ+12j=1NSj-0)+12j=1N1λ-λj(Sj0Sj-Sj+-Sj0)(L11(λ)L12(λ)L21(λ)-L11(λ)). Direct calculations yield that (17){L11(λ),L11(μ)}={L12(λ),L12(μ)}=0,{L21(λ),L21(μ)}=2L11(λ)-2L11(μ),{L11(λ),L12(μ)}=1μ-λ(L12(μ)-L12(λ)),{L12(λ),L21(μ)}=2μ-λ(L11(μ)-L11(λ)),{L11(λ),L21(μ)}=1μ-λ(L21(λ)-L21(μ))-L12(λ), which is equivalent to the r-matrix algebra.

Proposition 2.

L ( λ ) satisfies the r-matrix relation (18){L1(λ),L2(μ)}=[r12(λ,μ),L1(λ)]-[r21(λ,μ),L2(μ)], where (19)r12(λ,μ)=r21(μ,λ)=1μ-λP+S,P=(1000001001000001),S=(0000000000001000).

Expand detL(λ) as follows: (20)FλdetL(λ)=λ+m=0Fmλ-m-1, where (21)F0=-12{j=1NSj+-j=1NλjSj-+12(j=1NSj-)2},Fm=-12{j=1NλjmSj+-j=1Nλjm+1Sj-+12(j=1NSj-)(j=1NλjmSj-)}-14l+k=m-1[(j=1NλjlSj0)(j=1NλjkSj0)+(j=1NλjlSj-)(j=1NλjkSj+)],m1. Then, we have the involutive relation (22){Fj,Fk}=0,j,k=0,1,2,.

Under the realization of (4), we obtain the following Lax matrix (23)L(λ)=(01-λ-12q,q0)+12j=1N1λ-λj(qjpj-qj2pj2-qjpj). Then, (21) becomes (24)F0=-12{Aq,q+p,p+12q,q2},Fm=-12{Am+1q,q+Amp,p+12q,qAmq,q}-14l+k=m-1[Alq,pAkq,p-Alq,qAkp,p],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhm1. Here and after, A=diag(λ1,λ2,,λN) and , stands for the standard inner product in the Euclidean space. The Hamiltonian system with Hamiltonian H=-F0 reads (25)qj,x=pj,pj,x=-λjqj-q,qqj,1jN, which is nothing but the Garnier system [27, 28]. We can check directly that the Garnier system (25) allows the Lax representation: (26)ddxL(λ)=[U(λ),L(λ)], where L(λ) is given by (23), and (27)U(λ)=(01-λ-u0),u=q,q.

Example 3 (The Garnier-Rosochatius System).

From the realization of (7), we arrive at the Lax matrix (28)L~(λ)=(01-λ-12q,q0)+12j=1N1λ-λj(qjpj-qj2pj2+γjqj-2-qjpj), and (21) becomes (29)F~0=-12{Aq,q+p,p+γq-1,q-1+12q,q2},F~m=-12{Am+1q,q+Amp,p+Amγq-1,q-1(Akγq-1,q-1)]+12q,qAmq,q+12l+k=m-1[Alq,pAkq,p-Alq,q×(Akp,p+Akγq-1,q-1)]},hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhm1, where γ=diag(γ1,γ2,,γN). Choosing a Hamiltonian H~=-F~0, we arrive at (30)qj,x=pj,pj,x=-λjqj-q,qqj+γjqj-3,hhhhhhhhhhhhhhhhhhhhhhhhh1jN, which is just the Garnier-Rosochatius system [8, 9, 13, 29]. It can be checked easily that (30) allows the Lax representation: (31)ddxL~(λ)=[U(λ),L~(λ)], where L~(λ) is given by (28) and U(λ) is given by (27).

Example 4 (The Second Rosochatius Deformation of the Garnier System).

Based on the realization of (8), we obtain the Lax matrix (32)L^(λ)=(01-λ-12(q,q+i=1Nβi(pi2+γiqi-2)-1)0)+12j=1N1λ-λj×(qjpj-qj2-βj(pj2+γjqj-2)-1pj2+γjqj-2-qjpj), and the integrals of motion (33)F^0=-12{Aq,q+j=1Nλjβj(pj2+γjqj-2)-1+p,p+γq-1,q-1+12(q,q+j=1Nβj(pj2+γjqj-2)-1)2},(34)F^m=-12{Am+1q,q+j=1Nλjm+1βj(pj2+γjqj-2)-1+Amp,p+Amγq-1,q-1+12(q,q+j=1Nβj(pj2+γjqj-2)-1)×(Amq,q+j=1Nλjmβj(pj2+γjqj-2)-1)+12l+k=m-1[Alq,pAkq,p-(Alq,qj=1N+j=1Nλjlβj(pj2+γjqj-2)-1)×(Akp,p+Akγq-1,q-1)]}. Choosing a Hamiltonian H^=-F^0, we obtain an integrable Hamiltonian system (35)qj,x=pj-(λj+q,q+i=1Nβi(pi2+γiqi-2)-1)×βj(pj2+γjqj-2)-2pj,pj,x=γjqj-3-(λj+q,q+i=1Nβi(pi2+γiqi-2)-1)×(qj+βjγj(pj2+γjqj-2)-2qj-3), which is the second Rosochatius deformation of the Garnier system. With direct calculations, we find that (35) admits the Lax representation: (36)ddxL^(λ)=[U^(λ),L^(λ)], where (37)U^(λ)=(01-λ-u^0),u^=q,q+j=1Nβj(pj2+γjqj-2)-1.

There is no doubt that we can consecutively construct Rosochatius deformations of the Garnier system by applying the two kinds of realizations in Proposition 1 in turn and the recipe we described in Section 2.2. Here, we only present the above two examples.

4. Consecutive Rosochatius Deformations of the Hénon-Heiles System

Now, we begin with the Lax matrix of the form (38)L(λ)=(2pN+18λ-4qN+1-8λ2-4qN+1λ-2qN+12+12j=1NSj--2pN+1)+12j=1N1λ-λj(Sj0Sj-Sj+-Sj0). Defining a generating function (39)Fλ=detL(λ)=64λ3+P0+m=0Fmλ-m-1, we have (40)P0=-4pN+12-8qN+13-4j=1NSj++4qN+1j=1NSj-+4j=1NλjSj-,F0=-4j=1NλjSj++4j=1Nλj2Sj-+2qN+1(j=1NSj++j=1NλjSj-)+qN+12j=1NSj--2pN+1j=1NSj0-14(j=1NSj-)2,Fm=-4j=1Nλjm+1Sj++4j=1Nλjm+2Sj-+2qN+1(j=1NλjmSj++j=1Nλjm+1Sj-)+qN+12j=1NλjmSj--2pN+1j=1NλjmSj0-14(j=1NSj-)(j=1NλjmSj-)-14l+k=m-1[(j=1NλjlSj0)(j=1NλjkSj0)+(j=1NλjlSj-)(j=1NλjkSj+)],m1. We may check directly that (38) satisfies the same r-matrix relation as (18). Thus, we have the involutive relation: (41){P0,Fk}={Fj,Fk}=0,j,k=0,1,2,.

Now, we discuss the integrable Hamiltonian system generated by the Lax matrix (38) and its realizations. Firstly, with the realization of (4), we arrive at the following Lax matrix: (42)L(λ)=(2pN+18λ-4qN+1-8λ2-4qN+1λ-2qN+12-12q,q-2pN+1)+12j=1N1λ-λj(qjpj-qj2pj2-qjpj), and (40) becomes (43)P0=-4pN+12-8qN+13-4p,p-4qN+1q,q-4q,Aq,F0=-4p,Ap-4q,A2q+2qN+1(p,p-q,Aq)-qN+12q,q-2pN+1q,p-14q,q2,Fm=-4p,Am+1p-4q,Am+2q+2qN+1(p,Amp-q,Am+1q)-qN+12q,Amq-2pN+1q,Amp-14q,qq,Amq-14l+k=m-1[Alq,pAkq,p-Alq,qAkp,p],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhm1.

The Hamiltonian system with Hamiltonian H=-(1/8)P0 reads (44)qj,x=pj,pj,x=-λjqj-qN+1qj,qN+1,x=pN+1,pN+1,x=-3qN+12-12q,q,1jN, which is just the Hénon-Heiles system , and it allows the Lax representation: (45)ddxL(λ)=[U(λ),L(λ)], where L(λ) is given by (42), and (46)U(λ)=(01-λ-qN+10).

Example 5 (The Rosochatius Deformation of the Hénon-Heiles System).

Under realization of (7), we arrive at the Lax matrix (47)L~(λ)=(2pN+18λ-4qN+1-8λ2-4qN+1λ-2qN+12-12q,q-2pN+1)+12j=1N1λ-λj(qjpj-qj2pj2+γjqj-2-qjpj), and (40) becomes (48)P~0=-4pN+12-8qN+13-4(p,p+q-1,γq-1)-4qN+1q,q-4q,Aq,F~0=-4(p,Ap+q-1,Aγq-1)-4q,A2q+2qN+1(p,p+q-1,γq-1-q,Aq)-qN+12q,q-2pN+1q,p-14q,q2,F~m=-4(p,Am+1p+q-1,Am+1γq-1)-4q,Am+2q-2pN+1q,Amp-14q,qq,Amq+2qN+1(p,Amp+q-1,Amγq-1-q,Am+1q)-qN+12q,Amq-14l+k=m-1[Alq,pAkq,p-Alq,q(Akp,p+Akγq-1,q-1)],hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhm1. Taking the Hamiltonian as H~=-(1/8)P~0, we have (49)qj,x=pj,pj,x=-λjqj-qN+1qj+γjqj-3,qN+1,x=pN+1,pN+1,x=-3qN+12-12q,q,1jN, which is exactly the Rosochatius deformation of Hénon-Heiles system [12, 15]. It can be checked directly that (49) allows the Lax representation: (50)ddxL~(λ)=[U(λ),L~(λ)], where L~(λ) is given by (47) and U(λ) is given by (46).

Example 6 (The Second Rosochatius Deformation of the Hénon-Heiles System).

Based on the realization of (8), we obtain the Lax matrix(51)L^(λ)=(2pN+18λ-4qN+1-8λ2-4qN+1λ-2qN+12-12(q,q+j=1Nβj(pj2+γjqj-2)-1)-2pN+1)+12j=1N1λ-λj(qjpj-qj2-βj(pj2+γjqj-2)-1pj2+γjqj-2-qjpj).The integrals of motion P^0, F^0, and F^m, m1, can be generated from detL^(λ). In particular, we have (52)P^0=-4pN+12-8qN+13-4(p,p+q-1,γq-1)-4qN+1(q,q+j=1Nβj(pj2+γjqj-2)-1)-4(q,Aq+j=1Nλjβj(pj2+γjqj-2)-1). Choosing a Hamiltonian as H^=-(1/8)P^0, we arrive at an integrable Hamiltonian system (53)qj,x=pj-(λj+qN+1)βj(pj2+γjqj-2)-2pj,pj,x=-λjqj-qN+1qj+γjqj-3-(λj+qN+1)βjγj(pj2+γjqj-2)-2qj-3,qN+1,x=pN+1,pN+1,x=-3qN+12-12(q,q+i=1Nβi(pi2+γiqi-2)-1),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh1jN, which is the second Rosochatius deformation of Hénon-Heiles system. Again, we may check that (53) allows the Lax representation: (54)ddxL^(λ)=[U(λ),L^(λ)], where L^(λ) is given by (51) and U(λ) is given by (46).

5. Concluding Remarks

We have shown how to consecutively generate integrable Rosochatius deformations of the integrable Hamiltonian systems whose Lax matrices are of the form of the generalized Gaudin magnet. As applications, we obtained the consecutive Rosochatius deformations of the Garnier system and the Hénon-Heiles system together with their Lax representations. Our method is performed in a unified way. There is no doubt that our method can be applied to other constrained soliton flows [28, 32] whose Lax matrices are of the form of the generalized Gaudin magnet or the generalized Gaudin magnet with boundary. Also, we remark that our method can be used to construct consecutive Rosochatius deformations of the integrable symplectic maps and the soliton equations with self-consistent sources.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (Grant nos. 11301229 and 11271168), the Natural Science Foundation of the Jiangsu Province (Grant no. BK20130224), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant no. 13KJB110009).

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