Under investigation in this paper is a discrete modified exponential Toda lattice equation which may describe vibration of particles in a lattice. Firstly, we construct an integrable lattice hierarchy associated with this equation from a \(2\times 2\) matrix spectral problem, and some related integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws of this hierarchy are discussed. Secondly, we present the discrete generalized \((m, N-m)\)-fold Darboux transformation of the modified exponential Toda lattice equation on the basis of its known Lax representation. As applications of the obtained discrete generalized Darboux transformation, multi-kink-soliton solutions on an exponential surface background and an inclined plane background are obtained when \(m=2N\), the discrete rational and semi-rational solutions are derived when \(m=1\), and the mixed solutions of usual soliton solutions and rational solutions are given when \(m=2\). Based on the asymptotic and graphic analysis, soliton elastic interaction phenomena and limit states related to rational solutions are discussed and analyzed. Furthermore, some mathematical features of rational solutions are summarized. Finally, numerical simulations are used to explore the dynamical behaviors of such soliton solutions which show the soliton evolutions are robust against a small noise. These results and properties given in this paper might be useful for understanding nonlinear lattice dynamics.

本文研究的是一个离散的修正指数 Toda 晶格方程，它可以描述晶格中粒子的振动。首先，我们从一个\(2\times 2\)矩阵谱问题构造了一个与该方程相关的可积格子层次，并讨论了一些相关的可积性质，如哈密顿结构、刘维尔可积性和该层次的守恒定律。其次，我们提出离散广义\((m, Nm)\)基于已知 Lax 表示的修正指数 Toda 格子方程的 -fold Darboux 变换。作为所得到的离散广义达布变换的应用，当\(m=2N\)时得到指数面背景和斜面背景上的多扭结孤子解，当\( m=1\)，当\(m=2\)时给出通常孤子解和有理解的混合解. 在渐近和图形分析的基础上，讨论和分析了与有理解相关的孤子弹性相互作用现象和极限状态。此外，还总结了有理解的一些数学特征。最后，数值模拟用于探索这种孤子解的动力学行为，这表明孤子演化对小噪声具有鲁棒性。本文中给出的这些结果和性质可能有助于理解非线性晶格动力学。