Semidirect sums of Lie algebras and discrete integrable couplings

A relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.

[1]  P. Sorba,et al.  Dictionary on lie algebras and superalgebras , 2000 .

[2]  Wenxiu Ma,et al.  Algebraic Structure of Discrete Zero Curvature Equations and Master Symmetries of Discrete Evolution Equations , 1998, solv-int/9809009.

[3]  Yufeng Zhang,et al.  A simple method for generating integrable hierarchies with multi-potential functions , 2005 .

[4]  O. Bogoyavlensky.  On perturbations of the periodic Toda lattice , 1976 .

[5]  MA W.X.,et al.  Integrable Theory of the Perturbation Equations , 2004 .

[6]  Wen-Xiu Ma.  Integrable couplings of vector AKNS soliton equations , 2005 .

[7]  M. Wadati,et al.  Integrable semi-discretization of the coupled nonlinear Schr\ , 1999, solv-int/9903013.

[8]  Mark J. Ablowitz,et al.  Nonlinear differential−difference equations , 1975 .

[9]  Yufeng Zhang.  A generalized multi-component Glachette–Johnson (GJ) hierarchy and its integrable coupling system , 2004 .

[10]  Wen-Xiu Ma,et al.  Complexiton solutions of the Toda lattice equation , 2004 .

[11]  Xi-Xiang Xu,et al.  A modified Toda spectral problem and its hierarchy of bi-Hamiltonian lattice equations , 2004 .

[12]  On Integrability of a (2+1)-Dimensional Perturbed KdV Equation , 1998, solv-int/9805012.

[13]  Wen-Xiu Ma,et al.  THE BI-HAMILTONIAN STRUCTURE OF THE PERTURBATION EQUATIONS OF THE KDV HIERARCHY , 1996 .

[14]  Wen-Xiu Ma,et al.  Enlarging spectral problems to construct integrable couplings of soliton equations , 2003 .

[15]  A. Perelomov,et al.  Classical integrable finite-dimensional systems related to Lie algebras , 1981 .

[16]  T Gui-zhang,et al.  A trace identity and its applications to the theory of discrete integrable systems , 1990 .

[17]  Yufeng Zhang,et al.  A new algebraic system and its applications , 2005 .

[18]  Xi-Xiang Xu,et al.  New hierarchies of integrable positive and negative lattice models and Darboux transformation , 2005 .

[19]  Yufeng Zhang,et al.  A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling , 2003 .

[20]  W. Ma.  A bi-Hamiltonian formulation for triangular systems by perturbations , 2001, nlin/0112009.

[21]  Xi-Xiang Xu,et al.  Positive and Negative Hierarchies of Integrable Lattice Models Associated with a Hamiltonian Pair , 2004 .

[22]  Wenxiu Ma,et al.  Rational solutions of the Toda lattice equation in Casoratian form , 2004 .

[23]  H. Flaschka.  The Toda lattice. II. Existence of integrals , 1974 .

[24]  Dengyuan Chen,et al.  The conservation laws of some discrete soliton systems , 2002 .

[25]  S. Manakov.  Complete integrability and stochastization of discrete dynamical systems , 1974 .

[26]  Wen-Xiu Ma,et al.  Binary nonlinearization of spectral problems of the perturbation AKNS systems , 2002 .

[27]  M. Wadati,et al.  Integrable semi-discretization of the coupled modified KdV equations , 1998 .

[28]  Sergei Yu. Sakovich.  Coupled KdV Equations of Hirota-Satsuma Type , 1999 .

[29]  Wen-Xiu Ma,et al.  Semi-direct sums of Lie algebras and continuous integrable couplings , 2006, nlin/0603064.

[30]  Morikazu Toda,et al.  Theory Of Nonlinear Lattices , 1981 .

[31]  W. Ma.  Integrable couplings of soliton equations by perturbations I: A general theory and application to the KDV hierarchy , 1999, solv-int/9912004.

[32]  Ruguang Zhou,et al.  Nonlinearization of spectral problems for the perturbation KdV systems , 2001 .