Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T16:24:26.006Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Semisimple Lie Algebras of Block Matrices and Applications to Bi-Integrable Couplings

Published online by Cambridge University Press:  03 June 2015

Jinghan Meng  and
Wen-Xiu Ma
Jinghan Meng*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
Wen-Xiu Ma*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA
*
Corresponding author. Email: [email protected]
Email: [email protected]
Get access

Abstract

We propose a class of non-semisimple matrix loop algebras consisting of 3 × 3 block matrices, and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings. Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.

Keywords

37K0537K1035Q53Bi-integrable couplingsnon-semisimple matrix loop algebrasAKNS hierarchyHamiltonian structuresymmetry
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 5 , Issue 5 , October 2013 , pp. 652 - 670
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]Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H., The inverse scattering transform–fourier analysis for nonlinear problems, Stud. Appl. Math., 53 (1974), pp. 249–315.CrossRefGoogle Scholar
[2]Drinfel’d, V.G. and Sokolov, V.V., Equations of Korteweg-de Vries type and simple Lie algebras, Soviet Math. Dokl., 23 (1981), pp. 457–462.Google Scholar
[3]Fuchssteiner, B., Application of hereditary symmetries to nonlinear evolution equations, Nonlinear Anal. TMA, 3 (1979), pp. 849–862.Google Scholar
[4]Fuchssteiner, B. and Fokas, A. S., Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D, 4 (1981), pp. 47–66.Google Scholar
[5]Guo, F. G. and Zhang, Y. F., A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling, J. Math. Phys., 44 (2003), pp. 5793–5803.Google Scholar
[6]Lax, P., Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure Appl. Math., 21 (1968), pp. 467–490.Google Scholar
[7]Li, Z., Zhang, Y. J. and Dong, H. H., Integrable couplings of the TC hierarchy and its Hamiltonian structure, Modern Phys. Lett. B, 21 (2007), pp. 595–602.Google Scholar
[8]Ma, W. X., Integrable couplings of soliton equations by perturbations I. a general theory and application to the KdV hierarchy, Methods Appl. Anal., 7 (2000), pp. 21–55.Google Scholar
[9]Ma, W. X., A bi-Hamiltonian formulation for triangular systems by perturbations, J. Math. Phys., 43 (2002), pp. 1408–1421.CrossRefGoogle Scholar
[10]Ma, W. X., Enlarging spectral problems to construct integrable couplings of soliton equations, Phys. Lett. A, 316 (2003), pp. 72–76.Google Scholar
[11]Ma, W. X., Integrable couplings of vector AKNS soliton equations, J. Math. Phys., 46 (2005), 033507.Google Scholar
[12]Ma, W. X., A discrete variational identity on semi-direct sums of Lie algebras, J. Phys. A Math. Theoret., 40 (2007), pp. 15055–15069.CrossRefGoogle Scholar
[13]Ma, W. X., Variational identities and applications to Hamiltonian structures of soliton equations, Nonlinear Anal., 71 (2009), pp. e1716–e1726.Google Scholar
[14]Ma, W. X. and Fuchssteiner, B., Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations, J. Math. Phys., 40 (1999), pp. 2400–2418.Google Scholar
[15]Ma, W. X. and Fuchssteiner, B., The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A, 213 (1996), pp. 49–55.Google Scholar
[16]Ma, W. X. and Fuchssteiner, B., Integrable theory of the perturbation equations, Chaos Solitons Fractals, 7 (1996), pp. 1227–1250.CrossRefGoogle Scholar
[17]Ma, W. X. and Gao, L., Coupling integrable couplings, Modern Phys. Lett. B, 23 (2009), pp. 1847–1860.CrossRefGoogle Scholar
[18]Ma, W. X. and Guo, F. K., Lax representations and zero-curvature representations by the Kronecker product, Int. J. Theoret. Phys., 36 (1997), pp. 697–704.Google Scholar
[19]Ma, W. X., Xu, X. X. and Zhang, Y. F., Semi-direct sums of Lie algebras and continuous integrable couplings, Phys. Lett. A, 351 (2006), pp. 125–130.Google Scholar
[20]Ma, W. X., Xu, X. X. and Zhang, Y. F., Semi-direct sums of Lie algebras and discrete integrable couplings, J. Math. Phys., 47 (2006), 053501.Google Scholar
[21]Ma, W. X., Loop algebras and bi-integrable couplings, Chin. Ann. Math. Ser. B, 33 (2012), pp. 207–224.Google Scholar
[22]Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19 (1978), pp. 1156–1162.Google Scholar
[23]Olver, P. J., Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18 (1977), pp. 1212–1215.Google Scholar
[24]Tu, G. Z., The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems, J. Math. Phys., 30 (1989), pp. 330–338.Google Scholar
[25]Tu, G. Z. and Meng, D. Z., The trace identity, a powerful tool for constructing the hamiltonian structure of integrable systems (II), Acta Math. Appl. Sinica (English Series), 5 (1989), pp. 89–96.Google Scholar
[26]Xia, T. C., Chen, X. H. and Chen, D. Y., A new Lax integrable hierarchy, N Hamiltonian structure and its integrable couplings system, Chaos Solitons Fractals, 23 (2005), pp. 451–458.Google Scholar
[27]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, Phys. A, 343 (2004), pp. 238–246.Google Scholar
[28]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. Theor., 42 (2009), 395201.Google Scholar
[29]Yu, F. J. and Zhang, H. Q., Hamiltonian structure of the integrable couplings for the multicom-ponent Dirac hierarchy, Appl. Math. Comput., 197 (2008), pp. 828–835.Google Scholar
[30]Zhang, Y. F., A generalized multi-component Glachette-Johnson (GJ) hierarchy and its integrable coupling system, Chaos Solitons Fractals, 21 (2004), pp. 305–310.Google Scholar
[31]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 (2011), pp. 3045–3061.Google Scholar
[32]Zhang, Y. F., On integrable couplings of the dispersive long wave hierarchy and their Hamiltonian structure, Modern Phys. Lett. B, 21 (2007), pp. 37–44.Google Scholar