Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T17:09:57.416Z Has data issue: false hasContentIssue false
  • English
  • Français

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: mawx@cas.usf.edu
Corresponding author. Email Address: zhanghuiqun@qdu.edu.cn
Corresponding author. Email Address: jmeng@mail.usf.edu
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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