# zbMATH — the first resource for mathematics

A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order. (English) Zbl 1209.37070
Summary: Starting from a specific matrix iso-spectral problem, an associated hierarchy of multi-component Hamiltonian equations is constructed, based on zero curvature equations. The key point is to choose appropriate time parts of Lax pairs which can yield evolution equations, and the existence of a Hamiltonian structure for the obtained hierarchy is established by means of the trace identity. An example with five components is computed, along with its Hamiltonian structure.

##### MSC:
 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text:
##### References:
 [1] Ma, W.X., (), 450 [2] Degasperis, A., Am. J. phys., 66, 486, (1988) [3] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, London mathematical society lecture note series, vol. 149, (1991), Cambridge Univ. Press Cambridge · Zbl 0762.35001 [4] Conte, R.; Magri, F.; Musette, M.; Satsuma, J.; Winternitz, P., () [5] Boiti, M.; Pempinelli, F.; Pogrebkov, A.K.; Prinari, B., J. math. phys., 44, 3309, (2003) [6] Ma, W.X.; Fuchssteiner, B., Chaos solitons fractals, 7, 1227, (1996) [7] Ma, W.X., Methods appl. anal., 7, 21, (2000) [8] Guo, F.K.; Zhang, Y., Chaos solitons fractals, 22, 1063, (2004) [9] Ma, W.X., J. math. phys., 46, 033507, (2005) [10] Ma, W.X., J. math. phys., 43, 1408, (2002) [11] Ma, W.X.; Xu, X.X.; Zhang, Y.; Ma, W.X.; Xu, X.X.; Zhang, Y., Phys. lett. A, J. math. phys., 47, 053501, (2006) [12] W.X. Ma, Chaos Solitons Fractals (2006), in press [13] Lax, P.D., Commun. pure appl. math., 21, 467, (1968) [14] Tu, G.Z., J. phys. A: math. gen., 22, 2375, (1989) [15] Gelfand, I.M.; Dorfman, I.Ja., Funckt. anal. prilozhen., 13, 13, (1979) [16] Olver, P.J., Applications of Lie groups to differential equations, Graduate texts in mathematics, vol. 107, (1986), Springer-Verlag New York · Zbl 0656.58039 [17] Tu, G.Z., Sci. sinica ser. A, 29, 138, (1986) [18] Tu, G.Z., J. math. phys., 30, 330, (1989) [19] Guo, F.K.; Zhang, Y., J. phys. A: math. gen., 38, 8537, (2005) [20] Ma, W.X.; Chen, M., J. phys. A: math. gen., 39, 10787, (2006) [21] Fuchssteiner, B.; Fokas, A.S., Physica D, 4, 47, (1981/82)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.