1 Introduction

Studies of the lattice systems have been seen of current interest due to their applications in optics, condensed matter physics and plasma physics [1,2,3,4]. The theory of relativity has not been separated from the rest of physics, for every physical theory is supposed to conform to the basic relativistic principles, and any concrete physical problem involves a synthesis of relativity and some specific physical theory [5, 6]. The lattice systems and relativistic lattice systems have served as a useful guide in the recent studies on nonlinear wavesFootnote 1 [7,8,9,10].

The Toda lattice system has been introduced to study the vibrations in a chain with nonlinear interactions [21] and waves in an anharmonic lattice [22]. Numerical investigations have indicated that the Toda lattice system behaves remarkably like an integrable nonlinear system, where the integrability means that the system’s Hamiltonian can be brought to an integrable form [23]. Subsequently, the complete integrability of the Toda lattice system has been confirmed [24, 25]. With a change of the variables, the Toda lattice system has been written in the Lax form [25,26,27]. As for the solutions of the Toda lattice system, inverse-scattering solutions have been derived via the inverse-scattering method [26], multi-soliton solutions have been constructed through the Bäcklund transformation method [28], positon solutions have been generated via the Darboux transformation (DT) [29], soliton solutions in the Wronskian form has been obtained [30] and complexiton solutions have been determined through the Casoratian formulation [31].

It has been said that certain relativistic particle systems are not only completely integrable at the classical level, but can also be quantized in such a fashion that the integrability survives [32, 33]. The relativistic Toda lattice system has been derived via the requirement of Poincaré invariance for the Toda lattice system [33]. Lax representation, complete integrability and scattering problem of the relativistic Toda lattice system have been studied [33]. A hereditary recursion operator and an alternative proof of the complete integrability of the relativistic Toda lattice system have been provided via the Lax representation [34]. Three different Lax representations and complete integrability of the relativistic Toda lattice system have been investigated [35]. The direct and inverse spectral problems of the relativistic Toda lattice system have been solved [36]. Conserved quantities, bi-Hamiltonian formulation and recursive structure of the relativistic Toda lattice system have been obtained through the master symmetries [37]. Relation between the discrete time Toda lattice system and relativistic Toda lattice system has been found in the context of the Hamiltonian dynamics [38]. Bilinear forms of the relativistic Toda lattice system have been carried out via the auxiliary dependent variables, based on which the multi-soliton solutions have been constructed in the form of the Casorati determinant [39]. Other relativistic lattice systems such as the discrete-time relativistic Toda lattice system, relativistic Volterra lattice system and relativistic Lotka-Volterra lattice system have also been seen [40, 41]. Bilinear forms and multi-soliton solutions in the form of the Casorati determinant of the discrete-time relativistic Toda lattice system and relativistic Lotka-Volterra lattice system have been derived via the Hirota method and Casoratian technique [42, 43]. N-fold DT and some solutions of the relativistic Volterra lattice system have been obtained through a gauge transformation and Lax pair, where N is a positive integer [44].

Based on the work of Ref. [45], Ref. [46] has presented the following hybrid relativistic and modified Toda lattice-type system:

$$\begin{aligned}&p_{n,t}=\alpha \left( \Lambda -1 \right) \textrm{e}^{q_n-q_{n-1}} + \beta ^2 \left( \Lambda -1 \right) \textrm{e}^{q_n-q_{n-1} + p_{n-1}}, \end{aligned}$$
(1a)
$$\begin{aligned}&q_{n,t}=\textrm{e}^{p_n} \left( 1 + \beta ^2 \textrm{e}^{q_{n+1}-q_{n}} \right) , \end{aligned}$$
(1b)

where the integer n stands for a discrete spatial variable, \(q_n\) and \(p_n\) mean two differentiable functions of the continuous temporal variable t, \(q_{n,t}=\frac{\textrm{d} q_n }{\textrm{d} t} \) and \(p_{n,t}=\frac{\textrm{d} p_n }{\textrm{d} t} \), \(\Lambda \) is a shift operator defined via \(\Lambda p_n=p_{n+1}\), while \(\alpha \) and \(\beta \) denote two real constants which are respectively modified-Toda-lattice related and relativistic-Toda-lattice related. Dynamic issues with respect to System (1) have been seen in Refs. [45, 46]. Lax pair, one-fold DT and certain solutions of System (1) have been obtained [46].

However, as far as we know, equivalent form of System (1) has not been reported. In addition, N-fold DT of that equivalent form and some analytic solutions which are different from those in Ref. [46] have not been discussed. In Sect. 2, an equivalent form of System (1) will be presented via certain transformations. In Sect. 3, N-fold Darboux matrix and N-fold DT of that equivalent form will be determined via the equivalent-form Lax pair. In Sect. 4, some analytic solutions of that equivalent form will be obtained based on our N-fold DT. In Sect. 5, we shall give the conclusions.

2 Equivalent form of System (1)

For the convenience of deriving the N-fold DT and some analytic solutions, we take the transformations

$$\begin{aligned} r_n=\textrm{e}^{p_n}, \qquad s_n=\textrm{e}^{q_n}, \end{aligned}$$
(2)

and then System (1) is equivalent to the following system:

$$\begin{aligned}&r_{n,t}= r_n \left[ \alpha \left( \Lambda -1 \right) \frac{s_n}{s_{n-1}} + \beta ^2 \left( \Lambda -1 \right) \frac{r_{n-1} s_n}{s_{n-1}} \right] , \end{aligned}$$
(3a)
$$\begin{aligned}&s_{n,t}=r_n \left( s_n + \beta ^2 s_{n+1} \right) , \end{aligned}$$
(3b)

whose Lax pair is expressed as

$$\begin{aligned}&\Lambda \Psi _n=U_n \Psi _n,\qquad U_n= \begin{pmatrix} r_n \lambda - \lambda ^{-1} &{}s_n \\ -\frac{\beta ^2 r_n + \alpha }{s_n} &{} \alpha \lambda \end{pmatrix}, \end{aligned}$$
(4a)
$$\begin{aligned}&\Psi _{n,t} = V_n \Psi _n, \qquad V_n= \begin{pmatrix} \frac{s_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} &{} -s_n \lambda ^{-1} \\ \frac{ \beta ^2 r_{n-1} + \alpha }{s_{n-1}} \lambda ^{-1} &{} - \lambda ^{-2} \end{pmatrix}, \end{aligned}$$
(4b)

which is consistent with the Lax pair in Ref. [46] under Transformations (2), where \(\Psi _n=(\Psi _{1,n}, \Psi _{2,n})^T\), \(\Psi _{1,n}\) and \(\Psi _{2,n}\) are two differentiable functions of t, \(\lambda \) is a spectral parameter and the superscript “T” refers to the transpose for a vector/matrix.

3 N-fold DT of System (3)

For the purpose of constructing an N-fold DT of System (3), we first introduce a gauge transformation [47, 48]

$$\begin{aligned} \widetilde{\Psi }_n = M_n \Psi _n, \end{aligned}$$
(5)

where \(M_n\) is a reversible matrix and \(\widetilde{\Psi }_n\) is necessary to satisfy

$$\begin{aligned}&\Lambda \widetilde{\Psi }_n = \widetilde{U}_n \widetilde{\Psi }_n, \qquad \widetilde{U}_n = M_{n+1} U_n M_n^{-1}, \end{aligned}$$
(6a)
$$\begin{aligned}&\widetilde{\Psi }_{n,t} = \widetilde{V}_n \widetilde{\Psi }_n, \qquad \widetilde{V}_n = \left( M_{n,t} + M_n V_n \right) M_n^{-1}, \end{aligned}$$
(6b)

while \(\widetilde{U}_n\) and \(\widetilde{V}_n\) have the same structures as \(U_n\) and \({V}_n\), respectively, with the exception that the previous potentials \(r_n\), \(s_n\) have been substituted by new ones \(\widetilde{r}_n\), \(\widetilde{s}_n\), and the superscript “\(-1\)” means the inverse of a matrix. We assume that N-fold Darboux matrix \(M_n\) takes the form of a polynomial matrix of \(\lambda \), as shown below:

$$\begin{aligned} M_n = \begin{pmatrix} A_n &{} B_n \\ C_n &{} D_n \end{pmatrix} = \begin{pmatrix} \lambda ^{-N} + \sum \limits _{j=1}\limits ^{N}{A_{n}^{(-N+2j)} \lambda ^{-N+2j}} &{} \sum \limits _{j=1}\limits ^{N}{B_{n}^{(N-2j+1)} \lambda ^{N-2j+1}} \\ \sum \limits _{j=1}\limits ^{N}{C_{n}^{(N-2j+1)} \lambda ^{N-2j+1}} &{} \lambda ^{N} + \sum \limits _{j=1}\limits ^{N}{D_{n}^{(N-2j)} \lambda ^{N-2j}} \end{pmatrix}, \end{aligned}$$
(7)

where \(A_n\), \(B_n\), \(C_n\) and \(D_n\) are some to-be-determined functions of n and t. Assuming that \(\lambda _{\imath }\)’s (\(\lambda _{\imath } \ne 0\), \(\imath = 1, 2, \ldots , 2N \)) are the 2N roots of \( \text {det} M_n\), we get

$$\begin{aligned} \text {det} M_n = A_n^{(N)}\lambda ^{-2N} \prod \limits _{\imath =1}^{2N} \left( \lambda ^2 - \lambda _{\imath }^2 \right) . \end{aligned}$$
(8)

Consequently, we can use the linear algebraic system shown below to find \(A_n^{(-N+2j)}\)’s, \(B_n^{(N-2j+1)}\)’s, \(C_n^{(N-2j+1)}\)’s and \(D_n^{(N-2j)}\)’s uniquely:

$$\begin{aligned} \sum \limits _{j=1}\limits ^{N}{A_{n}^{(-N+2j)} \lambda _{\imath }^{-N+2j}} + \delta _{\imath ,n} \sum \limits _{j=1}\limits ^{N}{B_{n}^{(N-2j+1)} \lambda _{\imath }^{N-2j+1}} = -\lambda _{\imath }^{-N}, \end{aligned}$$
(9a)
$$\begin{aligned} \sum \limits _{j=1}\limits ^{N}{C_{n}^{(N-2j+1)} \lambda _{\imath }^{N-2j+1}} + \delta _{\imath ,n} \sum \limits _{j=1}\limits ^{N}{D_{n}^{(N-2j)} \lambda _{\imath }^{N-2j}} = -\delta _{\imath ,n} \lambda _{\imath }^{N}, \end{aligned}$$
(9b)

where

$$\begin{aligned} \delta _{\imath ,n} = \frac{\phi _{2,n} \left( \lambda _{\imath } \right) + \theta _{\imath } \psi _{2,n} \left( \lambda _{\imath } \right) }{\phi _{1,n} \left( \lambda _{\imath } \right) + \theta _{\imath } \psi _{1,n} \left( \lambda _{\imath } \right) }, \end{aligned}$$
(10)

\(\phi _n \left( \lambda _{\imath } \right) = \left( \phi _{1,n} \left( \lambda _{\imath } \right) , \phi _{2,n} \left( \lambda _{\imath } \right) \right) ^T\) and \(\psi _n \left( \lambda _{\imath } \right) = \left( \psi _{1,n} \left( \lambda _{\imath } \right) , \psi _{2,n} \left( \lambda _{\imath } \right) \right) ^T\) are the solutions of Lax Pair (4), while the 4N parameters \(\lambda _{\imath }\)’s, \(\theta _{\imath }\)’s (\(\lambda _{\imath } \ne \lambda _{\jmath }, \imath \ne \jmath , \theta _{\imath } \ne 0 \)) are set in such a way that the determinant of the coefficients for Eq. (9) is non-zero.

Proposition 1

The matrix \(\widetilde{U}_n\) defined via Eq. (6a) has the same structure as that of \(U_n\), i.e.,

$$\begin{aligned} \widetilde{U}_n = \begin{pmatrix} \widetilde{r}_n \lambda - \lambda ^{-1} &{} \widetilde{s}_n \\ -\frac{ \beta ^2 \widetilde{r}_n + \alpha }{\widetilde{s}_n} &{} \alpha \lambda \end{pmatrix}, \end{aligned}$$

where the transformations from the old potentials \(r_n\), \(s_n\) into the new ones \(\widetilde{r}_n\), \(\widetilde{s}_n\) are given by

$$\begin{aligned}&\widetilde{r}_n = \frac{A_{n+1}^{(N)}}{A_n^{(N)}} r_n, \end{aligned}$$
(11a)
$$\begin{aligned}&\widetilde{s}_n = \frac{s_n + B_n^{(-N+1)}}{D_n^{(-N)}}. \end{aligned}$$
(11b)

Proof

Let \(M_n^{-1}=M_n^{*}/ \text {det} M_n\) and

$$\begin{aligned} M_{n+1} U_n M_n^{*} = \begin{pmatrix} f_{11,n} \left( \lambda \right) &{} f_{12,n} \left( \lambda \right) \\ f_{21,n} \left( \lambda \right) &{} f_{22,n} \left( \lambda \right) \end{pmatrix}, \end{aligned}$$

in which

$$\begin{aligned} \begin{aligned}&f_{11,n} \left( \lambda \right) =-A_{n+1}D_n \lambda ^{-1} + \left( r_n A_{n+1} D_n -\alpha B_{n+1} C_n \right) \lambda - \frac{\left( \beta ^2 r_n +\alpha \right) B_{n+1} D_n}{s_n} - s_n A_{n+1} C_n, \\&f_{12,n} \left( \lambda \right) = A_{n+1} B_n \lambda ^{-1} + \left( \alpha A_n B_{n+1} - r_n A_{n+1} B_n \right) \lambda + \frac{\left( \beta ^2 r_n +\alpha \right) B_n B_{n+1}}{s_n}+s_n A_n A_{n+1}, \\&f_{21,n} \left( \lambda \right) = -C_{n+1}D_n \lambda ^{-1} + \left( r_n C_{n+1}D_n -\alpha C_n D_{n+1} \right) \lambda -\frac{\left( \beta ^2 r_n +\alpha \right) D_n D_{n+1}}{s_n} -s_n C_n C_{n+1} , \\&f_{22,n} \left( \lambda \right) = B_n C_{n+1} \lambda ^{-1} + \left( \alpha A_n D_{n+1} -r_n B_n C_{n+1} \right) \lambda + \frac{\left( \beta ^2 r_n +\alpha \right) B_n D_{n+1}}{s_n} +s_n A_n C_{n+1} , \end{aligned} \end{aligned}$$
(12)

and the superscript “\(*\)” means the adjoint of a matrix. According to Eq. (4a), (9) and (10), we have

$$\begin{aligned} A_n \left( \lambda _{\imath } \right) = - \delta _{\imath ,n} B_n \left( \lambda _{\imath } \right) , \quad C_n \left( \lambda _{\imath } \right) = - \delta _{\imath ,n} D_n \left( \lambda _{\imath } \right) , \quad \delta _{\imath ,n+1} = \frac{-\left( \beta ^2 r_n +\alpha \right) + \alpha s_n \lambda _{\imath } \delta _{\imath , n} }{s_n\left( r_n \lambda _{\imath } -\lambda _{\imath }^{-1} + s_n \delta _{\imath , n} \right) }. \end{aligned}$$
(13)

After that, using Eqs. (12) and (13), we can confirm that \( \lambda _{\imath }\)’s are the roots of \(f_{11,n} \left( \lambda \right) \), \(f_{12,n} \left( \lambda \right) \), \(f_{21,n} \left( \lambda \right) \) and \(f_{22,n} \left( \lambda \right) \).

According to Eq. (12), we realize that \(f_{11,n} \left( \lambda \right) \) is the polynomial in \(\lambda \) with the highest order \(2N+1\) and lowest order \(-2N-1\), \(f_{12,n} \left( \lambda \right) \) and \(f_{21,n} \left( \lambda \right) \) are the polynomials in \(\lambda \) with the highest order 2N and lowest order \(-2N\), while \(f_{22,n} \left( \lambda \right) \) is the polynomial in \(\lambda \) with the highest order \(2N+1\) and lowest order \(-2N+1\). Making use of Eq. (8), it is possible to verify the existence of a matrix \(H_n\) such that

$$\begin{aligned} M_{n+1} U_n M_n^{*} = \text {det} M_n \cdot H_n, \end{aligned}$$
(14)

where

$$\begin{aligned} H_n = \begin{pmatrix} h_{11,n}^{(1)} \lambda + h_{11,n}^{(-1)} \lambda ^{-1} &{} h_{12,n}^{(0)} \\ h_{21,n}^{(0)} &{} h_{22,n}^{(1)} \lambda \end{pmatrix}, \end{aligned}$$

while \(h_{11,n}^{(1)}\), \(h_{11,n}^{(-1)}\), \(h_{12,n}^{(0)}\), \(h_{21,n}^{(0)}\) and \(h_{22,n}^{(1)}\) are some to-be-determined functions of n and t. Following that, rewriting Eq. (14) as

$$\begin{aligned} M_{n+1} U_n = H_n M_n, \end{aligned}$$
(15)

and equating the same powers of \(\lambda \) in Eq. (15), we are able to derive

$$\begin{aligned} \begin{aligned}&h_{11,n}^{(1)}=\frac{A_{n+1}^{(N)}}{A_n^{(N)}} r_n=\widetilde{r}_n, \qquad h_{11,n}^{(-1)}=-1, \qquad h_{12,n}^{(0)} = \frac{s_n + B_n^{(-N+1)}}{D_n^{(-N)}} = \widetilde{s}_n, \qquad h_{22,n}^{(1)}=\alpha , \end{aligned} \end{aligned}$$
(16)

and

$$\begin{aligned} \begin{aligned}&s_n C_{n+1}^{(-N+1)} + \alpha \left( D_{n+1}^{(-N)}-D_n^{(-N)} \right) - h_{21,n}^{(0)} B_n^{(-N+1)} =0,\\&h_{21,n}^{(0)} + C_{n+1}^{(-N+1)} + \frac{\beta ^2 r_n + \alpha }{s_n} D_{n+1}^{(-N)}=0. \end{aligned} \end{aligned}$$
(17)

Solving Eqs. (9) gives rise to

$$\begin{aligned} D_n^{(-N)}=\prod \limits _{\imath =1}^{2N} \lambda _{\imath }^2 A_n^{(N)}, \end{aligned}$$
(18)

and then Eqs. (17) and (18) bring about

$$\begin{aligned} h_{21,n}^{(0)}=-\frac{ \beta ^2 \widetilde{r}_n + \alpha }{\widetilde{s}_n}. \end{aligned}$$
(19)

From Eqs. (6a), (16) and (19), we obtain the conclusion of Proposition 1. The proof is completed. \(\hfill\square \)

Proposition 2

The matrix \(\widetilde{V}_n\) defined via Eq. (6b) has the same structure as that of \(V_n\), i.e.,

$$\begin{aligned} \widetilde{V}_n = \begin{pmatrix} \frac{\widetilde{s}_{n} \left( \beta ^2 \widetilde{r}_{n-1} + \alpha \right) }{\widetilde{s}_{n-1}} &{} -\widetilde{s}_{n} \lambda ^{-1} \\ \frac{ \beta ^2 \widetilde{r}_{n-1} + \alpha }{\widetilde{s}_{n-1}} \lambda ^{-1} &{} -\lambda ^{-2} \end{pmatrix}, \end{aligned}$$

under Transformations (11).

Proof

Let

$$\begin{aligned} \left( M_{n,t} + M_{n} V_n \right) M_n^{*} = \begin{pmatrix} g_{11,n} \left( \lambda \right) &{} g_{12,n} \left( \lambda \right) \\ g_{21,n} \left( \lambda \right) &{} g_{22,n} \left( \lambda \right) \end{pmatrix}, \end{aligned}$$

in which

$${\begin{aligned} g_{11,n} \left( \lambda \right) = &B_n C_n \lambda ^{-2} + \left[ s_n A_n C_n+ \frac{B_n D_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} \right] \lambda ^{-1} + \frac{ s_n A_n D_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} + A_{n,t} D_n - B_{n,t} C_n, \\ g_{12,n} \left( \lambda \right) =&- A_n B_n \lambda ^{-2} - \left[ s_n A_n^2 + \frac{B_n^2 \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} \right] \lambda ^{-1} - \frac{s_n A_n B_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} + A_n B_{n,t} - A_{n,t} B_n, \\ g_{21,n} \left( \lambda \right) =&C_n D_n \lambda ^{-2} + \left[ s_n C_n^2 + \frac{D_n^2 \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} \right] \lambda ^{-1} + \frac{ s_n C_n D_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} + C_{n,t}D_n - C_n D_{n,t},\\ g_{22,n} \left( \lambda \right) =&- A_n D_n \lambda ^{-2} - \left[ s_n A_n C_n + \frac{B_n D_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} \right] \lambda ^{-1} - \frac{ s_n B_n C_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} + A_n D_{n,t}-B_n C_{n,t}. \end{aligned}} $$
(20)

With the aid of Eqs. (4b), (9) and (10), we obtain that

$$\begin{aligned} \begin{aligned}&A_{n,t} \left( \lambda _{\imath } \right) = - \delta _{\imath ,n,t} B_n \left( \lambda _{\imath } \right) - \delta _{\imath ,n} B_{n,t} \left( \lambda _{\imath } \right) , \\&C_{n,t} \left( \lambda _{\imath } \right) = - \delta _{\imath ,n,t} D_n \left( \lambda _{\imath } \right) - \delta _{\imath ,n} D_{n,t} \left( \lambda _{\imath } \right) , \\&\delta _{\imath ,n,t} = \frac{\beta ^2 r_{n-1} + \alpha }{s_{n-1}} \lambda _{\imath }^{-1} - \left[ \lambda _{\imath }^{-2} + \frac{s_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} \right] \delta _{\imath ,n} + s_n \lambda _{\imath }^{-1} \delta _{\imath ,n}^2. \end{aligned} \end{aligned}$$
(21)

In terms of Eqs. (20) and (21), it is possible to check that \( \lambda _{\imath }\) are all the roots of \(g_{11,n} \left( \lambda \right) \), \(g_{12,n} \left( \lambda \right) \), \(g_{21,n} \left( \lambda \right) \) and \(g_{22,n} \left( \lambda \right) \).

On the basis of Eqs. (20), we know that \(g_{11,n} \left( \lambda \right) \) is the polynomial in \(\lambda \) with the highest order 2N and lowest order \(-2N\), \(g_{12,n} \left( \lambda \right) \) and \(g_{21,n} \left( \lambda \right) \) are the polynomials in \(\lambda \) with the highest order \(2N-1\) and lowest order \(-2N-1\), while \(g_{22,n} \left( \lambda \right) \) is the polynomial in \(\lambda \) with the highest order \(2N-2\) and lowest order \(-2N-2\). With the help of Eq. (8), it can be demonstrated that there exists a matrix \(K_n\) such that

$$\begin{aligned} \left( M_{n,t} + M_{n} V_n \right) M_n^{*} = \text {det} M_n \cdot K_n, \end{aligned}$$
(22)

where

$$\begin{aligned} K_n = \begin{pmatrix} k_{11,n}^{(0)} &{} k_{12,n}^{(-1)} \lambda ^{-1} \\ k_{21,n}^{(-1)} \lambda ^{-1} &{} k_{22,n}^{(-2)} \lambda ^{-2} \end{pmatrix}, \end{aligned}$$

while \(k_{11,n}^{(0)} \), \(k_{12,n}^{(-1)}\), \(k_{21,n}^{(-1)}\) and \(k_{22,n}^{(-2)}\) are some to-be-determined functions of n and t. To calculate \(K_n\), we rewrite Eq. (22) as

$$\begin{aligned} M_{n,t} + M_{n} V_n = K_n M_n. \end{aligned}$$
(23)

By equating the same powers of \(\lambda \) in Eq. (23), we derive

$$\begin{aligned} k_{12,n}^{(-1)} = -\frac{s_n + B_n^{(-N+1)}}{D_n^{(-N)}} = -\widetilde{s}_n, \qquad k_{22,n}^{(-2)}=-1, \end{aligned}$$
(24)

and

$$\begin{aligned} \begin{aligned}&k_{11,n}^{(0)} - \frac{ s_n + B_n^{(-N+1)} }{D_n^{(-N)}}C_n^{(-N+1)} - \frac{s_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} - \frac{ \beta ^2 r_{n-1} + \alpha }{s_{n-1}} B_n^{(-N+1)}=0, \\&k_{21,n}^{(-1)}-\frac{ \beta ^2 r_{n-1} + \alpha }{s_{n-1}} D_n^{(-N)} - C_n^{(-N+1)}=0,\\&A_{n,t}^{(N)} - k_{11,n}^{(0)} A_n^{(N)} + \frac{s_n \left( \beta ^2 r_{n-1} + \alpha \right) }{s_{n-1}} A_n^{(N)}=0,\\&D_{n,t}^{(-N)} - k_{21,n}^{(-1)} B_n^{(-N+1)} - s_n C_n^{(-N+1)}=0. \end{aligned} \end{aligned}$$
(25)

Taking Eqs. (17), (18) and (25) into account, we obtain

$$\begin{aligned} k_{11,n}^{(0)} = \frac{\widetilde{s}_{n} \left( \beta ^2 \widetilde{r}_{n-1} + \alpha \right) }{\widetilde{s}_{n-1}}, \qquad k_{21,n}^{(-1)}=\frac{ \beta ^2 \widetilde{r}_{n-1} + \alpha }{\widetilde{s}_{n-1}}. \end{aligned}$$
(26)

From Eqs. (6b), (24) and (26), we arrive at the conclusion of Proposition 2. The proof is completed. \(\hfill\square \)

Further, we provide the following theorem in light of Propositions 1 and 2:

Theorem 1

Let \(r_n\) and \(s_n\) be the seed solutions of System (3), \(\phi _n \left( \lambda \right) = \left( \phi _{1,n} \left( \lambda \right) , \phi _{2,n} \left( \lambda \right) \right) ^T\) and \(\psi _n \left( \lambda \right) = \left( \psi _{1,n} \left( \lambda \right) , \psi _{2,n} \left( \lambda \right) \right) ^T\) be two linearly independent solutions of Lax Pair (4), then the N-Fold DT is given by Transformation (5) and

$$\begin{aligned} \widetilde{r}_n = \frac{A_{n+1}^{(N)}}{A_n^{(N)}} r_n, \qquad \widetilde{s}_n = \frac{s_n + B_n^{(-N+1)}}{D_n^{(-N)}}, \end{aligned}$$

where

$$\begin{aligned}{} & {} A_{n}^{(N)}=\frac{\Delta A_{n}^{(N)}}{\Delta _1},\qquad B_{n}^{(-N+1)} = \frac{\Delta B_{n}^{(-N+1)}}{\Delta _1},\qquad D_{n}^{(-N)}=\frac{\Delta D_{n}^{(-N)}}{\Delta _2}, \\{} & {} \begin{aligned}&\Delta _1= \begin{vmatrix} \lambda _1^{-N+2}&\lambda _1^{-N+4}&\cdots&\lambda _1^{N}&\lambda _1^{-N+1} \delta _{1,n}&\lambda _1^{-N+3} \delta _{1,n}&\cdots&\lambda _1^{N-1} \delta _{1,n} \\ \lambda _2^{-N+2}&\lambda _2^{-N+4}&\cdots&\lambda _2^{N}&\lambda _2^{-N+1} \delta _{2,n}&\lambda _2^{-N+3} \delta _{2,n}&\cdots&\lambda _2^{N-1} \delta _{2,n} \\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots \\ \lambda _{2N}^{-N+2}&\lambda _{2N}^{-N+4}&\cdots&\lambda _{2N}^{N}&\lambda _{2N}^{-N+1} \delta _{{2N},n}&\lambda _{2N}^{-N+3} \delta _{{2N},n}&\cdots&\lambda _{2N}^{N-1} \delta _{{2N},n} \end{vmatrix}, \\&\Delta _2= \begin{vmatrix} \lambda _1^{-N+1}&\lambda _1^{-N+3}&\cdots&\lambda _1^{N-1}&\lambda _1^{-N} \delta _{1,n}&\lambda _1^{-N+2} \delta _{1,n}&\cdots&\lambda _1^{N-2} \delta _{1,n} \\ \lambda _2^{-N+1}&\lambda _2^{-N+3}&\cdots&\lambda _2^{N-1}&\lambda _2^{-N} \delta _{2,n}&\lambda _2^{-N+2} \delta _{2,n}&\cdots&\lambda _2^{N-2} \delta _{2,n}\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots \\ \lambda _{2N}^{-N+1}&\lambda _{2N}^{-N+3}&\cdots&\lambda _{2N}^{N-1}&\lambda _{2N}^{-N} \delta _{2N,n}&\lambda _{2N}^{-N+2} \delta _{2N,n}&\cdots&\lambda _{2N}^{N-2} \delta _{2N,n} \\ \end{vmatrix}, \end{aligned} \end{aligned}$$

\(\Delta A_{n}^{(N)}\) and \(\Delta B_{n}^{(-N+1)}\) are respectively produced from \(\Delta _1\) by replacing the Nth and \((N+1)\)th columns with \((-\lambda _1^{-N}, -\lambda _2^{-N}, \ldots , -\lambda _{2N}^{-N})^{T}\), while \(\Delta D_{n}^{(-N)}\) is produced from \(\Delta _2\) by replacing the \((N+1)\)th column with \((-\lambda _1^N \delta _{1,n}, -\lambda _2^N \delta _{2,n}, \ldots , -\lambda _{2N}^N \delta _{2N,n})^{T}\).

4 Analytic solutions of System (3)

To determine some analytic solutions of System (3), we choose the non-zero seed solutions of System (3) as

$$\begin{aligned} r_n=1, \qquad s_n=\textrm{e}^{\left( 1+\beta ^2 \right) t}. \end{aligned}$$
(27)

Utilizing Seed Solutions (27), we obtain the two solutions of Lax Pair (4) as

$$\begin{aligned}&\phi _n = \begin{pmatrix} \phi _{1,n} \left( \lambda _{\imath } \right) \\ \phi _{2,n} \left( \lambda _{\imath } \right) \end{pmatrix}= \begin{pmatrix} \chi _1 \left( \lambda _{\imath } \right) \left[ \kappa _1\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{\mu _1\left( \lambda _{\imath } \right) t} \\ \left[ \kappa _1\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{ \left[ \mu _1\left( \lambda _{\imath } \right) - \left( 1+\beta ^2 \right) \right] t} \end{pmatrix}, \end{aligned}$$
(28a)
$$\begin{aligned}&\psi _n = \begin{pmatrix} \psi _{1,n} \left( \lambda _{\imath } \right) \\ \psi _{2,n} \left( \lambda _{\imath } \right) \end{pmatrix}= \begin{pmatrix} \chi _2 \left( \lambda _{\imath } \right) \left[ \kappa _2\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{\mu _2\left( \lambda _{\imath } \right) t} \\ \left[ \kappa _2\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{\left[ \mu _2\left( \lambda _{\imath } \right) - \left( 1+\beta ^2 \right) \right] t} \end{pmatrix}, \end{aligned}$$
(28b)

where

$$\begin{aligned}&\chi _1 \left( \lambda _{\imath } \right) =\frac{1}{2\left( \beta ^2 + \alpha \right) } \left[ \lambda _{\imath }^{-1} + \left( \alpha -1 \right) \lambda _{\imath } + \lambda _{\imath }^{-1} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] ,\\&\chi _2 \left( \lambda _{\imath } \right) =\frac{1}{2\left( \beta ^2 + \alpha \right) } \left[ \lambda _{\imath }^{-1} + \left( \alpha -1 \right) \lambda _{\imath } - \lambda _{\imath }^{-1} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] ,\\&\kappa _1 \left( \lambda _{\imath } \right) = \frac{1}{2} \left[ -\lambda _{\imath }^{-1} + \left( \alpha +1 \right) \lambda _{\imath } - \lambda _{\imath }^{-1} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] ,\\&\kappa _2 \left( \lambda _{\imath } \right) = \frac{1}{2} \left[ -\lambda _{\imath }^{-1} + \left( \alpha +1 \right) \lambda _{\imath } + \lambda _{\imath }^{-1} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] ,\\&\mu _1 \left( \lambda _{\imath } \right) = \frac{1}{2} \left[ -\lambda _{\imath }^{-2} + \left( 2\beta ^2 + \alpha +1 \right) + \lambda _{\imath }^{-2} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] ,\\&\mu _2 \left( \lambda _{\imath } \right) = \frac{1}{2} \left[ -\lambda _{\imath }^{-2} + \left( 2\beta ^2 + \alpha +1 \right) - \lambda _{\imath }^{-2} \sqrt{1-2\left( 1+ 2\beta ^2 +\alpha \right) \lambda _{\imath }^2 + \left( \alpha -1 \right) ^2\lambda _{\imath }^4} \right] . \end{aligned}$$

According to Eqs. (10), (13) and (28), we have

$$\begin{aligned} \begin{aligned}&\delta _{\imath ,n} = \frac{ \left[ \kappa _1\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{ \left[ \mu _1\left( \lambda _{\imath } \right) - \left( 1+\beta ^2 \right) \right] t} + \theta _{\imath } \left[ \kappa _2\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{ \left[ \mu _2\left( \lambda _{\imath } \right) - \left( 1+\beta ^2 \right) \right] t }}{\chi _1 \left( \lambda _{\imath } \right) \left[ \kappa _1\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{\mu _1\left( \lambda _{\imath } \right) t} + \theta _{\imath } \chi _2 \left( \lambda _{\imath } \right) \left[ \kappa _2\left( \lambda _{\imath } \right) \right] ^n \textrm{e}^{\mu _2\left( \lambda _{\imath } \right) t} }, \\&\delta _{\imath ,n+1}=\frac{-\left( \beta ^2 + \alpha \right) + \alpha \textrm{e}^{\left( 1+\beta ^2 \right) t} \lambda _{\imath } \delta _{\imath ,n}}{\textrm{e}^{\left( 1+\beta ^2 \right) t} \left[ \lambda _{\imath } -\lambda _{\imath }^{-1} + \textrm{e}^{\left( 1+\beta ^2 \right) t} \delta _{\imath ,n} \right] }. \end{aligned} \end{aligned}$$
(29)

When \(N=2\), making use of Theorem 1, two-fold solutions of System (3) can be expressed as

$$\begin{aligned}&\widetilde{r}_n = \frac{A_{n+1}^{(2)}}{A_n^{(2)}}, \end{aligned}$$
(30a)
$$\begin{aligned}&\widetilde{s}_n = \frac{\textrm{e}^{\left( 1+\beta ^2 \right) t} + B_n^{(-1)}}{D_n^{(-2)}}, \end{aligned}$$
(30b)

where

$$\begin{aligned}&A_{n}^{(2)}=\frac{\Delta A_{n}^{(2)}}{\Delta _1},~~~B_{n}^{(-1)} = \frac{\Delta B_{n}^{(-1)}}{\Delta _1},~~~D_{n}^{(-2)}=\frac{\Delta D_{n}^{(-2)}}{\Delta _2}, \end{aligned}$$

with

$$\begin{aligned}&\Delta _1= \begin{vmatrix} 1&\lambda _1^2&\lambda _1^{-1} \delta _{1,n}&\lambda _1 \delta _{1,n} \\ 1&\lambda _2^2&\lambda _2^{-1} \delta _{2,n}&\lambda _2 \delta _{2,n} \\ 1&\lambda _3^2&\lambda _3^{-1} \delta _{3,n}&\lambda _3 \delta _{3,n} \\ 1&\lambda _4^2&\lambda _4^{-1} \delta _{4,n}&\lambda _4 \delta _{4,n} \end{vmatrix},~~~ \Delta _2= \begin{vmatrix} \lambda _1^{-1}&\lambda _1&\lambda _1^{-2} \delta _{1,n}&\delta _{1,n} \\ \lambda _2^{-1}&\lambda _2&\lambda _2^{-2} \delta _{2,n}&\delta _{2,n} \\ \lambda _3^{-1}&\lambda _3&\lambda _3^{-2} \delta _{3,n}&\delta _{3,n} \\ \lambda _4^{-1}&\lambda _4&\lambda _4^{-2} \delta _{4,n}&\delta _{4,n} \end{vmatrix}, ~~~\Delta A_{n}^{(2)}= \begin{vmatrix} 1&-\lambda _1^{-2}&\lambda _1^{-1} \delta _{1,n}&\lambda _1 \delta _{1,n} \\ 1&-\lambda _2^{-2}&\lambda _2^{-1} \delta _{2,n}&\lambda _2 \delta _{2,n} \\ 1&-\lambda _3^{-2}&\lambda _3^{-1} \delta _{3,n}&\lambda _3 \delta _{3,n} \\ 1&-\lambda _4^{-2}&\lambda _4^{-1} \delta _{4,n}&\lambda _4 \delta _{4,n} \end{vmatrix},\\& \Delta B_{n}^{(-1)}= \begin{vmatrix} 1&\lambda _1^2&-\lambda _1^{-2}&\lambda _1 \delta _{1,n} \\ 1&\lambda _2^2&-\lambda _2^{-2}&\lambda _2 \delta _{2,n} \\ 1&\lambda _3^2&-\lambda _3^{-2}&\lambda _3 \delta _{3,n} \\ 1&\lambda _4^2&-\lambda _4^{-2}&\lambda _4 \delta _{4,n} \end{vmatrix}, ~~~\Delta D_{n}^{(-2)}= \begin{vmatrix} \lambda _1^{-1}&\lambda _1&-\lambda _1^{2} \delta _{1,n}&\delta _{1,n} \\ \lambda _2^{-1}&\lambda _2&-\lambda _2^{2} \delta _{2,n}&\delta _{2,n} \\ \lambda _3^{-1}&\lambda _3&-\lambda _3^{2} \delta _{3,n}&\delta _{3,n} \\ \lambda _4^{-1}&\lambda _4&-\lambda _4^{2} \delta _{4,n}&\delta _{4,n} \end{vmatrix}. \end{aligned}$$

5 Conclusions

Studies of the lattice systems have been seen of current interest due to their applications in relativity, optics, condensed matter physics and plasma physics. In this paper, we have studied a hybrid relativistic and modified Toda lattice-type system, i.e., System (1). It has been known that System (1) is related to the relativistic Toda lattice and modified Toda lattice. Under Transformations (2), we have given an equivalent form of System (1), i.e., System (3), along with Lax Pair (4). With the aid of Lax Pair (4), we have constructed N-Fold DT (5) and (11) of System (3). Via N-Fold DT (5) and (11) with \(N=2\), we have obtained Two-Fold Solutions (30) of System (3).