Advances in Mathematical Physics

Volume 2014 (2014), Article ID 708603, 8 pages

http://dx.doi.org/10.1155/2014/708603

## New Neumann System Associated with a 3 × 3 Matrix Spectral Problem

^{1}College of Science, Henan University of Technology, 100 Lianhua Road, Zhengzhou, Henan 450001, China^{2}Department of Information Engineering, Henan College of Finance and Taxation, Zhengkai Road, Zhengzhou, Henan 451464, China

Received 7 April 2014; Revised 21 June 2014; Accepted 13 July 2014; Published 24 July 2014

Academic Editor: Manuel De León

Copyright © 2014 Fang Li and Liping Lu. 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

The nonlinearization approach of Lax pair is applied to the case of the Neumann constraint associated with a 3 × 3 matrix spectral problem, from which a new Neumann system is deduced and proved to be completely integrable in the Liouville sense. As an application, solutions of the first nontrivial equation related to the 3 × 3 matrix spectral problem are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

#### 1. Introduction

Soliton equations are nonlinear partial differential equations described by infinite-dimensional integrable systems and have various beautiful algebraic and geometric properties [1–5]. It has been shown that the nonlinearization of spectral problems (NSPs) approach is a powerful tool to study soliton equations. According to this method which was first introduced by Cao [6], each -dimensional soliton equation is decomposed into two ordinary differential equations: one is spatial and the other is temporal. The resulting decomposition not only inherits many integrable properties from soliton equations such as possessing Lax pairs, but also provides an effective way to derive explicit solutions of soliton equations.

During the 1990s, the method of NSPs has attracted great interest in the soliton field and has been applied to a large number of soliton equations associated with matrix spectral problems [7–14]. Furthermore, based on the Cao’s nonlinearization technique, the work by Zhou and Qiao is the first time to develop the nonlinearization approach to find the algebrogeometric solutions for integrable systems in both continuous and discrete cases [15–20]. However, due to the complexity of higher-order matrix spectral problems, there is not much research on NSPs for soliton equations associated with higher-order matrix spectral problems [21–27]. In addition, because of the limitation of the Riemann theory, the algebrogeometric solutions of the soliton equations associated with matrix spectral problems cannot be obtained with the aid of the nonlinearization technique.

In this paper, we will study the following soliton equation with the help of the method of NSPs: Equation (1) is first proposed in [28] and associated with the matrix spectral problem

In [28], Geng and Du have obtained some explicit solutions, which include soliton and periodic solutions. If , (1) can be reduced to a couple of equations in and , which can be presented as The corresponding Lax pair for the reduced system is as follows:

The aim of the present paper is to derive the corresponding finite-dimensional Hamiltonian system associated with the matrix spectral problem, which is proved to be completely integrable in the Liouville sense. As an application, solutions of (1) are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

The paper is organized as follows. In Section 2, we will introduce the Neumann constraint between the potentials and eigenfunctions of the matrix spectral problem (2). Under this constraint, we obtain a new Neumann system and a generating function of integrals of motion. In Section 3, the generating function approach is used to calculate the involutivity of integrals of motion, by which the Neumann system is further proved to be completely integrable in the Liouville sense. In Section 4, we will reveal the relation between the Neumann system and (1). Solutions of (1) are decomposed into solving two compatible Hamiltonian systems of ordinary differential equations.

#### 2. A New Neumann System

In this section, we first consider the stationary zero-curvature equation of the spectral problem (2) and its auxiliary problem; that is, where Substituting (6) into (5) yields the Lenard equation: where and are two skew-symmetric operators defined byin which we denote by for convenience. Expanding entries , , and as the Laurent expansions in then (7) leads to where . We can choose the first two members as In order to calculate the functional gradient of the eigenvalue with regard to the potentials, we introduce the spectral problem where , . Let be distinct nonzero eigenvalues; then the systems associated with (12) can be written in the form where , , , , are eigenfunctions. A direct calculation gives rise to the functional gradient of the eigenvalue with regard to the potentials , , and :

Now we consider the Neumann constraint which can be written as where is the standard inner-product in , , . From (13) and the third expression of (16), it is easy to see that where . Substituting (16) and (17) into (13), we obtain the following new Neumann system on a -dimensional manifold : with the manifold being defined by Consider the following function : Through a direct calculation we have in which the poisson bracket of two functions is defined as Introduce a modified function : where and are two Lagrangian multipliers: This means that is tangent to the manifold . Therefore, (18) can be represented as the standard canonical equation on :

On the other hand, through tedious calculations we obtain where Then the solution of the Lenard equation with parameter can be written as where , and satisfies under the Neumann constraint (16).

#### 3. The Liouville Integrability

Now we introduce a Lax matrix by Through a direct calculation we can prove that and are two solutions of , where is a unit matrix and is a parameter. Then and are independent of . It is easy to see that where In order to generate the Hamiltonians, we take the following notations Substituting the Laurent expansion of into (32) we have where We can prove the following assertion.

Proposition 1. *Suppose that , then
*

*Proof. *Through tedious calculation we can obtain
then we have
Substituting (30) and (38) into the left hand side of (35) shows that (35) holds. Moreover, relation (36) follows by comparison of power of in (33) with (37) taken into account.

*In order to guarantee that the Hamiltonians are tangent to the constrained manifold , we calculate that
The Lagrangian multipliers are given by
Thus the modified functions
are tangent to the manifold and are in involution in pairs on ; that is
*

*Proposition 2. The 1-forms , , () are linearly independent.*

*Proof. *Assume that there exist constants , so that
It is easy to see that
Then we have
which gives rise to
by substituting and into (45), respectively. Therefore, we have , , , , by utilizing that Vandermonde determinant is not zero. Then (43) is transformed to

According to (34), we obtain

Let
and noticing that
then we have
Thus (47) yields
where

To arrive at , one needs only to prove det . Therefore we introduce a matrix :

It is easy to verify that det, and
where represents the entries which may not be zero, and this proves Proposition 2.

*Resorting to the two propositions above and noticing that , we obtain the following assertion.*

*Proposition 3. The Neumann system defined by (18) is completely integrable in the Liouville sense on .*

*4. The Representation of Solutions*

*In this section, we will give the representation of solutions for (1). To this end, we denote the variable of -flow by , where
and is the Lagrangian multiplier
Then the canonical equation of the -flow on is
Using (58) and the Neumann constraints (16) and (17), we derive that
On the other hand, combining (18), (16), and (17) we have
Then we arrive at
which is (1). Therefore we obtain the following result.*

*Proposition 4. Let , , be a compatible solution of the Hamiltonian systems (18) and (58); then the functions
solve (1).*

*Proof. *We only need to prove that the Hamiltonian systems (18) and (58) are compatible. In fact, it is not difficult to verify that . Hence the proof is completed [29].

*Conflict of Interests*

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

*Acknowledgments*

*This work was supported by the National Natural Science Foundation of China (Grant no. 11301487), the Foundation for the Author of National Excellent Doctoral Dissertation of China (no. 201313), the Research Foundation of Henan University of Technology (Grant no. 2009BS041), and Soft Science Research Project of the Science and Technology Department of Henan Province (no. 142400410274).*

*References*

- A. C. Newell,
*Solitons in Mathematics and Physics*, SIAM, Philadelphia, Pa, USA, 1985. View at Publisher · View at Google Scholar · View at MathSciNet - C. H. Gu,
*Soliton Theory and Its Application*, Springer, Berlin, Germany, 1995. - X. G. Geng and C. W. Cao, “Decomposition of the $(2+1)$-dimensional Gardner equation and its quasi-periodic solutions,”
*Nonlinearity*, vol. 14, no. 6, pp. 1433–1452, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Zhou, “Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field,”
*Journal of Mathematical Physics*, vol. 43, no. 10, pp. 5002–5012, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. G. Geng and F. Li, “A hierarchy of new nonlinear evolution equations associated with a 3×3 matrix spectral problem,”
*Chinese Physics Letters*, vol. 26, Article ID 050201, 2009. View at Google Scholar - C. W. Cao, “Nonlinearization of the Lax system for AKNS hierarchy,”
*Science in China A*, vol. 33, no. 5, pp. 528–536, 1990. View at Google Scholar · View at MathSciNet - C. W. Cao and X. G. Geng, “C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy,”
*Journal of Physics. A: Mathematical and General*, vol. 23, no. 18, pp. 4117–4126, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. Geng, “Discrete Bargmann and Neumann systems and finite-dimensional integrable systems,”
*Physica A*, vol. 212, no. 1-2, pp. 132–142, 1994. View at Publisher · View at Google Scholar · View at Scopus - W. X. Ma and W. Strampp, “An explicit symmetry constraint for the LAX pairs and the adjoint LAX pairs of AKNS systems,”
*Physics Letters A*, vol. 185, no. 3, pp. 277–286, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. Du and C. Cao, “The Lie-Poisson representation of the nonlinearized eigenvalue problem of the Kac-van Moerbeke hierarchy,”
*Physics Letters A*, vol. 278, no. 4, pp. 209–224, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - C. W. Cao, “Parametric representation of the finite-band solution of the Heisenberg equation,”
*Physics Letters. A*, vol. 184, no. 4-5, pp. 333–338, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. G. Geng, Y. T. Wu, and C. W. Cao, “Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation,”
*Journal of Physics A: Mathematical and General*, vol. 32, no. 20, pp. 3733–3742, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W.-X. Ma and Z. Zhou, “Binary symmetry constraints of $N$-wave interaction equations in {$1+1$} and {$2+1$} dimensions,”
*Journal of Mathematical Physics*, vol. 42, no. 9, pp. 4345–4382, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. G. Zhou, “Nonlinearizations of spectral problems of the nonlinear Schrödinger equation and the real-valued modified Korteweg-de Vries equation,”
*Journal of Mathematical Physics*, vol. 48, no. 1, Article ID 013510, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Qiao and R. Zhou, “Discrete and continuous integrable systems possessing the same non-dynamical $r$-matrix,”
*Physics Letters A*, vol. 235, no. 1, pp. 35–40, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. G. Zhou, “The finite-band solution of the Jaulent-Miodek equation,”
*Journal of Mathematical Physics*, vol. 38, no. 5, pp. 2535–2546, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Qiao, “$r$-matrix and algebraic-geometric solution for integrable symplectic map,”
*Chinese Science Bulletin*, vol. 44, no. 2, pp. 114–118, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. J. Qiao, “Generalized r-matrix structure and algebro-geometric solution for integrable system,”
*Reviews in Mathematical Physics*, vol. 13, no. 5, pp. 545–586, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. J. Qiao, “R-matrix and algebro-geometric solution for the AKNS system,”
*Theoretical and Mathematical Physics*, vol. 127, pp. 827–834, 2001. View at Google Scholar - Z. Qiao, “The Camassa-Holm hierarchy, $N$-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold,”
*Communications in Mathematical Physics*, vol. 239, no. 1-2, pp. 309–341, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y. Wu and X. Geng, “A finite-dimensional integrable system associated with the three-wave interaction equations,”
*Journal of Mathematical Physics*, vol. 40, no. 7, pp. 3409–3430, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - H. H. Dai and X. G. Geng, “Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation,”
*Physics Letters A*, vol. 317, no. 5-6, pp. 389–400, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - D. L. Du, “Complex form, reduction and Lie-Poisson structure for the nonlinearized spectral problem of the Heisenberg hierarchy,”
*Physica A*, vol. 303, no. 3-4, pp. 439–456, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - R. Zhou, “Dynamical $r$-matrices for the constrained Harry-Dym flows,”
*Physics Letters A*, vol. 220, no. 6, pp. 320–330, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. G. Geng and H. H. Dai, “Nonlinearization of the $3\times 3$ matrix eigenvalue problem related to coupled nonlinear Schrödinger equations,”
*Journal of Mathematical Analysis and Applications*, vol. 233, no. 1, pp. 26–55, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - M. De la Sen, “Sufficiency-type stability and stabilization criteria for linear time-invariant systems with constant point delays,”
*Acta Applicandae Mathematica*, vol. 83, no. 3, pp. 235–256, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Z. Qiao and S. Li, “A new integrable hierarchy, parametric solutions and traveling wave solutions,”
*Mathematical Physics, Analysis and Geometry*, vol. 7, no. 4, pp. 289–308, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - X. G. Geng and D. L. Du, “Two hierarchies of new nonlinear evolution equations associated with 3 × 3 matrix spectral problems,”
*Chaos, Solitons and Fractals*, vol. 29, no. 5, pp. 1165–1172, 2006. View at Publisher · View at Google Scholar · View at Scopus - V. I. Arnold,
*Mathematical Methods of Classical Mechanics*, Springer, 1978. View at MathSciNet

*
*