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Research Article | Open Access

Volume 2013 |Article ID 595946 | https://doi.org/10.1155/2013/595946

Chao Yue, Tiecheng Xia, Guijuan Liu, Jianbo Liu, "The Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy", Journal of Applied Mathematics, vol. 2013, Article ID 595946, 7 pages, 2013. https://doi.org/10.1155/2013/595946

# The Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy

Revised04 Aug 2013
Accepted26 Aug 2013
Published05 Oct 2013

#### Abstract

A fractional quadratic-form identity is derived from a general isospectral problem of fractional order, which is devoted to constructing the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy. The method can be generalized to other fractional integrable couplings.

#### 1. Introduction

The theory of derivatives of noninteger order can go back to Leibniz, Liouville, Grunwald, Letnikov, and Riemann. And the fractional analysis has attracted increasing interest of many researchers, because fractional analysis has numerous applications: kinetic theories , such as statistical mechanics , dynamics in complex media [7, 8], and many others . In recent studies in physics, the researchers have found many applications of the derivatives and integrals of fractional order [16, 17]. They also pointed out that fractional-order models are more appropriate than integer-order models for various real materials. The main advantage of fractional derivative in comparison with classical integer-order models is that it provides an effective instrument for the description of memory and hereditary properties of various materials and progress. Also, the advantages of the fractional derivatives become apparent in modeling mechanical and electrical properties of real materials and in the description of rheological properties of rocks, as well as in many other fields.

The fractional calculus is a generalization of ordinary differentiation and integration to arbitrary order . Since Riewe [4, 21] presented a concept of nonconservation mechanics, fractional conservation laws , Lie symmetries , and fractional Hamiltonian systems  have been receiving more and more attention.

It is an important and interesting topic to search for new Hamiltonian hierarchies of soliton equations and their integrable couplings in soliton theory. Tu once proposed a simple and efficient method to construct the integrable systems and Hamiltonian structures , which was called the Tu scheme by Ma . Later, many integrable systems and their Hamiltonian structures were worked out . Recently, Wu and Zhang proposed the generalized Tu formula and searched for the Hamiltonian structure of fractional AKNS hierarchy . In , a generalized Hamiltonian structure of the fractional soliton equation hierarchy was presented. Very recently, Wang and Xia obtained the fractional supersoliton hierarchies and their super-Hamiltonian structures by using fractional supertrace identity [42, 43]. Then, how to generate integrable coupling system and Hamiltonian structure of fractional soliton equation?

In this paper, begining with a general isospectral problem of fractional order, we propose a fractional quadratic-form identity, from which the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy is derived.

#### 2. Brief Overview of Fractional Differentiable Functions

Several local versions have been presented , among which Jumarie’s derivative is defined as follows : some properties of the fractional differentiable functions are given as follows.

(a) The Leibniz product law.

Assuming that is an order differentiable function in the area of point , from the Jumarie-Kolwankar’s Taylor series , we can have

If is a differentiable function of order, the Leibniz product law can hold for the nondifferentiable functions [39, 44, 45]

(b) Denoting as the Riemann-Liouville integration in the following form: we can have a generalized Newton-Leibniz formulation

(c) With the properties (a) and (b), integration by parts for order differentiable functions and can be generated as

(d) From [31, 32, 55], the fractional variational derivative is written as where is a positive integer. In this paper, we propose a generalized quadratic-form identity for fractional soliton hierarchy from (7).

#### 3. Fractional Exterior Differential and Hamiltonian Equations

Since Adda proposed the fractional generalization of differential forms [56, 57], several versions of fractional exterior differential approaches and applications related to different forms of fractional derivatives appeared in some parts of the open literature [58, 59]. The properties of fractional derivatives are discussed in .

The exterior derivative is defined as The exterior derivative map forms into forms and has the following algebraic results. Let and be forms, and let be an form; we have The last identity is called the Poincaré lemma. A form is called closed if . A form is called exact if there exists a form such that . The order of is one less than the order of . Exact forms are always closed, closed forms are not always exact.

Next, we introduce the fractional exterior derivative A differential 1-form is defined by with the vector field that can be represented as and is a continuously differentiable function. Using (10), the exact fractional form can be expressed as Note that (11) is a fractional generalization of the differential form (8). It is easy to find that fractional 1-form can be closed when the differential 1-form is not closed.

Then, we define the fractional functional hence, we can readily derive the generalized Poincare-Cartan 1-form, which reads

From (14), one has In the previous derivation, and are fractional differentiable functions with respect to .

The fractional closed condition admits the fractional Hamilton’s equations  which can be generalized to the following case :

#### 4. The Fractional Quadratic-Form Identity

Guo and Zhang once proposed quadratic-form identity , which is very efficient tool to systematically generate integrable couplings and their Hamiltonian structures. In the following, the fractional quadratic-form identity is presented.

Set to be an -dimensional Lie algebra with the basis whose corresponding loop algebra possesses the following basis: In terms of , we construct the following isospectral problem: The compatibility condition of (20) gives rise to the generalized zero curvature equation: Taking (21) reduces to the classical zero curvature equation. For and in , defining , then , can be presented. If the ranks of are taken as , , then each term in has the homogeneous rank which is denoted by

Set , , as a solution of the stationary zero curvature equation and rank is assumed to be given so that rank, ; each team in has the same rank as follows: Let the two arbitrary solutions and of (23) with the same rank be linearly related by In the following, relation (25) will be used when deducing the fractional quadratic-form identity. For , the -order matrix is determined by and constant matrix is determined by Defining functional satisfies the symmetry and the bilinear relation In the sense of the local fractional derivative, the gradient of the functional is defined by where is variational derivative with respect to . With the fractional variational derivative (7), one can have where is a positive integer and . The communication relationship of can be given as

Introduce a functional where , meet (23), while is to be determined; using (7), we can obtain the following fractional variation constraint conditions: according to the Jacobi identity and the previous equations, we can have and are solutions of (23); using (25) and , due to satisfying (34), we can have . From (23) and (33), a fractional quadratic-form identity is firstly presented as follows:

#### 5. Application of the Fractional Quadratic-Form Identity

Introduce a loop algebra , with the commuting relations Consider the following spectral problem: Solving equation leads to

Set then the generalized zero curvature equation, , gives rise to a system where is a Hamiltonian operator. From (40), we have a recurrence operatorwhich meets . Hence, expression (42) can be written as From expression (37), we have Solving the matrix equation (27) for leads to Let we have Substituting the previous results into the fractional quadratic-form identity (36) gives Comparing the coefficients of on both sides of (49) yields It is easy to find that ; then we obtain the fractional Hamiltonian structure of (42) where is the fractional Hamiltonian function. When taking , we have an integrable coupling of a fractional BK hierarchy

Reduction Cases

Case 1. When , , ; (52) reduces to the BK hierarchy

Case 2. Let , , (53) is transformed to the classical Boussinesq equation

#### 6. Conclusion

A way to construct the Hamiltonian structure of integrable coupling of fractional soliton equation hierarchy is presented. As an application, the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy is obtained by use of the fractional quadratic-form identity. The method can be generalized to other fractional integrable couplings.

#### Acknowledgments

The Project is in part supported by the Natural Science Foundation of China under Grants nos. 11271008, 61072147, and 11071159, by a Grant of “The First-class Discipline of Universities in Shanghai” and the Shanghai Universty Leading Academic Discipline Project (A.13-0101-12-004), the Natural Science Foundation of Shandong Province (Grant Nos. ZR2012AM021, and ZR2012AQ011), and Scientific Research Reward Fund for Excellent Middle-Aged and Young Scientists in Shandong Province (Grants no. BS2011DX038).

1. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports, vol. 371, no. 6, pp. 461–580, 2002.
2. G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK, 2008, Reprint of the 2005 original. View at: MathSciNet
3. V. E. Tarasov and G. M. Zaslavsky, “Fractional ginzburg-landau equation for fractal media,” Physica A, vol. 354, pp. 249–261, 2005. View at: Publisher Site | Google Scholar
4. F. Riewe, “Mechanics with fractional derivatives,” Physical Review E, vol. 55, no. 3, pp. 3581–3592, 1997. View at: Publisher Site | Google Scholar | MathSciNet
5. V. E. Tarasov, “Fractional systems and fractional Bogoliubov hierarchy equations,” Physical Review E, vol. 71, no. 1, p. 12, 2005. View at: Publisher Site | Google Scholar | MathSciNet
6. V. E. Tarasov, “Fractional fokker-planck equation for fractal media,” Chaos, vol. 15, no. 2, p. 8, 2005.
7. V. E. Tarasov, “Fractional Liouville and BBGKI equations,” Journal of Physics, vol. 7, p. 17, 2005. View at: Google Scholar
8. R. R. Nigmatullin, “Realization of the generalized transfer equation in a medium with fractal geometry,” Physica Status Solidi B, vol. 133, no. 1, pp. 425–430, 1986. View at: Google Scholar
9. G. C. Wu, “A fractional lie group method for anomalous diffusion equations,” Communications in Fractional Calculus, vol. 1, pp. 27–31, 2010. View at: Google Scholar
10. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singarpore, 2000.
11. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A, vol. 37, no. 31, pp. R161–R208, 2004.
12. J. C. Liu and G. L. Hou, “New approximate solution for time-fractional coupled KdV equations by generalised differential transform method,” Chinese Physics B, vol. 19, no. 11, Article ID 110203, 2010. View at: Publisher Site | Google Scholar
13. S. B. Zhou, X. R. Lin, and H. Li, “Chaotic synchronization of a fractional-order system based on washout filter control,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1533–1540, 2011. View at: Publisher Site | Google Scholar
14. D. L. Qi, Q. Wang, and J. Yang, “Comparison between two different sliding mode controllers for a fractional-order unified chaotic system,” Chinese Physics B, vol. 20, no. 10, Article ID 100505, 2011. View at: Publisher Site | Google Scholar
15. H. X. Ge, Y. Q. Liu, and R. J. Cheng, “Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations,” Chinese Physics B, vol. 21, no. 1, Article ID 010206, 2012. View at: Publisher Site | Google Scholar
16. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. View at: MathSciNet
17. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, New York, NY, USA, 1993. View at: MathSciNet
18. D. Baleanu, Z. B. Guvenc, and J. A. Machado, Tenreiro, Springer, 2009.
19. I. Podlubny, Fractional Differential Equations, Academic, New York, NY, USA, 1999.
20. G. M. Zaslavsky, Ed., Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, New York, NY, USA, 2005.
21. F. Riewe, “Nonconservative Lagrangian and Hamiltonian mechanics,” Physical Review E, vol. 53, no. 2, pp. 1890–1899, 1996. View at: Publisher Site | Google Scholar | MathSciNet
22. G. S. F. Frederico and D. F. M. Torres, “Fractional conservation laws in optimal control theory,” Nonlinear Dynamics, vol. 53, no. 3, pp. 215–222, 2008. View at: Publisher Site | Google Scholar
23. D. Baleanu and S. I. Muslih, “Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives,” Physica Scripta, vol. 72, no. 2-3, pp. 119–121, 2005.
24. D. Baleanu and Om. P. Agrawal, “Fractional Hamilton formalism within Caputo's derivative,” Czechoslovak Journal of Physics, vol. 56, no. 10-11, pp. 1087–1092, 2006. View at: Publisher Site | Google Scholar | MathSciNet
25. R. A. El-Nabulsi, “Modifications at large distances from fractional and fractal arguments,” Fractals, vol. 18, no. 2, pp. 185–190, 2010. View at: Google Scholar
26. R. A. El-Nabulsi, “A fractional action-like variational approach of some classical, quantum and geometrical dynamics,” International Journal of Applied Mathematics, vol. 17, no. 3, pp. 299–317, 2005.
27. R. A. El-Nabulsi and D. F. M. Torres, “Fractional actionlike variational problems,” Journal of Mathematical Physics, vol. 49, no. 5, p. 053521, 2008.
28. O. P. Agrawal, “Fractional variational calculus and the transversality conditions,” Journal of Physics A, vol. 39, no. 33, pp. 10375–10384, 2006.
29. O. P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives,” Journal of Physics A, vol. 40, no. 24, pp. 6287–6303, 2007.
30. V. E. Tarasov, “Fractional variations for dynamical systems: hamilton and Lagrange approaches,” Journal of Physics A, vol. 39, no. 26, pp. 8409–8425, 2006.
31. G. Jumarie, “Lagrangian mechanics of fractional order, Hamilton-Jacobi fractional PDE and Taylor's series of nondifferentiable functions,” Chaos, Solitons and Fractals, vol. 32, no. 3, pp. 969–987, 2007.
32. R. Almeida, A. B. Malinowska, and D. F. M. Torres, “A fractional calculus of variations for multiple integrals with application to vibrating string,” Journal of Mathematical Physics, vol. 51, no. 3, p. 12, 2010. View at: Publisher Site | Google Scholar | MathSciNet
33. A. B. Malinowska, M. R. S. Ammi, and D. F. M. Torres, “Composition functional in fractional calculus of variations,” Communications in Fractional Calculus, vol. 1, p. 32, 2010. View at: Google Scholar
34. 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.
35. W. X. Ma, “A new family of Liouville integrable generalized Hamilton equations and its reduction,” Chinese Annals of Mathematics A, vol. 13, no. 1, pp. 115–123, 1992. View at: Google Scholar | MathSciNet
36. Y. Zhang and H. Zhang, “A direct method for integrable couplings of TD hierarchy,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 466–472, 2002.
37. T. Xia, F. You, and D. Chen, “A generalized AKNS hierarchy and its bi-Hamiltonian structures,” Chaos, Solitons & Fractals, vol. 23, no. 5, pp. 1911–1919, 2005.
38. W. X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,” Journal of Physics A, vol. 39, no. 34, Article ID 10787, 2006. View at: Publisher Site | Google Scholar
39. W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings,” Journal of Mathematical Physics, vol. 47, no. 5, p. 16, 2006.
40. G.-c. Wu and S. Zhang, “A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy,” Physics Letters A, vol. 375, no. 42, pp. 3659–3663, 2011.
41. F. Yu, “A generalized fractional KN equation hierarchy and its fractional Hamiltonian structure,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1522–1530, 2011.
42. H. Wang and T. C. Xia, “The fractional supertrace identity and its application to the super Ablowitz-Kaup-Newell-Segur hierarchy,” Journal of Mathematical Physics, vol. 54, no. 5, Article ID 043505, 2013. View at: Publisher Site | Google Scholar
43. H. Wang and T.-C. Xia, “The fractional supertrace identity and its application to the super Jaulent-Miodek hierarchy,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 10, pp. 2859–2867, 2013. View at: Publisher Site | Google Scholar | MathSciNet
44. K. M. Kolwankar and A. D. Gangal, “Fractional differentiability of nowhere differentiable functions and dimensions,” Chaos, vol. 6, no. 4, pp. 505–513, 1996.
45. K. M. Kolwankar and A. D. Gangal, “Hölder exponents of irregular signals and local fractional derivatives,” Pramana, vol. 48, no. 1, pp. 49–68, 1997. View at: Publisher Site | Google Scholar
46. K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2, pp. 214–217, 1998.
47. W. Chen, “Time-space fabric underlying anomalous diffusion,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 923–929, 2006. View at: Publisher Site | Google Scholar
48. W. Chen and H. G. Sun, “Multiscale statistical model of fully-developed turbulence particle accelerations,” Modern Physics Letters B, vol. 23, no. 3, p. 449, 2009. View at: Publisher Site | Google Scholar
49. J. Cresson, “Scale calculus and the Schrödinger equation,” Journal of Mathematical Physics, vol. 44, no. 11, pp. 4907–4938, 2003.
50. J. Cresson, “Non-differentiable variational principles,” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 48–64, 2005.
51. A. Parvate and A. D. Gangal, “Calculus on fractal subsets of real line—I. Formulation,” Fractals, vol. 17, no. 1, pp. 53–81, 2009.
52. G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
53. G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009. View at: Google Scholar
54. Y. Chen, Y. Yan, and K. Zhang, “On the local fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 362, no. 1, pp. 17–33, 2010.
55. X. J. Yang, Research on fractal mathematics and some applications in mechanics [M.S. thesis], China University of Mining and Technology, 2009, Chinese.
56. F. Ben Adda, “Geometric interpretation of the fractional derivative,” Journal of Fractional Calculus, vol. 11, pp. 21–51, 1997.
57. F. Ben Adda and J. Cresson, “About non-differentiable functions,” Journal of Mathematical Analysis and Applications, vol. 263, no. 2, pp. 721–737, 2001.
58. K. Cottrill-Shepherd and M. Naber, “Fractional differential forms,” Journal of Mathematical Physics, vol. 42, no. 5, pp. 2203–2212, 2001.
59. K. K. Kazbekov, “Fractional differential forms in euclidean space,” Vladikavkazskiĭ Matematicheskiĭ Zhurnal, vol. 7, no. 2, pp. 41–54, 2005. View at: Google Scholar | MathSciNet
60. C. Li and W. Deng, “Remarks on fractional derivatives,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 777–784, 2007.
61. F. Guo and Y. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems,” Journal of Physics A, vol. 38, no. 40, pp. 8537–8548, 2005.

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