N -fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation
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N-fold Darboux transformations and exact solutions of the combined Toda lattice and relativistic Toda lattice equation Fang-Cheng Fan1 · Zhi-Guo Xu2
· Shao-Yun Shi2,3
Received: 30 April 2020 / Revised: 16 May 2020 / Accepted: 11 June 2020 © Springer Nature Switzerland AG 2020
Abstract 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. Keywords Integrable lattice equation · N -fold Darboux transformation · Exact solutions Mathematics Subject Classification 35Q51 · 35Q53 · 37K40
1 Introduction Dating back to the work of Fermi, Pasta and Ulam in the 1950s [1], nonlinear lattice equations have been the focus of many nonlinear studies. Particularly, a great number of physically interesting phenomena can be modeled with nonlinear lattice equations,
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Zhi-Guo Xu [email protected]
1
School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, People’s Republic of China
2
School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
3
School of Mathematics and State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun 130012, People’s Republic of China 0123456789().: V,-vol
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such as particle vibrations in lattices, currents in electrical networks, pulses in biological chains [2–4]. One of the most famous and well studied lattice equations is the Toda lattice (TL) equation, which was first introduced by Toda [5]. The TL equation has an integrable generalization of relativistic version called the relativistic Toda lattice (RTL), which was first introduced by Ruijsenaars [6]. It has been shown that both the TL equation and RTL equation possess integrable properties such as Lax pair [7,8], bi-Hamiltonian structure [9], conservation laws [10,11], Darboux transformation (DT) [12–14] and so on. In 2004, Ma and Xu [15] introduced a discrete integrable hierarchy, and the typical equation in such hierarchy is
rn,t = rn (sn−1 − sn ) + αrn (rn−1 − rn+1 ), sn,t = αsn (rn − rn+1 ) + β(rn+1 − rn ),
(1)
where rn = r (n, t), sn,t = s(n, t) are functions of a discrete variable n ∈ Z and the dsn n time variable t ∈ R, rn,t = dr dt , sn,t = dt , α and β
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