In this paper, the N-fold Darboux transformation (DT) of the combined Toda lattice and relativistic Toda lattice equation is constructed in terms of determinants. Comparing with the usual 1-fold DT of equations, this kind of N-fold DT enables us to generate the multi-soliton solutions without complicated recursive process. As applications of the N-fold DT, we derive two kinds of N-fold explicit exact solutions from two different seed solutions and plot the figures with properly parameters to illustrate the propagation of solitary waves. What’s more, we present the relationships between the structures of exact solutions parameters with \(N=1\), from which we find the 1-fold solutions may be one soliton solutions or periodic solutions and the waves pass through without change of shapes, amplitudes, wavelengths and directions, etc. The results in this paper might be helpful for interpreting certain physical phenomena.

中文翻译：

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Toda格子与相对论Toda格子方程的N倍Darboux变换和精确解

本文采用行列式构造了Toda格和相对论Toda格方程的N倍Darboux变换（DT）。与通常的方程的1倍DT相比，这种N倍DT使我们能够生成多孤子解，而无需复杂的递归过程。作为N倍DT的应用，我们从两种不同的种子解中得出两种N倍显式精确解，并用适当的参数绘制图形以说明孤立波的传播。此外，我们用\（N = 1 \）表示精确解参数结构之间的关系，从中我们可以发现1阶解可能是一个孤子解或周期解，并且波通过时形状，振幅，波长和方向等都没有变化。本文的结果可能对解释某些物理现象有帮助。