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The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present an integrable coupling of lattice hierarchy and its continuous limits by using Lie algebra . By introducing a complex discrete spectral problem, the integrable coupling of Kac–Van Moerbeke lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable coupling of discrete soliton equation hierarchy, which has the integrable coupling system of MKdV hierarchy as a new kind of continuous limit. [Copyright &y& Elsevier]

Abstract: We present a new method to construct two (2+1)-dimensional lattice hierarchies. By considering a kind of spectral problems with two shift operators and , a new (2+1)-dimensional discrete soliton hierarchy and a new lattice hierarchy are constructed by using matrix spectral problem. [Copyright &y& Elsevier]

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak-Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation. [ABSTRACT FROM AUTHOR]

Abstract: A kind of non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with [Y.F. Zhang, E.G. Fan, Characteristic Numbers of Matrix Lie Algebras, Commun. Theor. Phys (China) 49 (2008) 845] is presented. As an application example, the hierarchy of non-isospectral cubic Volterra lattice hierarchy with self-consistent sources is derived. Furthermore, we construct a non-isospectral integrable couplings of cubic Volterra lattice hierarchy with self-consistent sources by using the loop algebra . [Copyright &y& Elsevier]

Abstract: In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra . By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit. [Copyright &y& Elsevier]

Abstract: In [E.G. Fan, Phys. Lett. A 372 (2008) 6368], Fan present a lattice hierarchy and its continuous limits. In this Letter, we extend this method, by introducing a complex discrete spectral problem, a coupling lattice hierarchy is derived. It is shown that a new sequence of combinations of complex lattice spectral problem converges to the integrable coupling couplings of soliton equation hierarchy, which has the integrable coupling system of AKNS hierarchy as a continuous limit. [Copyright &y& Elsevier]

Abstract: In [W.X. Ma, J. Phys. A: Math. Theor. 40 (2007) 15055], Prof. Ma gave a beautiful result (a discrete variational identity). In this Letter, based on a discrete block matrix spectral problem, a new hierarchy of Lax integrable lattice equations with four potentials is derived. By using of the discrete variational identity, we obtain Hamiltonian structure of the discrete soliton equation hierarchy. Finally, an integrable coupling system of the soliton equation hierarchy and its Hamiltonian structure are obtained through the discrete variational identity. [Copyright &y& Elsevier]

Abstract: In this Letter by considering a -dimensional discrete non-isospectral linear problem, a new -dimensional integrable lattice hierarchy is constructed. It shows that generalization of the Blaszak–Marciniak lattice hierarchy can be obtained as a reduction. Then an extended algebraic system of X is presented, from which the integrable coupling system of the -dimensional discrete non-isospectral Blaszak–Marciniak lattice equations are obtained. [Copyright &y& Elsevier]