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A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations

Published online by Cambridge University Press:  28 May 2015

Wen-Xiu Ma ,
Huiqun Zhang  and
Jinghan Meng
Wen-Xiu Ma*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
Huiqun Zhang*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA College of Mathematical Science, Qingdao University, Qingdao, Shandong 266071, P.R. China
Jinghan Meng*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
*
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
Corresponding author. Email Address: [email protected]
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Abstract

A non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

Keywords

37K0537K1035Q53Integrable couplingmatrix loop algebraHamiltonian structure
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 3 , August 2013 , pp. 171 - 189
Copyright
Copyright © Global-Science Press 2013

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References

[1]Ma, W.X., Xu, X.X. and Zhang, Y.F., Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. Lett. A 351, 125–130 (2006).Google Scholar
[2]Ma, W.X., Xu, X.X. and Zhang, Y.F., Semidirect sums of Lie algebras and discrete integrable couplings, J. Math. Phys. 47, 053501, 16 pp. (2006).Google Scholar
[3]Ma, W.X. and Chen, M., Hamiltonian and quasi-Hamiltonian structures associated with semi- direct sums of Lie algebras, J. Phys. A: Math. Gen. 39, 10787–10801 (2006).CrossRefGoogle Scholar
[4]Ma, W.X., A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A: Math. Theoret. 40, 15055–15069 (2007).Google Scholar
[5]Xia, T.C., Chen, H.X. and Chen, D.Y., A new Lax integrable hierarchy, N Hamiltonian structure and its integrable couplings system, Chaos, Solitons & Fractals 23, 451–458 (2005).Google Scholar
[6]Yu, F.Y. and Zhang, H.Q., Hamiltonian structure of the integrable couplings for the multicomponent Dirac hierarchy, Appl. Math. Comput. 197, 828–835 (2008).Google Scholar
[7]Zhang, Y.F. and Feng, B.L., A few Lie algebras and their applications for generating integrable hierarchies of evolution types, Commun. Nonlinear Sci. Numer. Simul. 16, 3045–3061 (2011).CrossRefGoogle Scholar
[8]Ma, W.X., Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal. 71, e1716–e1726 (2009).CrossRefGoogle Scholar
[9]Ma, W.X., Variational identities and Hamiltonian structures, in: Ma, W.X., Hu, X.B. and Liu, Q.P. (Eds.), Nonlinear and Modern Mathematical Physics, Vol. 1212, pp.1–27, AIP Conference Proceedings, American Institute of Physics, Melville, NY (2010).Google Scholar
[10]Ma, W.X. and Zhang, Y., Component-trace identities for Hamiltonian structures, Appl. Anal. 89, 457–472 (2010).Google Scholar
[11]Blaszak, M., Szablikowski, B.M. and Silindir, B., Construction and separability of nonlinear soliton integrable couplings, Appl. Math. Comput. 219, 1866–1873 (2012).Google Scholar
[12]Ma, W.X., Meng, J.H. and Zhang, H.Q., Tri-integrable couplings by matrix Lie algebras, to appear in Int. J. Nonlinear Sci. Numer. Simul. 14, (2013).CrossRefGoogle Scholar
[13]Ma, W.X. and Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos, Solitons & Fractals 7, 1227–1250 (1996).Google Scholar
[14]Ma, W.X., Integrable couplings of soliton equations by perturbations I - A general theory and application to the KdV hierarchy, Methods Appl. Anal. 7, 21–55 (2000).CrossRefGoogle Scholar
[15]Sakovich, S.Yu., On integrability of a (2 + 1)-dimensional perturbed KdV equation, J. Nonlinear Math. Phys. 5, 230–233 (1998).Google Scholar
[16]Sakovich, S.Yu., Coupled KdV equations of Hirota-Satsuma type, J. Nonlinear Math. Phys. 6, 255–262 (1999).Google Scholar
[17]Lax, P., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21, 467–490 (1968).Google Scholar
[18]Tu, G.Z., On Liouville integrability of zero-curvature equations and the Yang hierarchy, J. Phys. A: Math. Gen. 22, 2375–2392 (1989).Google Scholar
[19]Ma, W.X. and Zhu, Z.N., Constructing nonlinear discrete integrable Hamiltonian couplings, Comput. Math. Appl. 60, 2601–2608 (2010).Google Scholar
[20]Ma, W.X., Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput. 217, 7238–7244 (2011).Google Scholar
[21]Kupershmidt, B.A., Dark equations, J. Nonlinear Math. Phys. 8, 363–445 (2011).CrossRefGoogle Scholar
[22]Peebles, P.J.E. and Ratra, B., The cosmological constant and dark energy, Rev. Mod. Phys. 75, 559–606 (2003).CrossRefGoogle Scholar
[23]Hogan, J., Unseen universe: welcome to the dark side, Nature 448, 240–245 (2007).Google Scholar
[24]Erdmann, K. and Wildon, M.J., Introduction to Lie Algebras, Springer, London (2006).Google Scholar
[25]Ma, W.X., Loop algebras and bi-integrable couplings, in: Ma, W.X. (Ed.) Special Issue on Solitons, Chin. Ann. Math. B 33, 207–224 (2012).Google Scholar
[26]Ma, W.X., A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction, Chin. Ann. Math. A 13, 115–123 (1992).Google Scholar
[27]Ma, W.X., Binary nonlinearization for the Dirac systems, Chin. Ann. Math. B 18, 79–88 (1997).Google Scholar
[28]Olver, P.J., Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18, 1212–1215 (1977).Google Scholar
[29]Fuchssteiner, B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. 3, 849–862 (1979).Google Scholar
[30]Fuchssteiner, B. and Fokas, A.S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4, 47–66 (1981/1982).Google Scholar
[31]Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys. 19, 1156–1162 (1978).Google Scholar
[32]Olver, P.J., Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107, Springer-Verlag, New York (1986).Google Scholar
[33]Ma, W.X. and Fuchssteiner, B., The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A 213, 49–55 (1996).CrossRefGoogle Scholar
[34]Ma, W.X., A bi-Hamiltonian formulation for triangular systems by perturbations, J. Math. Phys. 43, 1408–1421 (2002).Google Scholar
[35]Ma, W.X., Integrable couplings of vector AKNS soliton equations, J. Math. Phys. 46, 033507, 19 pp. (2005).CrossRefGoogle Scholar
[36]Ma, W.X., Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. Lett. A 316, 72–76 (2003).CrossRefGoogle Scholar
[37]Ma, W.X. and Gao, L., Coupling integrable couplings Modern Phys. Lett. B 23, 1847–1860 (2009).Google Scholar
[38]Ma, W.X. and Guo, F.K., Lax representations and zero curvature representations by Kronecker product, Int. J. Theor. Phys. 36, 697–704 (1997).CrossRefGoogle Scholar
[39]Yu, F.J. and Li, L., A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product, Phys. Lett. A 372, 3548–3554 (2008).CrossRefGoogle Scholar
[40]Ma, W.X. and Zhou, R.G., Nonlinearization of spectral problems for the perturbation KdV systems, Physica A 296, 60–74 (2001).CrossRefGoogle Scholar
[41]Ma, W.X. and Zhou, R.G., Binary nonlinearization of spectral problems of the perturbation AKNS systems, Chaos, Solitons & Fractals 13, 1451–1463 (2002).Google Scholar
[42]Olver, P.J., Classical Invariant Theory, London Mathematical Society Student Texts 44, Cambridge University Press (1999).Google Scholar
[43]Ma, W.X., Gu, X. and Gao, L., A note on exact solutions to linear differential equations by the matrix exponential, Adv. Appl. Math. Mech. 1, 573–580 (2009).Google Scholar
[44]Ma, W.X., Integrability, in: Scott, A. (Ed.) Encyclopedia of Nonlinear Science, pp. 250–253, Taylor & Francis, New York (2005).Google Scholar
[45]Ma, W.X., Strampp, W., Bilinear forms and Bäcklund transformations of the perturbation systems, Phys. Lett. A 341, 441–449 (2005).Google Scholar
[46]Ma, W.X. and Fan, E.G., Linear superposition principle applying to Hirota bilinear equations, Comput. Math. Appl. 61, 950–959 (2011).CrossRefGoogle Scholar
[47]Ma, W.X., Zhang, Y., Tang, Y.N. and Tu, J.Y., Hirota bilinear equations with linear subspaces of solutions, Appl. Math. Comput. 218, 7174–7183 (2012).Google Scholar
[48]Ma, W.X., A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order, Phys. Lett. A 367, 473–477 (2007).Google Scholar
[49]Ma, W.X., Multi-component bi-Hamiltonian Dirac integrable equations, Chaos, Solitons & Fractals 39, 282–287 (2009).Google Scholar
[50]Xia, T.C., Yu, F.J. and Zhang, Y.F., The multi-component coupled Burgers hierarchy of soliton equations and its multi-component integrable couplings system with two arbitrary functions, Physica A 343, 238–246 (2004).Google Scholar
[51]Li, Z. and Dong, H.H., Two integrable couplings of the Tu hierarchy and their Hamiltonian structures, Comput. Math. Appl. 55, 2643–2652 (2008).Google Scholar
[52]Zhang, Y.F. and Tam, H.W., Coupling commutator pairs and integrable systems, Chaos, Solitons & Fractals 39, 1109–1120 (2009).Google Scholar
[53]Xu, X.X., Integrable couplings of relativistic Toda lattice systems in polynomial form and rational form, their hierarchies and bi-Hamiltonian structures, J. Phys. A: Math. Theoret. 42, 395201, 21 pp. (2009).Google Scholar
[54]Feng, B.L. and Liu, J.Q., A new Lie algebra along with its induced Lie algebra and associated with applications, Commun. Nonlinear Sci. Numer. Simul. 16, 1734–1741 (2011).Google Scholar
[55]Hirota, R. and Ito, M., Resonance of solitons in one dimension, J. Phys. Soc. Jpn. 52, 744–748 (1983).Google Scholar