Rational Solutions for Two Nonautonomous Lattice Korteweg–de Vries Type Equations

Abstract

Up to now, many works have been conducted to deal with the rational solutions for the autonomous discrete integrable systems, while there have been few works on the rational solutions to the nonautonomous discrete integrable systems. In this paper, we investigate two nonautonomous lattice Korteweg–de Vries type equations: nonautonomous lattice potential Korteweg–de Vries equation and nonautonomous lattice potential modified Korteweg–de Vries equation. By the bilinear method, we construct the rational solutions for the aforesaid equations. These solutions are presented in terms of the Casoratian. By setting special forms of the lattice parameters of the nonautonomous lattice potential Korteweg–de Vries equation, dynamical behaviors for the first two rational solutions of this equation are analyzed with graphical illustration.

1. Introduction

The nonautonomous discrete integrable systems are of considerable interest in various fields of mathematics and physics [1,2]. Particularly, most of the discrete Painlevé equations are nonautonomous ordinary difference equations, which are connected to the integrable nonautonomous partial difference equations through reductions. An autonomous system refers to a differential or difference model with constant coefficients, while the coefficients of a nonautonomous system are associated with independent variables. To date, numerous methods have been developed to construct soliton solutions for nonautonomous discrete integrable systems. These include the Darboux transformation, the bilinear method, and the reduction technique, among others (see Refs. [3,4,5,6,7,8,9]).

This paper focuses on deriving rational solutions for the following two nonautonomous lattice Korteweg–de Vries (nlKdV) type equations:

$$\begin{array}{c}\hfill (a_n-b_m+\widehat{u}-\tilde{u})(a_n+b_m+u-\widehat{\tilde{u}})=a_n^2-b_m^2,\end{array}$$

(1)

and

$$\begin{array}{c}\hfill p_n(v\widehat{v}-\tilde{v}\widehat{\tilde{v}})=q_m(v\tilde{v}-\widehat{v}\widehat{\tilde{v}}).\end{array}$$

(2)

Habitually, we refer to Equations (1) and (2) as the nonautonomous lattice potential KdV (nlpKdV) equation and the nonautonomous lattice potential modified KdV (nlpmKdV) equation, respectively. These two equations relate variables on the vertices of each quad on the regular ${\mathbb{Z}}^2$ lattice, whence $u=u(n,m)$ and $v=v(n,m)$. The operations $f\to \tilde{f}$ and $f\to \widehat{f}$ denote elementary shifts in the two directions of the lattice, i.e., $\tilde{f}=f(n+1,m)$ and $\widehat{f}=f(n,m+1)$ while for the combined shifts we have as follows: $\widehat{\tilde{f}}=f(n+1,m+1)$. And $a_n,b_m,p_n,q_m\in \mathbb{C}$ are lattice parameters of the equations. We refer the reader to the notations $\mathbb{Z}$ and $\mathbb{C}$: the sets of integers and complex numbers, respectively.

Both Equations (1) and (2) belong to the nonautonomous Adler–Bobenko–Suris (nABS) lattice list [10]. In fact, under simple point transformation $u=U-{\sum }_{i=n_0}^{n-1}{a}_i-{\sum }_{j=m_0}^{m-1}{b}_j$, the nlpKdV Equation (1) becomes the nonautonomous H1 equation

$$\begin{array}{c}\hfill (\widehat{U}-\tilde{U})(U-\widehat{\tilde{U}})=a_n^2-b_m^2,\end{array}$$

(3)

which is the lowest member in the nABS lattice list. The nlpmKdV Equation (2) itself is the nonautonomous H3$\left(0\right)$ equation. When $a_n=p_n$ and $b_m=q_m$, the nlpKdV Equation (1) and the nlpmKdV Equation (2) appeared in a same Cauchy matrix scheme (e.g., Equations (11) and (12) in [11]), which means they have solutions in terms of Cauchy matrix. These two equations are related to each other by Miura transformations

$$\begin{array}{cc}\hfill & p_n-q_m+\widehat{u}-\tilde{u}={\displaystyle \frac{1}{\widehat{\tilde{v}}}}(p_n\widehat{v}-q_m\tilde{v})\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{v}}(p_n\tilde{v}-q_m\widehat{v}),\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill & p_n+q_m+u-\widehat{\tilde{u}}={\displaystyle \frac{1}{\tilde{v}}}(p_{n}v+q_m\widehat{\tilde{v}})\hfill \end{array}$$

(4c)

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{1}{\widehat{v}}}(p_n\widehat{\tilde{v}}+q_{m}v).\hfill \end{array}$$

(4d)

Both Equations (1) and (2) possess a multidimensional consistency (MDC) property [12,13] and singularity confinement property [10] (for the sake of simplicity, we provide details in Appendix A for the MDC property of the nonautonomous H1 Equation (3) as an understanding of MDC property).

In recent decades, the search for rational solutions to integrable systems has been a central focus in applied mathematics and physics. This type of solution, expressed by a fraction of polynomials of independent variables, always coincides with the characteristics of the rogue wave [14]. This nonlinear phenomenon, also named as freak wave, extreme wave, etc., has been observed in nonlinear optics, Bose–Einstein condensates, atmospherics, superfluid and even finance. Superficially, rational solutions and soliton solutions differ in their expressions, but they are fundamentally deeply related. In general, rational solutions can be derived from soliton solutions through a special limit procedure (see Refs. [15,16,17] as examples). Compared with the case in continuous and semi-discrete integrable systems, it is more difficult to obtain rational solutions for discrete integrable systems, known as integrable difference-difference systems. For the rational solutions to the quad-graph lattice equations, much progress has been made. For the H3($\delta$) and Q1($\delta$) in the ABS lattice list [18], as well as a lattice Boussinesq equation [19], the existence of $\delta$ (i.e., $\delta \ne 0$) plays a crucial role in the procedure of obtaining rational solutions from their soliton solutions. Subsequently, the reduction technique [20] was applied to construct rational solutions for the lpKdV equation and two semi-discrete lpKdV equations [21]. In [22], Bäcklund transformations were adopted to derive rational solutions for the lpmKdV equation, and Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the ABS lattice list. All the obtained rational solutions were related to a unified function in Casoratian form which obeyed a bilinear superposition formula. Quite recently, rational solutions in Casoratian form for the Nijhoff–Quispel–Capel (NQC) equation have been discussed [23]. Moreover, rational solutions for the whole ABS lattice list except Q4 were given, based on the Miura transformation for the NQC equation and Q3($\delta$) and the degenerations from Q3($\delta$) to Q2, Q1($\delta$), H3($\delta$), H2 and H1 in the ABS lattice list.

In this paper, our primary focus is on exploring rational solutions for the nlKdV type Equations (1) and (2). Our approach to these equations closely follows the derivation of rational solutions for autonomous lattice KdV type equations and lattice potential Boussinesq equation as presented in references [19,23]. This approach is based on Hirota’s bilinear method [24] and Wronskian technique [25] introduced in 1971 and 1983, respectively. The idea was to make a dependent variable transform into new variables, for which the solution would be given by a Wronski/Casorati determinant. Compared with other methods, this approach has turned out to be quite successful in constructing rational solutions for integrable systems, which was shown in a series of papers [16,17,18,19,23,26].

The structure of the paper is as follows: Section 2 presents the bilinearization and Casoratian solutions for Equations (1) and (2).

Section 3 discusses the rational solutions derived from these equations.

Section 4 delves into the analysis of the dynamic behaviors exhibited by the obtained rational solutions.

Finally, Section 5 provides the conclusions of our study.

2. Bilinearization and Casoratian Solutions

The Casoratian technique enables the direct verification of a solution in Casoratian form for a discrete bilinear equation. Within this framework, solutions in Casoratian form and the condition equation set satisfied by the basic column vector are crucial elements.

In this section, we will explore the bilinearization and Casoratian solutions for the nlpKdV (1) and nlpmKdV (2) equations using this technique.

To begin, we would like to briefly review the Casoratian. Casoratian is the discrete version of Wronskian. For the given column vector $\mathit{\psi }:=\mathit{\psi }(n,m,l)=({\psi }_1(n,m,l),{\psi }_2(n,m,l),\cdots ,$${\psi }_N{(n,m,l))}^T$, we introduce up-shift operators $E^{\nu }(\nu =1,2,3)$ as

$$\begin{array}{c}\hfill E^1\mathit{\psi }=\tilde{\mathit{\psi }},E^2\mathit{\psi }=\widehat{\mathit{\psi }},E^3\mathit{\psi }=\overline{\mathit{\psi }},\end{array}$$

(5)

where $\overline{\phantom{a}}$ means the elementary shift in the lattice in the l-direction, i.e., $\overline{\mathit{\psi }}=\mathit{\psi }(n,m,l+1)$. Also, we use $E_{\nu }(\nu =1,2,3)$ to denote down-shift operators, namely, $E_1\mathit{\psi }=\underset{˜}{\mathit{\psi }}$, etc. Generally, the Casoratian composed of vector $\mathit{\psi }$ with $E^{\nu }$-shift can be described as

$$\begin{array}{c}\hfill C\left(\mathit{\psi }\right)=|\mathit{\psi },E^{\nu }\mathit{\psi },{\left(E^{\nu }\right)}^2\mathit{\psi },\cdots ,{\left(E^{\nu }\right)}^{N-1}{\mathit{\psi }|=|0,1,\cdots ,N-1|}_{\left[\nu \right]},\end{array}$$

(6)

which can be more compactly expressed as $C\left(\mathit{\psi }\right)={|\widehat{N-1}|}_{\left[\nu \right]}$ (cf. [25]). With this notation we also have ${|\widehat{N-2},N|}_{\left[\nu \right]}={|0,1,\cdots ,N-2,N|}_{\left[\nu \right]}$ and $|-1,\widehat{N-2}{|}_{\left[\nu \right]}={|-1,0,\cdots ,N-2|}_{\left[\nu \right]}$, etc. For the sake of simplicity, we denote $|\widehat{N-1}{|}_{\left[3\right]}=|\widehat{N-1}|$. Generally speaking, in Wronskian/Casoratian the order of the column vector N is usually related to the number of solitons for multiple collisions (see [16,17] and the references therein).

We now show the bilinearization and Casoratian solutions for the nlpKdV Equation (1) and the nlpmKdV Equation (2). Let us first consider the nlpKdV Equation (1):

$$\begin{array}{c}\hfill [(a_n-\delta )-(b_m-\delta )+\widehat{u}-\tilde{u}][(a_n-\delta )+(b_m+\delta )+u-\widehat{\tilde{u}}]=a_n^2-b_m^2.\end{array}$$

(7)

Here, and throughout, $\delta$ is a constant.

By the dependent variable transformation $u=g/f$, the nlpKdV Equation (1) is bilinearized as

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{11}\equiv \tilde{f}\widehat{g}-\widehat{f}\tilde{g}+[(a_n-\delta )-(b_m-\delta )](\widehat{f}\tilde{f}-f\widehat{\tilde{f}})=0,\hfill \end{array}$$

(8a)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{12}\equiv \widehat{\tilde{f}}g-f\widehat{\tilde{g}}+[(a_n-\delta )+(b_m+\delta )](f\widehat{\tilde{f}}-\widehat{f}\tilde{f})=0.\hfill \end{array}$$

(8b)

The Casoratian solutions to bilinear system (8) can be described by the following theorem (see [9] and for the autonomous case, one can refer to [23,27]):

Theorem 1.

The nonautonomous bilinear Equation (8) admit Casoratians

$$\begin{array}{c}\hfill f=|\widehat{N-1}|,g=|\widehat{N-2},N|,\end{array}$$

(9)

in which the column vector $\mathit{\psi }\left(l\right)$ satisfies condition equation set (CES)

$$\begin{array}{cc}\hfill & (a_n-\delta )\mathit{\psi }=\tilde{\mathit{\psi }}-\overline{\mathit{\psi }},(b_m-\delta )\mathit{\psi }=\widehat{\mathit{\psi }}-\overline{\mathit{\psi }},\hfill \end{array}$$

(10a)

$$\begin{array}{cc}\hfill & (a_n-\delta )\mathit{\varphi }=\tilde{\mathit{\varphi }}-\overline{\mathit{\varphi }},(b_m+\delta )\widehat{\mathit{\varphi }}=\mathit{\varphi }+\overline{\widehat{\mathit{\varphi }}},\hfill \end{array}$$

(10b)

$$\begin{array}{cc}\hfill & \mathit{\psi }={\mathbf{\Gamma }}_{\left[m\right]}\mathit{\varphi },\hfill \end{array}$$

(10c)

where $\mathit{\varphi }:=\mathit{\varphi }(n,m,l)={({\varphi }_1(n,m,l),{\varphi }_2(n,m,l),\cdots ,{\varphi }_N(n,m,l))}^T$ is an auxiliary vector, ${\mathbf{\Gamma }}_{\left[m\right]}$ is an $N\times N$ matrix only depends on m but is independent of $(n,l)$.

Remark 1.

In terms of the Equation (10), it is easy to show that

$$|\widehat{N-1}{|}_1=|\widehat{N-1}{|}_2=|\widehat{N-1}|.$$

To continue, we next consider the nlpmKdV Equation (2). For introducing the constant $\delta$, we reparameterize $p_n$ and $q_m$ in (2) as $p_n={(a_n^2-{\delta }^2)}^{{\displaystyle \frac{1}{2}}}$ and $q_m={(b_m^2-{\delta }^2)}^{{\displaystyle \frac{1}{2}}}$ and introduce a new dependent variable

$$\begin{array}{c}\hfill w=\prod _{i=n_0}^{n-1}{\left({\displaystyle \frac{a_i+\delta }{a_i-\delta }}\right)}^{-{\displaystyle \frac{1}{2}}}\prod _{j=m_0}^{m-1}{\left({\displaystyle \frac{b_i+\delta }{b_i-\delta }}\right)}^{-{\displaystyle \frac{1}{2}}}v,\end{array}$$

(11)

then we arrive at

$$\begin{array}{c}\hfill w[(a_n-\delta )\widehat{w}-(b_m-\delta )\tilde{w}]=\widehat{\tilde{w}}[(a_n+\delta )\tilde{w}-(b_m+\delta )\widehat{w}],\end{array}$$

(12)

where $n_0$ and $m_0$ are fixed integers. Through the transformation $w=h/f$, then we obtain the bilinear form of Equation (12), given by

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{21}\equiv (a_n-\delta )\widehat{\tilde{f}}h+(b_m+\delta )f\widehat{\tilde{h}}-[(a_n-\delta )+(b_m+\delta )]\widehat{f}\tilde{h}=0,\hfill \end{array}$$

(13a)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{22}\equiv (a_n+\delta )f\widehat{\tilde{h}}+(b_m-\delta )\widehat{\tilde{f}}h-[(a_n+\delta )+(b_m-\delta )]\tilde{f}\widehat{h}=0.\hfill \end{array}$$

(13b)

With regard to Casoratian solutions, we have the following result (For the autonomous case, one can refer to [23]):

Theorem 2.

The nonautonomous bilinear Equation (13) admit Casoratians

$$\begin{array}{c}\hfill f=|\widehat{N-1}|,h=|-1,\widehat{N-2}|,\end{array}$$

(14)

in which the column vector ψ satisfies (10) together with

$$\begin{array}{cc}\hfill & (a_n+\delta )\tilde{\mathit{\phi }}=\mathit{\phi }+\overline{\tilde{\mathit{\phi }}},(b_m-\delta )\mathit{\phi }=\widehat{\mathit{\phi }}-\overline{\mathit{\phi }},\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & \mathit{\psi }={\mathbf{\Gamma }}_{\left[n\right]}\mathit{\phi },\hfill \end{array}$$

(15b)

where $\mathit{\phi }:=\mathit{\phi }(n,m,l)={({\phi }_1(n,m,l),{\phi }_2(n,m,l),\cdots ,{\phi }_N(n,m,l))}^T$ is an auxiliary vector, ${\mathbf{\Gamma }}_{\left[n\right]}$ is the same as ${\mathbf{\Gamma }}_{\left[m\right]}$ just with replacement $(m,b)\to (n,a)$.

We assume matrices $({\mathbf{\Gamma }}_{\left[m\right]},{\mathbf{\Gamma }}_{\left[n\right]})$ are similar to lower triangular canonical forms $({\mathbf{\Lambda }}_{\left[m\right]},{\mathbf{\Lambda }}_{\left[n\right]})$ by a transform matrix $\mathit{T}$, i.e., ${\mathbf{\Gamma }}_{\left[m\right]}={\mathit{T}}^{-1}{\mathbf{\Lambda }}_{\left[m\right]}\mathit{T}$ and ${\mathbf{\Gamma }}_{\left[n\right]}={\mathit{T}}^{-1}{\mathbf{\Lambda }}_{\left[n\right]}\mathit{T}$. Defining ${\mathit{\psi }}^{\prime }=\mathit{T}\mathit{\psi }$, ${\mathit{\varphi }}^{\prime }=\mathit{T}\mathit{\varphi }$, as well as ${\mathit{\phi }}^{\prime }=\mathit{T}\mathit{\phi }$, CES (10) and (15) lead to their canonical form and the Casoratians composed by $\mathit{\psi }$ and ${\mathit{\psi }}^{\prime }$ are related to each other as $f\left({\mathit{\psi }}^{\prime }\right)=|\mathit{T}|f\left(\mathit{\psi }\right)$, $g\left({\mathit{\psi }}^{\prime }\right)=|\mathit{T}|g\left(\mathit{\psi }\right)$ and $h\left({\mathit{\psi }}^{\prime }\right)=|\mathit{T}|h\left(\mathit{\psi }\right)$. Apparently, if $\left(f\right(\mathit{\psi }),g(\mathit{\psi }),h(\mathit{\psi }\left)\right)$ solves the bilinear Equations (8) and (13), so does $(f\left({\mathit{\psi }}^{\prime }\right),g\left({\mathit{\psi }}^{\prime }\right),h\left({\mathit{\psi }}^{\prime }\right))$. Since CES (10) and (15) are the same as their canonical form, we in the following, take ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$ as Jordan canonical forms.

3. Some Examples of Solutions

In this section, we construct rational solutions for the nlpKdV (1) and nlpmKdV (2) by solving systems (10) and (15). We will find that the parameter $\delta$ plays a crucial role in the procedure of obtaining rational solutions. In the Wronskian/Casoratian scheme, the lower triangular Toeplitz (LTT) matrix is essential. This type of matrix is of form

$$\begin{array}{c}\hfill \mathcal{A}={\left(\begin{array}{cccccc}{a}_0& 0& 0& \cdots & 0& 0\\ a_1& a_0& 0& \cdots & 0& 0\\ a_2& a_1& a_0& \cdots & 0& 0\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮\\ a_{N-1}& a_{N-2}& a_{N-3}& \cdots & a_1& a_0\end{array}\right)}_{N\times N},a_j\in \mathbb{C},\end{array}$$

which commutes with a Jordan block. Note, that all the LTT matrices of the same order compose a commutative set $F^N$ in terms of matrix product and the subset $G^N=\left\{\mathcal{A}|.\mathcal{A}\in F^N,|\mathcal{A}|\ne 0\right\}$ is an Abelian group. The canonical form of such a matrix is a Jordan matrix.

In terms of the eigenvalues of ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$, we can categorize solutions to the Equations (1) and (2) as solitons, Jordan-block solutions and rational solutions. In the subsequent section, we list them case by case.

3.1. Soliton Solutions

Let ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$ be the diagonal matrices given by

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[m\right]}=\mathrm{diag}\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_N\right),{\alpha }_s=\prod _{j=m_0}^{m-1}(b_j^2-k_s^2),\hfill \end{array}$$

(16a)

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[n\right]}=\mathrm{diag}\left({\beta }_1,{\beta }_2,\dots ,{\beta }_N\right),{\beta }_s=\prod _{i=n_0}^{n-1}(a_i^2-k_s^2),\hfill \end{array}$$

(16b)

with distinct constants $\left\{k_s\right\}$ and fixed integers $n_0$ and $m_0$, then (9) and (14) composed by

$$\begin{array}{cc}\hfill & {\psi }_s(n,m,l)={\psi }_s^{0^{+}}\rho \left(k_s\right)+{\psi }_s^{0^{-}}\rho (-k_s),\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill & \rho \left(k\right)={(\delta +k)}^l\prod _{i=n_0}^{n-1}(a_i+k)\prod _{j=m_0}^{m-1}(b_j+k),\hfill \end{array}$$

(17b)

generate the usual N-soliton solutions, where $\left\{{\psi }_s^{0^{\pm }}\right\}$ are constants. Meanwhile, ${\varphi }_s(n,m,l)$ and ${\phi }_s(n,m,l)$ are of the form

$$\begin{array}{c}\hfill {\varphi }_s(n,m,l)={\psi }_s(n,m,l)/{\alpha }_s,{\phi }_s(n,m,l)={\psi }_s(n,m,l)/{\beta }_s.\end{array}$$

(18)

The one-soliton solution for nlpKdV Equation (1) is

$$\begin{array}{c}\hfill u=\delta +{\displaystyle \frac{k_1(1-{\chi }_1)}{1+{\chi }_1}},\end{array}$$

(19)

and the one-soliton solution for nlpmKdV Equation (12) is

$$\begin{array}{c}\hfill w={\displaystyle \frac{\delta }{{\delta }^2-k_1^2}}+{\displaystyle \frac{k_1({\chi }_1-1)}{({\delta }^2-k_1^2)(1+{\chi }_1)}},\end{array}$$

(20)

where ${\chi }_1$ is the nonautonomous plane-wave factor defined by

$$\begin{array}{c}\hfill {\chi }_1={\chi }_1^0{\left({\displaystyle \frac{\delta -k_1}{\delta +k_1}}\right)}^l\prod _{i=n_0}^{n-1}\left({\displaystyle \frac{a_i-k_1}{a_i+k_1}}\right)\prod _{j=m_0}^{m-1}\left({\displaystyle \frac{b_j-k_1}{b_j+k_1}}\right),\end{array}$$

(21)

with ${\chi }_1^0={\psi }_1^{0^{-}}/{\psi }_1^{0^{+}}$. When $\delta =0$, these soliton solutions have been reported in [9].

3.2. Jordan-Block Solutions

Let ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$ be the LTT matrices given by

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[m\right]}={\left({\alpha }_{s,j}\right)}_{N\times N},{\alpha }_{s,j}=\{\begin{array}{cc}{\displaystyle \frac{{\partial }_{k_1}^{s-j}{\alpha }_1}{(s-j)!}},\hfill & s\ge j,\hfill \\ 0,\hfill & s<j,\hfill \end{array}\hfill \end{array}$$

(22a)

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[n\right]}={\left({\beta }_{s,j}\right)}_{N\times N},{\beta }_{s,j}=\{\begin{array}{cc}{\displaystyle \frac{{\partial }_{k_1}^{s-j}{\beta }_1}{(s-j)!}},\hfill & s\ge j,\hfill \\ 0,\hfill & s<j,\hfill \end{array}\hfill \end{array}$$

(22b)

where ${\alpha }_1$ and ${\beta }_1$ are defined by (16). The Casoratian column vector $\mathit{\psi }$ can then be taken as

$$\begin{array}{c}\hfill \mathit{\psi }={\mathcal{A}}^{+}{\mathcal{P}}^{+}+{\mathcal{A}}^{-}{\mathcal{P}}^{-},\end{array}$$

(23a)

where

$$\begin{array}{c}\hfill {\mathcal{P}}^{\pm }={(P_0^{\pm },P_1^{\pm },\dots ,P_{N-1}^{\pm })}^T,P_s^{\pm }={\displaystyle \frac{1}{s!}}{\partial }_{k_1}^s\rho (\pm k_1),\end{array}$$

(23b)

and ${\mathcal{A}}^{\pm }$ are arbitrary N-th-order LTT matrices. The corresponding vectors $\mathit{\varphi }$ and $\mathit{\phi }$ are

$$\begin{array}{cc}\hfill & \mathit{\varphi }={\mathcal{A}}^{+}{\mathcal{Q}}_1^{+}+{\mathcal{A}}^{-}{\mathcal{Q}}_1^{-},\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill & \mathit{\phi }={\mathcal{A}}^{+}{\mathcal{R}}_1^{+}+{\mathcal{A}}^{-}{\mathcal{R}}_1^{-},\hfill \end{array}$$

(24b)

where

$$\begin{array}{cc}\hfill & {\mathcal{Q}}_1^{\pm }={(Q_0^{\pm },Q_1^{\pm },\dots ,Q_{N-1}^{\pm })}^T,Q_s^{\pm }={\displaystyle \frac{1}{s!}}{\partial }_{k_1}^s\sigma (\pm k_1),\hfill \end{array}$$

(25a)

$$\begin{array}{cc}\hfill & {\mathcal{R}}_1^{\pm }={(R_0^{\pm },R_1^{\pm },\dots ,R_{N-1}^{\pm })}^T,R_s^{\pm }={\displaystyle \frac{1}{s!}}{\partial }_{k_1}^s\varrho (\pm k_1),\hfill \end{array}$$

(25b)

$$\begin{array}{cc}\hfill & \sigma (\pm k_1)=\rho (\pm k_1)/{\alpha }_1,\varrho (\pm k_1)=\rho (\pm k_1)/{\beta }_1.\hfill \end{array}$$

(25c)

The solutions, in this case, are named Jordan-block solutions, which can be derived from soliton solutions by taking limiting procedures $\{k_s\to k_1\}$ successively (see [16,17]). For convenience, we take ${\mathcal{A}}^{\pm }={\psi }_1^{0^{\pm }}I$ with N-th order unit matrix I. Then, the simplest Jordan-block solution for nlpKdV Equation (1) reads

$$\begin{array}{c}\hfill u={\displaystyle \frac{2\left(k_1+\delta +(k_1-\delta ){\chi }_1^2+2k_1{\chi }_1-2\delta k_1{\partial }_{k_1}{\chi }_1\right)}{1-{\chi }_1^2-2k_1{\partial }_{k_1}{\chi }_1}},\end{array}$$

(26)

where ${\chi }_1$ is given by (21). The simplest Jordan-block solution for nlpmKdV Equation (12) is of form

$$\begin{array}{c}\hfill w={\displaystyle \frac{{(k_1-\delta )}^2-{(k_1+\delta )}^2-4\delta k_1{\chi }_1+2k_1(k_1^2-{\delta }^2){\partial }_{k_1}{\chi }_1}{{(k_1^2-{\delta }^2)}^2(1-{\chi }_1^2-2k_1{\partial }_{k_1}{\chi }_1)}}.\end{array}$$

(27)

In (26) and (27), ${\partial }_{k_1}{\chi }_1$ can be explicitly expressed as

$$\begin{array}{c}\hfill {\partial }_{k_1}{\chi }_1=-2{\chi }_1\left({\displaystyle \frac{l\delta }{{\delta }^2-k_1^2}}+\sum _{i=n_0}^{n-1}{\displaystyle \frac{a_i}{a_i^2-k_1^2}}+\sum _{j=m_0}^{m-1}{\displaystyle \frac{b_j}{b_j^2-k_1^2}}\right).\end{array}$$

(28)

In order to obtain the rational solutions, in what follows we suppose $\delta \ne 0$.

3.3. Rational Solutions

Rational solutions are formally generated from Jordan-block solutions by taking $k_1=0$ in (22) and (23). To reach nontrivial rational solutions, let us start from

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[m\right]}={\left({\gamma }_{s,j}\right)}_{N\times N},{\gamma }_{s,j}=\{\begin{array}{cc}{\displaystyle \frac{{\partial }_{k_1}^{2(s-j)}{\alpha }_1}{(2s-2j)!}}{|}_{k_1=0},\hfill & s\ge j,\hfill \\ 0,\hfill & s<j,\hfill \end{array}\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill & {\mathbf{\Gamma }}_{\left[n\right]}={\left({\zeta }_{s,j}\right)}_{N\times N},{\zeta }_{s,j}=\{\begin{array}{cc}{\displaystyle \frac{{\partial }_{k_1}^{2(s-j)}{\beta }_1}{(2s-2j)!}}{|}_{k_1=0},\hfill & s\ge j,\hfill \\ 0,\hfill & s<j.\hfill \end{array}\hfill \end{array}$$

(29b)

In this case, solution $\mathit{\psi }$ can be taken as

$$\begin{array}{c}\hfill \mathit{\psi }={\mathcal{A}}^{+}{\mathcal{Q}}^{+}+{\mathcal{A}}^{-}{\mathcal{Q}}^{-},\end{array}$$

(30)

in which ${\mathcal{Q}}^{\pm }={(X_0^{\pm },X_2^{\pm },\dots ,X_{2N-2}^{\pm })}^T$ or ${\mathcal{Q}}^{\pm }={(X_1^{\pm },X_3^{\pm },\dots ,X_{2N-1}^{\pm })}^T$, where

$$\begin{array}{c}\hfill X_s^{\pm }={\displaystyle \frac{1}{s!}}{\partial }_{k_1}^s\rho (\pm k_1,l+1/2){|}_{k_1=0},\end{array}$$

(31)

and here we replace ${(\delta \pm k_1)}^l$ by ${(\delta \pm k_1)}^{l+1/2}$ to guarantee the derivative ${\partial }_{k_1}^s{(\delta \pm k_1)}^{l+1/2}$${|}_{k_1=0}\ne 0$. We list explicit expressions on $X_s$ for $s=0,1,2,3,4$, which are

$$\begin{array}{cc}\hfill & X_0^{\pm }=\rho (0,l+1/2),X_1^{\pm }=\pm \rho (0,l+1/2)A_1,\hfill \end{array}$$

(32a)

$$\begin{array}{cc}\hfill & X_2^{\pm }=\rho (0,l+1/2)(A_1^2-A_2)/2,X_3^{\pm }=\pm \rho (0,l+1/2)(A_1^3-3A_1{A}_2+A_3)/6,\hfill \end{array}$$

(32b)

$$\begin{array}{cc}\hfill & X_4^{\pm }=\rho (0,l+1/2)(A_1^4-6A_1^2{A}_2+3A_2^2+4A_1{A}_3-A_4)/24,\hfill \end{array}$$

(32c)

where $A_{\lambda }=(\lambda -1)!(B_{\lambda }+(l+1/2){\delta }^{-\lambda })$ with $B_{\lambda }={\displaystyle \sum _{i=n_0}^{n-1}}{a}_i^{-\lambda }+{\displaystyle \sum _{j=m_0}^{m-1}}{b}_j^{-\lambda }$. If omitting the sign before $X_{2\lambda -1}^{\pm }$, one can easily find that $X_{\lambda }^{\pm }={\partial }_{A_1}{X}_{\lambda +1}^{\pm }$ for arbitrary positive integer $\lambda$.

To proceed, we must consider the Casoratian solutions composed by $X_{2\lambda }^{\pm }$. For the nlpKdV (1) one can obtain the first two simplest rational solutions as

$$\begin{array}{cc}\hfill & u=1/A_1,\hfill \end{array}$$

(33a)

$$\begin{array}{cc}\hfill & u=-12A_1^2{\delta }^3/[1+2l-(4A_1^3-A_3){\delta }^3].\hfill \end{array}$$

(33b)

For the nlpmKdV (2), the rational solutions can be recovered through (11), where the first two simplest rational solutions for (12) read

$$\begin{array}{cc}\hfill & w=1-1/\left(2A_1\right),\hfill \end{array}$$

(34a)

$$\begin{array}{cc}\hfill & w=1-12A_1\delta (1-A_1\delta )/[1+2l-(4A_1^3-A_3){\delta }^3].\hfill \end{array}$$

(34b)

4. Dynamics of Rational Solutions

In this section, we just discuss the dynamical behaviour of the obtained rational solutions (33). Without loss of generality, we take $n_0=m_0=0$. In what follows, we take two cases as examples: (1) $a_i=2^{i}a$ and $b_j=2^{j}b$ and (2) $a_i=csc\left(ai\right)$ and $b_j=csc\left(bj\right)$, where a and b are real constants.

In the first case, the nlpKdV Equation (1) becomes

$$\begin{array}{c}\hfill (2^{n}a-2^{m}b+\widehat{u}-\tilde{u})(2^{n}a+2^{m}b+u-\widehat{\tilde{u}})=4^n{a}^2-4^m{b}^2,\end{array}$$

(35)

and solution (33a) gives rise to

$$\begin{array}{c}\hfill u={\displaystyle \frac{2\delta }{1+2l+4\delta [(1-2^{-n})a^{-1}+(1-2^{-m})b^{-1}]}}.\end{array}$$

(36)

It can be noticed that $u\to {\displaystyle \frac{2\delta }{1+2l+4\delta (a^{-1}+b^{-1})}}$ as $(n,m)\to (+\infty ,+\infty )$ and $u\to 0$ as $(n,m)\to (-\infty ,-\infty )$, see Figure 1a. For fixed m, u appears as a kink shape and one can identify the solution (36) by its moving inflexion point trace $1+2l+4\delta [(1+2^{-n})a^{-1}+(1-2^{-m})b^{-1}]=0$, and the slope and the value of u at the inflection point, which are

$$\begin{array}{c}\hfill {\displaystyle \frac{ab\delta ln2}{8b\delta +2a[b(1+2l)+4\delta (1-2^{-m})]}},\end{array}$$

(37)

and

$$\begin{array}{c}\hfill {u|}_{\mathrm{inflection}}={\displaystyle \frac{ab\delta }{4\delta (a+b)+a[b(1+2l)-2^{2-m}\delta ]}},\end{array}$$

(38)

respectively. These are enough to understand how the shape of solution (36) is related to $\delta$ (see Figure 1b).

For the solution (33b), the formula is too complicated to be analyzed here. So, we only depict this solution in Figure 2.

In the second case, the nlpKdV Equation (1) becomes

$$\begin{array}{c}\hfill (csc\left(an\right)-csc\left(bm\right)+\widehat{u}-\tilde{u})(csc\left(an\right)+csc\left(bm\right)+u-\widehat{\tilde{u}})={csc}^2\left(an\right)-{csc}^2\left(bm\right),\end{array}$$

(39)

and solution (33a) gives rise to

$$\begin{array}{c}\hfill u={\displaystyle \frac{2\delta }{1+2l-2\delta \left(csc\frac{a}{2}sin\frac{an}{2}sin\frac{a(1-n)}{2}+csc\frac{b}{2}sin\frac{bm}{2}sin\frac{b(1-m)}{2}\right)}},\end{array}$$

(40)

which is oscillatory due to the sine in the denominator, and reaches its extrema along points $(n,m)=\left(\right(2\kappa \pi +a)/(2a),(2\ell \pi +b)/(2b\left)\right),\kappa ,\ell \in \mathbb{Z}$. For fixed m (or n), solution (40) is a travelling wave with period $2\pi /a$ (or $2\pi /b$). We depict such a wave in Figure 3. Solution (33b) in the second case is depicted in Figure 4.

5. Conclusions

In this paper, we consider rational solutions of the nlpKdV Equation (1) and the nlpmKdV Equation (2) by using the bilinear method. To this end, we introduce a parameter $\delta$ and derive the equivalent forms of these two equations, which are Equations (7) and (12). Inserting the rational transformations $u=g/f$ and $w=h/f$ into Equations (7) and (12), bilinear forms and Casoratian solutions are then derived. According to the different cases of the eigenvalue set of the matrices ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$, we have obtained three types of exact solutions of the objective equations, where the rational solutions are constructed by considering the case of auxiliary matrices ${\mathbf{\Gamma }}_{\left[m\right]}$ and ${\mathbf{\Gamma }}_{\left[n\right]}$ being formed (21) with $k_1=0$. Compared with the autonomous integrable lattice equations, the nonautonomous equations exhibit richer structures by setting different forms of lattice parameters $p_n$ and $q_m$. To understand the dynamical behaviors of the obtained rational solutions, we take two forms of $p_n$ and $q_m$, where one is an exponential function and the other is related to a trigonometric function. Rational solutions arising from the first and second forms show kink shape and periodic structures, respectively. This approach shown in the present paper can also be used to discuss rational solutions of the nonautonomous ABS lattice list.