Advances in Mathematical Physics

Volume 2016 (2016), Article ID 6347961, 13 pages

http://dx.doi.org/10.1155/2016/6347961

## The Bi-Integrable Couplings of Two-Component Casimir-Qiao-Liu Type Hierarchy and Their Hamiltonian Structures

Department of Applied Mathematics, China Agricultural University, Beijing 100083, China

Received 13 March 2016; Accepted 29 June 2016

Academic Editor: Boris G. Konopelchenko

Copyright © 2016 Juhui Zhang and Yuqin Yao. 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

A new type of two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is produced from a new spectral problem and their bi-Hamiltonian structures are constructed. Particularly, a new completely integrable two-component Casimir-Qiao-Liu type equation (2-CQLTE) is presented. Furthermore, based on the semidirect sums of matrix Lie algebras consisting of block matrix Lie algebra, the bi-integrable couplings of the 2-CQLTH are constructed and their bi-Hamiltonian structures are furnished.

#### 1. Introduction

Solitons and integrable systems play an important role in nonlinear wave and dynamics systems. It has been significant in soliton theory to find more new integrable systems. In 1996, Olver and Rosenau obtained a Casimir equation [1]:which is an integrable case of the general class of equations . In 2009, Qiao and Liu proposed a third-order integrable peakon equation [2, 3]:which possesses Lax representation and bi-Hamiltonian structures. Here we can find that (2) can be obtained from the Casimir equation (1) by setting . We will use Casimir-Qiao-Liu equation (CQLE) to denote (2) in this paper. The CQLE (2) represents a third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids. Moreover, it may be reduced from the two-dimensional Euler equation by an approximation procedure, and its solutions may be useful to construct new solvable potentials in Newtonian dynamics and to model electrophysiological phenomena in neuroscience. It has attracted many scholars’ attention in recent years [4–6].

In recent years, the construction of soliton hierarchies and integrable couplings have become important research fields in soliton theory [7–10]. The soliton hierarchies are generated from the zero curvature equations [11, 12] which are based on semisimple Lie algebras, while the integrable couplings are generated from the zero curvature equations based on semidirect sums of Lie algebras [13–15].

The trace identity proposed by Tu is a useful tool for constructing the Hamiltonian structures for both continuous and discrete integrable systems [11, 12]. Many integrable Hamiltonian systems of infinite dimensions with various physics and mathematical backgrounds have been obtained [16–18]. But when it comes to the integrable couplings, since they are based on semidirect sums of Lie algebras, the trace identity cannot be used properly. In order to solve this problem, Ma proposes the variational identity [19] while Guo and Zhang propose the quadratic-form identity [20]. By using them, the Hamiltonian structures of many integrable couplings systems have been furnished [21–23].

Integrable couplings correspond to nonsemisimple Lie algebras , and such Lie algebras can be written as semidirect sums [24]:The notion of semidirect sums means that the two Lie subalgebras and satisfywhere , with denoting the Lie bracket of . We also require the closure property between and under the matrix multiplication:where .

Now we make the following assumptions:where and satisfying (3), (4), and (5). In this condition, we can construct the enlarged spectral problem asFrom the enlarged zero curvature equationswe haveThe first equation of (9) is the original soliton hierarchy while the second equation is the integrable couplings.

The bi-Hamiltonian structures of (9)can be obtained by using the following variational identity:where is a bilinear form [19, 20, 25].

In this paper, starting from a new eigenvalue problem, the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is worked out. It is proved that the 2-CQLTH has Lax pairs and bi-Hamiltonian structures, so it is completely integrable. Particularly, the two-component Casimir-Qiao-Liu type equation (2-CQLTE) is given. Then, by constructing the enlarged spectral problem, we obtain the the bi-integrable couplings of the 2-CQLTH. Similarly, we have proved the integrability of the bi-integrable couplings by constructing their Lax pairs and bi-Hamiltonian structures.

This paper is organized as follows. In Section 2, a new 2-CQLTH is given. In Section 3, the bi-Hamiltonian structures of the 2-CQLTH are constructed. In Section 4, we obtain the bi-integrable couplings of the 2-CQLTH. In Section 5, we generate the bi-Hamiltonian structures of the integrable couplings. The conclusion is given in Section 6.

#### 2. The Two-Component Casimir-Qiao-Liu Type Hierarchy

Consider a spatial matrix isospectral problem:where , are potential functions and is the spectral parameter.

Assuming that has the form then the stationary zero curvature equationgives rise toFurther, let possess the Laurent expansions: where the initial values are as follows: In this case, system (15) becomesImposing the following conditions on constants of integration: the sequence of can be uniquely determined and the first two sets are as follows: Now we introducewhere

By consideringthe compatibility of (12) and (23) gives the zero curvature equation:From the zero curvature equation (24), we can obtain the 2-CQLTH:where

*Remark 1. *The Lax pair for the 2-CQLTH (see (25)) is given by (12) and (23). This implies that the 2-CQLTH (see (25)) is integrable in Lax sense.

Particularly, when we take , we obtain

*Remark 2. *Taking , the second equation of (27) can be reduced to the CQLE (2). Thus, (27) is called 2-CQLTE.

#### 3. Bi-Hamiltonian Structures and Liouville Integrability

By direct computation, we can get so we have Now, the trace identity [11]givesComparing coefficients of all powers of in the equality (31), we haveChecking a particular case with in (32), we have .

Thus, we obtainwhere the Hamiltonian functionals are defined by It now follows that the soliton hierarchy (25) has the following Hamiltonian structures:

From the recursion relations (18), we can get where It is easy to verify that . Actually, we know that the hierarchy (25) is bi-Hamiltonian:where the second Hamiltonian operator isand and constitute the Hamiltonian pairs. Thus, the soliton hierarchy (25) is Liouville integrable.

*Remark 3. *Based on [26, 27], a matrix isospectral problem is constructed: with the spectral matrix as By a standard procedure similar to Sections 2 and 3, another 2-QLTH can be worked out:with the Hamiltonian operators and the Hamiltonian functionals Particularly, when we take , a new 2-CQLTE is presented asEquation (45) can be transformed into (27) in complex field, but they are different equations in real field.

#### 4. Bi-Integrable Couplings of the 2-CQLTH

Supposing that the triangular block matrices have the following form [28, 29]: where is a constant, and defining , , and , we know that satisfy (3), (4), and (5).

Considering the enlarged spectral matrix where and are dependent variables, if we substitute into the corresponding enlarged stationary zero curvature equation, we can getWhen are assumed to be and the initial values are taken as we can write (49) in the following forms:Furthermore, by imposing the sequence of has the unique expressions

In order to get the integrable couplings systems from the enlarged zero curvature equationswe define the Lax matrices and modification terms aswhere and denotes the polynomial part of .

In this condition, the integrable couplings system can be obtained aswhich is equal towhere

*Remark 4. * and are the Lax pairs of the integrable couplings of the 2-CQLTH; this implies that the integrable couplings are integrable in Lax sense.

#### 5. Bi-Hamiltonian Structures and Liouville Integrability of the Bi-Integrable Couplings

Because the associated matrix Lie algebras are nonsemisimple, we should use the variational identity (11) to furnish the Hamiltonian structures of the integrable couplings system (58) (or (59)). Before doing this, we should find out the nondegenerate, symmetric, and adinvariant bilinear form corresponding to the nonsemisimple matrix loop algebras first [22].

By defining a mapping from to aswhere we have the following results according to [22, 28, 30].

The mapping (61) is a Lie algebra isomorphism from to and the Lie bracket can be calculated as where

Assuming that the bilinear form on possesses the following form: where is a constant matrix, the symmetric property and adinvariance property of give rise toSo the form of the constant matrix can be written aswhere , , and are arbitrary constants. Thus, the bilinear form on the semidirect sums of and can be defined as where , .

In order to guarantee that the bilinear form is nondegenerate, we should require that is to say, and .

By calculation, we have Further, we have Substituting the above expressions into the variational identity (11), we can obtainComparing the coefficients of all powers of in (73), we know thatwhere Setting in (74), we know that the constant , so we haveThus, the Hamiltonian functions can be expressed as and the Hamiltonian structures of (58) as

Furthermore, by using (52), we can getwhere By a complicated computation, we know that namely, the hierarchy (58) possess bi-Hamiltonian structureswhere and are the Hamiltonian pairs of the integrable couplings of the 2-CQLTH. So the integrable couplings (58) are Liouville integrable.

#### 6. Conclusion

In this paper, we construct a new eigenvalue problem and give the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH). The 2-CQLTH has Lax representation and possesses bi-Hamiltonian structures, which implies that it is integrable in both Lax and Liouville sense. Furthermore, we obtain the integrable couplings of the integrable hierarchy as well as its Lax pairs and bi-Hamiltonian structures and prove that the integrable couplings are also integrable in both Lax and Liouville senses.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

The author Yuqin Yao is grateful to Professor Wen-Xiu Ma for his stimulating discussions. This work is supported in part by the National Natural Science Foundation of China (Grants no. 11301179) and China Scholarship Council.

#### References

- P. J. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,”
*Physical Review E. Statistical, Nonlinear, and Soft Matter Physics*, vol. 53, no. 2, pp. 1900–1906, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - Z. J. Qiao and L. P. Liu, “A new integrable equation with no smooth solitons,”
*Chaos, Solitons & Fractals*, vol. 41, no. 2, pp. 587–593, 2009. View at Publisher · View at Google Scholar · View at Scopus - Z. J. Qiao, “New integrable hierarchy, its parametric solutions, cuspons, one-peak solitons, and M/W-shape peak solitons,”
*Journal of Mathematical Physics*, vol. 48, no. 8, Article ID 082701, 20 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Li and Z. Qiao, “Bifurcations of traveling wave solutions for an integrable equation,”
*Journal of Mathematical Physics*, vol. 51, article 042703, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - V. Marinakis, “Higher-order equations of the KdV type are integrable,”
*Advances in Mathematical Physics*, vol. 2010, Article ID 329586, 5 pages, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - S. Sakovich, “Smooth soliton solutions of a new integrable equation by Qiao,”
*Journal of Mathematical Physics*, vol. 52, article 023509, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. X. Ma, “Integrable couplings of vector AKNS soliton equations,”
*Journal of Mathematical Physics*, vol. 46, no. 3, pp. 033507–033525, 2005. View at Publisher · View at Google Scholar - T.-C. Xia, F.-J. Yu, and D.-Y. Chen, “The multi-component Yang hierarchy and its multi-component integrable coupling system with two arbitrary functions,”
*Chaos, Solitons and Fractals*, vol. 24, no. 1, pp. 235–240, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - L. Luo and E. Fan, “The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems,”
*Science in China. Series A. Mathematics*, vol. 52, no. 1, pp. 147–159, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Zhang and E. Fan, “Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti-Pempinelli-Tu hierarchy,”
*Journal of Mathematical Physics*, vol. 51, article 083506, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. Z. Tu, “The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems,”
*Journal of Mathematical Physics*, vol. 30, no. 2, pp. 330–338, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - G. Z. Tu, “A trace identity and its applications to the theory of discrete integrable systems,”
*Journal of Physics A: Mathematical and General*, vol. 23, no. 17, pp. 3903–3922, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,”
*Journal of Physics A: General Physics*, vol. 39, no. 34, pp. 10787–10801, 2006. View at Google Scholar - W.-X. Ma, X.-X. Xu, and Y. F. Zhang, “Semi-direct sums of Lie algebras and continuous integrable couplings,”
*Physics Letters A*, vol. 351, no. 3, pp. 125–130, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W.-X. Ma, X.-X. Xu, and Y. F. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings,”
*Journal of Mathematical Physics*, vol. 47, no. 5, Article ID 053501, 5 pages, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W. X. Ma, “A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction,”
*Chinese Annals of Mathematics, Series A*, vol. 13, no. 1, pp. 115–123, 1992 (Chinese). View at Google Scholar · View at MathSciNet - Y. B. Zeng and X. Cao, “Separation of variables for higher-order binary constrained flows of the Tu hierarchy,”
*Advance in Mathematics*, vol. 2, no. 2, pp. 135–147, 2002. View at Google Scholar - X. X. Xu, “A generalized Wadati-Konno-Ichikawa hierarchy and new finite-dimensional integrable systems,”
*Physics Letters A*, vol. 301, no. 3-4, pp. 250–262, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W.-X. Ma, “Variational identities and applications to Hamiltonian structures of soliton equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 12, pp. e1716–e1726, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - F. Guo and Y. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems,”
*Journal of Physics. A. Mathematical and General*, vol. 38, no. 40, pp. 8537–8548, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - W.-X. Ma, “Integrable couplings of vector AKNS soliton equations,”
*Journal of Mathematical Physics*, vol. 46, no. 3, pp. 033507–033525, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. M. Yu, Y. Q. Yao, S. F. Shen, and W.-X. Ma, “Bi-integrable couplings of a Kaup-Newell type soliton hierarchy and their bi-Hamiltonian structures,”
*Communications in Nonlinear Science & Numerical Simulation*, vol. 23, no. 1–3, pp. 366–377, 2015. View at Publisher · View at Google Scholar · View at Scopus - Y. Zhang, H. Dong, and Z. Li, “Integrable couplings of the TB hierarchy and its Hamiltonian structure,”
*Annals of Differential Equations*, vol. 24, no. 1, pp. 112–116, 2008. View at Google Scholar · View at MathSciNet - W. X. Ma, J. H. Meng, and H. Q. Zhang, “Integrable couplings, variationmal identities and Hamiltonian formulations,” in