N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation

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.