Starting from a more generalized discrete 2×2 matrix spectral problem and using the Tu scheme, some integrable lattice hierarchies (ILHs) are presented which include the well-known relativistic Toda lattice hierarchy and some new three-field ILHs. Taking one of the hierarchies as example, the corresponding Hamiltonian structure is constructed and the Liouville integrability is illustrated. For the first nontrivial lattice equation in the hierarchy, the N-fold Darboux transformation (DT) of the system is established basing on its Lax pair. By using the obtained DT, we generate the discrete N-soliton solutions in determinant form and plot their figures with proper parameters, from which we get some interesting soliton structures such as kink and anti-bell-shaped two-soliton, kink and anti-kink-shaped two-soliton and so on. These soliton solutions are much stable during the propagation, the solitary waves pass through without change of shapes, amplitudes, wave-lengths and directions. Finally, we derive infinitely many conservation laws of the system and give the corresponding conserved density and associated flux formulaically.

中文翻译：

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与相对论 Toda 晶格相关的广义可积层次：哈密顿结构、Darboux 变换、孤子解和守恒定律

从更广义的离散开始2×2矩阵谱问题，并使用 Tu 方案，提出了一些可积格层级（ILHs），包括著名的相对论 Toda 格子层级和一些新的三场 ILHs。以其中一个层次结构为例，构造了相应的哈密顿量结构，并说明了刘维尔可积性。对于层次结构中的第一个非平凡晶格方程，ñ系统的-fold Darboux变换(DT)是基于其Lax对建立的。通过使用获得的DT，我们生成离散的ñ-孤子的行列式解，并用适当的参数绘制它们的图形，从中我们得到一些有趣的孤子结构，例如扭结和反钟形双孤子，扭结和反扭结形双孤子等。这些孤子解在传播过程中非常稳定，孤波通过时不会改变形状、幅度、波长和方向。最后，我们推导出系统的无穷多个守恒定律，并公式化地给出相应的守恒密度和相关通量。