Two integrable lattice hierarchies and their respective Darboux transformations

Two integrable lattice hierarchies associated with two discrete matrix spectral problems are derived, and the infinitely many conservation laws of the first integrable model is obtained. Moreover, the Darboux transformations based on different Darboux matrixes (2.3.4) and (3.2.4) for the above-mentioned integrable hierarchies are established with the help of two different gauge transformations of lax pairs in this paper. As an application, the explicit solutions of the two integrable hierarchies are presented.

[1]  K. M. Tamizhmani,et al.  Master Symmetries from Lax Operators for Certain Lattice Soliton Hierarchies. , 2000 .

[2]  R. Willox,et al.  Darboux transformations for the two-dimensional Toda system , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  Abdul-Majid Wazwaz,et al.  Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota's method, tanh-coth method and Exp-function method , 2008, Appl. Math. Comput..

[4]  Zhijun Qiao,et al.  Darboux transformation of the special (2+1)-dimensional Toda lattice and its explicit solution , 2010 .

[5]  V. M. Babich,et al.  A binary Darboux transformation for the Toda lattice , 1986 .

[6]  Hong-Xiang Yang,et al.  Soliton solutions by Darboux transformation for a Hamiltonian lattice system , 2009 .

[7]  Geng Xian-Guo,et al.  Darboux Transformation for Tzitzeica Equation , 2006 .

[8]  Xi-Xiang Xu,et al.  Darboux transformation of a coupled lattice soliton equation , 2007 .

[9]  Yunbo Zeng,et al.  A new constrained mKP hierarchy and the generalized Darboux transformation for the mKP equation with self-consistent sources , 2004, nlin/0412066.

[10]  Brian Wesley Williams,et al.  New Potentials for Old: The Darboux Transformation in Quantum Mechanics , 2008 .

[11]  Wen-Xiu Ma,et al.  COUPLING INTEGRABLE COUPLINGS , 2009 .

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

[13]  S. Y. Lou,et al.  New interaction property of (2 + 1)-dimensional localized excitations from Darboux transformation , 2005 .

[14]  Wenxiu Ma,et al.  Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras , 2006 .

[15]  Yi Zhang,et al.  Hirota bilinear equations with linear subspaces of solutions , 2012, Appl. Math. Comput..

[16]  Wenxiu Ma.  Darboux Transformations for a Lax Integrable System in 2n Dimensions , 1996, solv-int/9605002.

[17]  Qiu-Lan Zhao,et al.  Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation , 2009, J. Comput. Appl. Math..

[18]  Geng Xian-Guo,et al.  Darboux Transformation of a Differential–Difference Equation and Its Explicit Solutions , 2006 .

[19]  Abdelhalim Ebaid,et al.  Exact solutions for the generalized Klein-Gordon equation via a transformation and Exp-function method and comparison with Adomian's method , 2009 .

[20]  Zhijun Qiao,et al.  Darboux transformation and explicit solutions for two integrable equations , 2011 .

[21]  J. Nimmo,et al.  The Relation between a 2D Lotka-Volterra equation and a 2D Toda Lattice , 2005 .

[22]  Gu Chaohao,et al.  On the Darboux matrices of Bäcklund transformations for AKNS systems , 1987 .

[23]  Guang Zhang,et al.  Lax Pairs for Discrete Integrable Equations via Darboux Transformations , 2012 .

[24]  X. Geng,et al.  A new hierarchy of integrable differential-difference equations and Darboux transformation , 1998 .

[25]  Wen-Xiu Ma,et al.  A discrete variational identity on semi-direct sums of Lie algebras , 2007, 0711.1147.

[26]  Wen-Xiu Ma,et al.  Variational identities and applications to Hamiltonian structures of soliton equations , 2009 .

[27]  Yufeng Zhang,et al.  A direct method for integrable couplings of TD hierarchy , 2002 .

[28]  Qiu-Lan Zhao,et al.  THE INTEGRABLE PROPERTY OF LOTKA–VOLTERRA TYPE DISCRETE NONLINEAR LATTICE SOLITON SYSTEMS , 2008 .

[29]  Ke Chen,et al.  Applied Mathematics and Computation , 2022 .

[30]  Hon-Wah Tam,et al.  Application of Hirota's bilinear formalism to a two-dimensional lattice by Leznov , 2000 .

[31]  Xianguo Geng,et al.  Some new integrable nonlinear evolution equations and Darboux transformation , 2010 .

[32]  Wenxiu Ma,et al.  A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation , 2009, 0903.5337.

[33]  Jingsong He,et al.  A supertrace identity and its applications to superintegrable systems , 2008 .