A vector generalization of Volterra type differential-difference equations

Abstract A discrete ( N + 1 ) × ( N + 1 ) matrix spectral problem and the corresponding hierarchy of Volterra type differential–difference equations are proposed. It is also shown that the hierarchy of differential–difference equations possesses the Hamiltonian structures. Infinite conservation laws for the first nontrivial member in the hierarchy are given.

[1]  H. Dai,et al.  Algebro-geometric constructions of semi-discrete Chen–Lee–Liu equations , 2010 .

[2]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[3]  Zuo-nong Zhu,et al.  Nonisospectral negative Volterra flows and mixed Volterra flows: Lax pairs, infinitely many conservation laws and integrable time discretization , 2004 .

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

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

[6]  Kimiaki Konno,et al.  Conservation Laws of Nonlinear-Evolution Equations , 1974 .

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

[8]  O. Ragnisco,et al.  A NOVEL HIERARCHY OF INTEGRABLE LATTICES , 1994 .

[9]  Abdul-Majid Wazwaz,et al.  A modified KdV-type equation that admits a variety of travelling wave solutions: kinks, solitons, peakons and cuspons , 2012 .

[10]  M. Wadati,et al.  The Coupled Modified Korteweg-de Vries Equations , 1998, solv-int/9812003.

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

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

[13]  Xianguo Geng,et al.  A Hierarchy of New Nonlinear Differential-Difference Equations(General) , 2006 .

[14]  M. Wadati,et al.  Transformation Theories for Nonlinear Discrete Systems , 1976 .

[15]  S. Novikov,et al.  Theory of Solitons: The Inverse Scattering Method , 1984 .

[16]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[17]  Xianguo Geng,et al.  Algebro-geometric solutions of the coupled modified Korteweg–de Vries hierarchy , 2014 .

[18]  A Hamiltonian structure associated with a matrix spectral problem of arbitrary-order , 2007 .

[19]  Xianguo Geng,et al.  Algebro-geometric solution of the 2+1 dimensional Burgers equation with a discrete variable , 2002 .

[20]  Lihua Wu,et al.  A new super-extension of the KdV hierarchy , 2010, Appl. Math. Lett..

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

[22]  Abdul-Majid Wazwaz,et al.  A KdV6 hierarchy: Integrable members with distinct dispersion relations , 2015, Appl. Math. Lett..

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

[24]  Xiaoyong Wen.  A New Integrable Lattice Hierarchy Associated with a Discrete 3 × 3 Matrix Spectral Problem: N-Fold Darboux Transformation And Explicit Solutions , 2013 .

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