1. Introduction
Recently, the discrete nonlinear differential difference equations, taken as the spatially discrete counterparts of nonlinear partial differential equations, have received widespread attention due to their reach applicability in different physical phenomena [1–14]. The motivation of studying discrete equations is that they can maintain certain properties of original corresponding continuous equations, and may be applied in numerical simulation [15]. In 1977, the following discrete mKdV lattice equation has been proposed as [16]:1 ) is related to the continuous mKdV equation. Since equation (1 ) was proposed, some extended discrete mKdV lattice equations have also been presented and studied [17–27]. In [18, 19], an extended discrete mKdV lattice equation is proposed as1 ) is just equation (2 ). When α = −1, we only replace t with − t, at this time, the above two equations are completely equivalent. In [17], the author has compared two special cases of equation (2 ) when α = 1 and α = 0, and has given their different Lax pairs and solitary wave solutions, and also has found that the first-order solitary wave solutions of the two cases are very different, the solitary wave solutions of the case α = 1 are closer to the continuous case which has sech-shaped solitary wave solutions, while the case α = 0 is interesting because it has non-vanishing waves with a sech'shape [17]. In [18], many explicit rational exact solutions including some new solitary wave solutions of equation (2 ) have been obtained by using a new generalized ansätz method. In [19, 20], the periodic solutions and solitary solutions of equation (2 ) have been gained by the modified Jacobian elliptic function method. In [21, 22], the new solitary wave solutions have been derived by the hyperbolic function method. In [23], the rational formal solutions have been investigated by the rational expansion method. In [24, 25], when α ≠ 0, equation (2 ) has been mapped to the famous mKdV and KdV equations by the continuous conditions2 ) and its corresponding Lax-integrable coupled extension, which is corresponding to the coupled KdV and mKdV equations by using the continuous limit, have been proposed, and some periodic and solitary wave solutions have been constructed via the function-expansion method. However, when α = 0, equations (2 ) and (5 ) are equivalent by just replacing t with − t, and equation (5 ) can not correspond to the continuous mKdV equation under the transformation (3 ), but it still corresponds to the continuous KdV equation under the transformation (4 ). In [26], an exact two-soliton solution of equation (2 ) has been derived by using the Hirota direct approach when α is an arbitrary real constant, and the completely elastic interaction between two solitons has been discussed. In [27], the one-fold Darboux transformation (DT) and explicit solutions of equation (2 ) have been obtained by one author of this paper when α is an arbitrary real constant, and when α ≠ 1, its DT is difficult to be extended to higher levels, so that it is difficult for us to obtain its higher-order soliton solution.
$\begin{eqnarray}{u}_{n,t}=(1\pm {u}_{n}^{2})({u}_{n+1}-{u}_{n-1}),\end{eqnarray}$
where un = u(n, t) is the function of variables n and t, and in the continuous limit, equation ( $\begin{eqnarray}{u}_{n,t}=(\alpha -{u}_{n}^{2})({u}_{n+1}-{u}_{n-1}),\end{eqnarray}$
where α is an arbitrary real constant. When α = ±1, equation ( $\begin{eqnarray}\begin{array}{rcl}{u}_{n} & = & \delta u((n+2\alpha t)\delta ,\displaystyle \frac{1}{3}\alpha {\delta }^{3}t)\\ & & +O({\delta }^{2})\triangleq \delta u(x,\tau )+O({\delta }^{2}),\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{u}_{n} & = & 1+\sqrt{1+\alpha }+{\delta }^{2}u((n-4-4\sqrt{1+\alpha })\delta ,\\ & & \displaystyle \frac{2}{3}(1+\sqrt{1+\alpha }){\delta }^{3}t)+O({\delta }^{3})\\ & & \triangleq 1+\sqrt{1+\alpha }+{\delta }^{2}u(x,\tau )+O({\delta }^{3}),\end{array}\end{eqnarray}$
respectively, and moreover, the Lax pair of equation (Based on previous studies, we know that there has been considerable work done on considering equation (2 ) with the case α = 1 [18–25] and the case α ≠ 0 [26, 27]. From the known results in [17], we can clearly see that the two cases α = 1 and α = 0 of equation (2 ) are very different. However, little work has been done to apply the generalized (m, 2N − m)-fold DT technique to give various exact solutions of equation (2 ) with α = 0. Therefore, it is necessary to do further research on this case, in what follows, we will discuss equation (2 ) with α = 0 [17], that is5 ) is the second equation in [17], for the convenience of our later discussion, we have changed the second equation in [17] and its corresponding Lax pair, which admits5 ) is equivalent to the compatibility condition of Lax pair (6 ) and (7 ).
$\begin{eqnarray}{u}_{n,t}={u}_{n}^{2}({u}_{n+1}-{u}_{n-1}),\end{eqnarray}$
which is a little different from the equation in the form in [17], in fact, if we replace un with − (1 + un), equation ( $\begin{eqnarray}E{\phi }_{n}={U}_{n}{\phi }_{n}=\left(\begin{array}{cc}\lambda & -{u}_{n}\\ {u}_{n} & 0\end{array}\right){\phi }_{n},\end{eqnarray}$
$\begin{eqnarray}{\phi }_{n,t}={V}_{n}{\phi }_{n}=\left(\begin{array}{cc}{u}_{n-1}{u}_{n} & -\lambda {u}_{n}\\ \lambda {u}_{n-1} & {u}_{n}{u}_{n-1}-{\lambda }^{2}\end{array}\right){\phi }_{n},\end{eqnarray}$
where λ is the spectral parameter unrelated to t, E is the shift operator satisfying Ef(n, t) = f(n + 1, t), E−1f(n, t) = f(n − 1, t), and ${\phi }_{n}={\left({\varphi }_{n},{\psi }_{n}\right)}^{{\rm{T}}}$ is an eigenfunction vector. With the aid of symbolic computation, we can easily find that equation (It should be noted that equation (2 ) can be continuous to the mKdV and KdV equations through the continuous limit when α is a non-zero real constant [24, 25]. However, when α = 0, equation (2 ) is just equation (5 ), which can only be continuous to the KdV equation by using the continuous limit, not related to the continuous mKdV equation. Therefore, it is more appropriate for us to describe equation (5 ) as a discrete KdV equation when α = 0, hereafter referred to as the discrete KdV equation. And later, by solving equation (5 ), we will find that the properties of soliton and rational solutions of equation (5 ) are very similar to the continuous KdV equation, so that is the main reason why we call it the discrete KdV equation. The discrete generalized (m, 2N − m)-fold DT [28, 29] or generalized (m, N − m)-fold DT [30] is an effective method to solve the discrete integrable equation, and the main advantage of this method is that it can not only give soliton solutions and rational solutions but also give a variety of mixed solutions. According to the information we have, the relevant hierarchy and generalized (m, 2N − m)-fold DT of equation (5 ) have not been studied, and its soliton, rational and hybrid solutions, and relevant asymptotic state analysis have not been considered before.
Therefore in this work, we will further study equation (5 ) and give some new results and physical phenomena. The article is organized as follows. In section 2 , the hierarchy related to equation (5 ) will be constructed via Tu scheme technique [5]. In section 3 , we will study the continuous limit of two new higher-order discrete equations in this hierarchy. In section 4 , the discrete generalized (m, 2N − m)-fold DT will be constructed. In section 5 , the soliton solutions, rational solutions, and their mixed solution of equation (5 ) are obtained via the resulting DT. Meanwhile, we will also discuss their physical and asymptotic properties through asymptotic analysis. The last section is a brief summary.
2. A discrete hierarchy related to equation (5 )
In this section, we will construct the discrete hierarchy related to equation (5 ) from the discrete matrix spectral problem (6 ) via Tu scheme technique [5]. First, starting from (7 ), we take8 ). The first few expressions of them are written as8 ), we obtain6 ), from which we can obtain the following discrete equation hierarchy as
$\begin{eqnarray*}\begin{array}{l}{P}_{n}=\left(\begin{array}{cc}{A}_{n} & {B}_{n}\\ {C}_{n} & -{A}_{n}\end{array}\right)\\ \quad =\left(\begin{array}{cc}\sum _{j=0}^{\infty }{A}_{n}^{(j)}{\lambda }^{-2j} & \sum _{j=0}^{\infty }{B}_{n}^{(j)}{\lambda }^{-2j+1}\\ \sum _{j=0}^{\infty }{C}_{n}^{(j)}{\lambda }^{-2j+1} & -\sum _{j=0}^{\infty }{A}_{n}^{(j)}{\lambda }^{-2j}\end{array}\right).\end{array}\end{eqnarray*}$
By seeking the solution of the stationary zero curvature equation Pn+1Un − UnPn = 0, we have $\begin{eqnarray}\left\{\begin{array}{l}{A}_{n+1}^{(j)}-{A}_{n}^{(j)}+{u}_{n}({B}_{n+1}^{(j)}+{C}_{n}^{(j)})=0,\\ {u}_{n}({A}_{n+1}^{(j)}+{A}_{n}^{(j)})+{B}_{n}^{(j+1)}=0,\\ -{u}_{n}({A}_{n+1}^{(j)}+{A}_{n}^{(j)})+{C}_{n+1}^{(j+1)}=0,\\ {B}_{n}^{(j)}+{C}_{n+1}^{(j)}=0,\end{array}\right.\end{eqnarray}$
where ${A}_{n}^{(j)},$ ${B}_{n}^{(j)}$ and ${C}_{n}^{(j)}$ are functions related to un. Next, we take ${A}_{n}^{(0)}=\tfrac{1}{2}$, then the rest of ${A}_{n}^{(j)}$, ${B}_{n}^{(j)}$, ${C}_{n}^{(j)}$ will be gained by recursive relation ( $\begin{eqnarray*}\begin{array}{rcl}{B}_{n}^{(0)} & = & 0,{C}_{n}^{(0)}=0,{A}_{n}^{(1)}={u}_{n}{u}_{n-1},\\ {B}_{n}^{(1)} & = & -{u}_{n},{C}_{n}^{(1)}={u}_{n-1},\\ {A}_{n}^{(2)} & = & {u}_{n-1}^{2}{u}_{n}^{2}\\ & & +{u}_{n-1}{u}_{n}^{2}{u}_{n+1}+{u}_{n}{u}_{n-1}^{2}{u}_{n-2},\\ {B}_{n}^{(2)} & = & -{u}_{n}^{2}({u}_{n+1}+{u}_{n-1}),\\ {C}_{n}^{(2)} & = & {u}_{n-1}^{2}({u}_{n}+{u}_{n-2}),\\ {A}_{n}^{(3)} & = & {u}_{n}^{3}{u}_{n-1}^{3}+{u}_{n}{u}_{n-1}^{3}{u}_{n-2}^{2}+2{u}_{n}^{2}{u}_{n-1}^{3}{u}_{n-2}\\ & & +2{u}_{n}^{3}{u}_{n-1}^{2}{u}_{n+1}+{u}_{n}{u}_{n-1}^{2}{u}_{n-2}^{2}{u}_{n-3}\\ & & +{u}_{n-1}{u}_{n}^{3}{u}_{n+1}^{2}+{u}_{n-1}{u}_{n}^{2}{u}_{n+1}^{2}{u}_{n+2}\\ & & +{u}_{n-2}{u}_{n-1}^{2}{u}_{n}^{2}{u}_{n+1},\\ {B}_{n}^{(3)} & = & -{u}_{n}^{2}({u}_{n}{\left({u}_{n+1}+{u}_{n-1}\right)}^{2}\\ & & +{u}_{n+1}^{2}{u}_{n+2}+{u}_{n-2}{u}_{n-1}^{2}),\\ {C}_{n}^{(3)} & = & {u}_{n-1}^{2}({u}_{n-1}{\left({u}_{n}+{u}_{n-2}\right)}^{2}\\ & & +{u}_{n}^{2}{u}_{n+1}+{u}_{n-3}{u}_{n-2}^{2}),\\ {B}_{n}^{(4)} & = & -{u}_{n}^{2}({\left({u}_{n+1}+{u}_{n-1}\right)}^{3}{u}_{n}^{2}+2({u}_{n+1}+{u}_{n-1})\\ & & \times ({u}_{n-2}{u}_{n-1}^{2}+{u}_{n+1}^{2}{u}_{n+2}){u}_{n}\\ & & +{u}_{n-2}^{2}{u}_{n-1}^{3}+{u}_{n-3}{u}_{n-2}^{2}{u}_{n-1}^{2}\\ & & +{u}_{n+1}^{2}{u}_{n+2}^{2}({u}_{n+1}+{u}_{n+3})),\\ {C}_{n}^{(4)} & = & {u}_{n-1}^{2}({\left({u}_{n}+{u}_{n-2}\right)}^{3}{u}_{n-1}^{2}\\ & & +2({u}_{n}+{u}_{n-2})({u}_{n-3}{u}_{n-2}^{2}+{u}_{n}^{2}{u}_{n+1}){u}_{n-1}\\ & & +{u}_{n-3}^{2}{u}_{n-2}^{3}+{u}_{n-4}{u}_{n-3}^{2}{u}_{n-2}^{2}\\ & & +{u}_{n}^{2}{u}_{n+1}^{2}({u}_{n}+{u}_{n+2})),....\end{array}\end{eqnarray*}$
Then by using the matrix Pn we define a new matrix ${P}_{n}^{(m)}$ as $\begin{eqnarray*}\begin{array}{l}{P}_{n}^{(m)}={\lambda }^{2m}{P}_{n}\\ \quad =\left(\begin{array}{cc}\sum _{j=0}^{m}{A}_{n}^{(j)}{\lambda }^{2m-2j} & \sum _{j=0}^{m}{B}_{n}^{(j)}{\lambda }^{2m-2j+1}\\ \sum _{j=0}^{m}{C}_{n}^{(j)}{\lambda }^{2m-2j+1} & -\sum _{j=0}^{m}{A}_{n}^{(j)}{\lambda }^{2m-2j}\end{array}\right), m\geqslant 0.\end{array}\end{eqnarray*}$
According to the relation ( $\begin{eqnarray*}\begin{array}{l}{{EP}}_{n}^{(m)}{U}_{n}-{U}_{n}{P}_{n}^{(m)}\\ \quad =\left(\begin{array}{cc}0 & {B}_{n}^{(m+1)}\\ -{C}_{n+1}^{(m+1)} & 0\end{array}\right).\end{array}\end{eqnarray*}$
In order to gain the discrete KdV hierarchy we want, we need to find a matrix to modify ${P}_{n}^{(m)}$, so a specially modified matrix can be defined as $\begin{eqnarray*}{{\rm{\Delta }}}_{n}^{(m)}=\left(\begin{array}{cc}0 & 0\\ 0 & 2{A}_{n}^{(m)}\end{array}\right),\end{eqnarray*}$
and ${V}_{n}^{(m)}$ is defined as ${V}_{n}^{(m)}={P}_{n}^{(m)}+{{\rm{\Delta }}}_{n}^{(m)}$. Then we have $\begin{eqnarray*}\begin{array}{l}{{EV}}_{n}^{(m)}{U}_{n}-{U}_{n}{V}_{n}^{(m)}\\ \quad =\,\left(\begin{array}{cc}0 & 2{u}_{n}{A}_{n}^{(m)}+{B}_{n}^{(m+1)}\\ 2{u}_{n}{A}_{n+1}^{(m)}-{C}_{n+1}^{(m+1)} & 0\end{array}\right).\end{array}\end{eqnarray*}$
Now the time evolution of φn can be expressed as ${\phi }_{n,{t}_{m}}={V}_{n}^{(m)}{\phi }_{n}$, and it meets the compatibility condition $\begin{eqnarray*}{U}_{n,{tm}}=({{EV}}_{n}^{(m)}){U}_{n}-{U}_{n}{V}_{n}^{(m)},m\geqslant 0,\end{eqnarray*}$
with equation ( $\begin{eqnarray}{u}_{n,{t}_{m}}=2{u}_{n}{A}_{n+1}^{(m)}-{C}_{n+1}^{(m+1)}.\end{eqnarray}$
| I | (I)When m = 1, equation ( $\begin{eqnarray}{u}_{n,{t}_{1}}=2{u}_{n}{A}_{n+1}^{(1)}-{C}_{n+1}^{(2)}={u}_{n}^{2}({u}_{n+1}-{u}_{n-1}),\end{eqnarray}$ which is the same as equation ( $\begin{eqnarray}{\phi }_{n,{t}_{1}}={V}_{n}^{(1)}{\phi }_{n}=\left(\begin{array}{cc}\displaystyle \frac{{\lambda }^{2}}{2}+{u}_{n}{u}_{n-1} & -\lambda {u}_{n}\\ \lambda {u}_{n-1} & -\displaystyle \frac{{\lambda }^{2}}{2}+{u}_{n}{u}_{n-1}\end{array}\right){\phi }_{n}.\end{eqnarray}$ However, in fact, equations ( |
| II | (II)When m = 2, equation ( $\begin{eqnarray}\begin{array}{rcl}{u}_{n,{t}_{2}} & = & 2{u}_{n}{A}_{n+1}^{(2)}-{C}_{n+1}^{(3)}={u}_{n}^{2}({u}_{n}{u}_{n+1}^{2}\\ & & +{u}_{n+1}^{2}{u}_{n+2}-{u}_{n}{u}_{n-1}^{2}-{u}_{n-1}^{2}{u}_{n-2}),\end{array}\end{eqnarray}$ and the time part of its Lax pair is $\begin{eqnarray}\begin{array}{l}{\phi }_{n,{t}_{2}}={V}_{n}^{(2)}{\phi }_{n}\\ \quad =\left(\begin{array}{cc}{A}_{n}^{(0)}{\lambda }^{4}+{A}_{n}^{(1)}{\lambda }^{2}+{A}_{n}^{(2)} & {B}_{n}^{(0)}{\lambda }^{5}+{B}_{n}^{(1)}{\lambda }^{3}+{B}_{n}^{(2)}\lambda \\ {C}_{n}^{(0)}{\lambda }^{5}+{C}_{n}^{(1)}{\lambda }^{3}+{C}_{n}^{(2)}\lambda & -{A}_{n}^{(0)}{\lambda }^{4}-{A}_{n}^{(1)}{\lambda }^{2}+{A}_{n}^{(2)}\end{array}\right){\phi }_{n}\\ \quad \equiv \left(\begin{array}{cc}{V}_{11}(n) & {V}_{12}(n)\\ {V}_{13}(n) & {V}_{14}(n)\end{array}\right){\phi }_{n},\end{array}\end{eqnarray}$ with $\begin{eqnarray*}\begin{array}{l}{V}_{11}(n)=\displaystyle \frac{{\lambda }^{4}}{2}+{u}_{n}{u}_{n-1}{\lambda }^{2}+{u}_{n}{u}_{n-1}\\ \quad \times ({u}_{n}{u}_{n-1}+{u}_{n}{u}_{n+1}+{u}_{n-1}{u}_{n-2}),\\ {V}_{12}(n)=-{\lambda }^{3}{u}_{n}-\lambda {u}_{n}^{2}({u}_{n+1}+{u}_{n-1}),\\ {V}_{13}(n)={\lambda }^{3}{u}_{n-1}+\lambda {u}_{n-1}^{2}({u}_{n}+{u}_{n+2}),\\ {V}_{14}(n)=-\displaystyle \frac{{\lambda }^{4}}{2}-{u}_{n}{u}_{n-1}{\lambda }^{2}+{u}_{n}{u}_{n-1}\\ \quad \times ({u}_{n}{u}_{n-1}+{u}_{n}{u}_{n+1}+{u}_{n-1}{u}_{n-2}).\end{array}\end{eqnarray*}$ |
| III | (III)When m = 3, equation ( $\begin{eqnarray}\begin{array}{rcl}{u}_{n,{t}_{3}} & = & 2{u}_{n}{A}_{n+1}^{(3)}-{C}_{n+1}^{(4)}\\ & = & {u}_{n}^{2}[({u}_{n+1}-{u}_{n-1}){\left({u}_{n+1}+{u}_{n-1}\right)}^{2}{u}_{n}^{2}\\ & & +2{u}_{n}({u}_{n+1}^{3}{u}_{n+2}-{u}_{n-1}^{3}{u}_{n-2})\\ & & +{u}_{n+1}^{2}{u}_{n+2}^{2}({u}_{n+1}+{u}_{n+3})\\ & & -{u}_{n-1}^{2}{u}_{n-2}^{2}({u}_{n-1}+{u}_{n-3})],\end{array}\end{eqnarray}$ and the time part of its Lax pair is $\begin{eqnarray}\begin{array}{l}{\phi }_{n,{t}_{3}}={V}_{n}^{(3)}{\phi }_{n}\\ =\left(\begin{array}{cc}{A}_{n}^{(0)}{\lambda }^{6}+{A}_{n}^{(1)}{\lambda }^{4}+{A}_{n}^{(2)}{\lambda }^{2}+{A}_{n}^{(3)} & {B}_{n}^{(0)}{\lambda }^{7}+{B}_{n}^{(1)}{\lambda }^{5}+{B}_{n}^{(2)}{\lambda }^{3}+{B}_{n}^{(3)}\lambda \\ {C}_{n}^{(0)}{\lambda }^{7}+{C}_{n}^{(1)}{\lambda }^{5}+{C}_{n}^{(2)}{\lambda }^{3}+{C}_{n}^{(3)}\lambda & -{A}_{n}^{(0)}{\lambda }^{6}-{A}_{n}^{(1)}{\lambda }^{4}-{A}_{n}^{(2)}{\lambda }^{2}+{A}_{n}^{(3)}\end{array}\right){\phi }_{n}\\ \equiv \left(\begin{array}{cc}{V}_{21}(n) & {V}_{22}(n)\\ {V}_{23}(n) & {V}_{24}(n)\end{array}\right){\phi }_{n},\end{array}\end{eqnarray}$ with $\begin{eqnarray*}\begin{array}{l}{V}_{21}(n)=\displaystyle \frac{{\lambda }^{6}}{2}+{\lambda }^{4}{u}_{n}{u}_{n-1}+{\lambda }^{2}\\ \times \,({u}_{n-1}^{2}{u}_{n}^{2}+{u}_{n-1}{u}_{n}^{2}{u}_{n+1}+{u}_{n}{u}_{n-1}^{2}{u}_{n-2})\\ +{u}_{n}{u}_{n-1}[{\left({u}_{n}+{u}_{n-2}\right)}^{2}{u}_{n-1}^{2}\\ +(2{u}_{n}^{2}{u}_{n+1}+{u}_{n}{u}_{n-2}{u}_{n+1}\\ +{u}_{n-3}{u}_{n-2}^{2}){u}_{n-1}+{u}_{n}{u}_{n+1}^{2}({u}_{n}+{u}_{n+2})],\\ {V}_{22}(n)=-{\lambda }^{5}{u}_{n}-{\lambda }^{3}{u}_{n}^{2}({u}_{n+1}+{u}_{n-1})\\ -\lambda {u}_{n}^{2}[{u}_{n}{\left({u}_{n+1}+{u}_{n-1}\right)}^{2}+{u}_{n+1}^{2}{u}_{n+2}+{u}_{n-2}{u}_{n-1}^{2}],\\ {V}_{23}(n)={\lambda }^{5}{u}_{n-1}+{\lambda }^{3}{u}_{n-1}^{2}({u}_{n}+{u}_{n-2})\\ +\lambda {u}_{n-1}^{2}[{u}_{n-1}{\left({u}_{n}+{u}_{n-2}\right)}^{2}+{u}_{n}^{2}{u}_{n+1}+{u}_{n-3}{u}_{n-2}^{2}],\\ {V}_{24}(n)=-\displaystyle \frac{{\lambda }^{6}}{2}-{\lambda }^{4}{u}_{n}{u}_{n-1}-{\lambda }^{2}\\ \times \,({u}_{n-1}^{2}{u}_{n}^{2}+{u}_{n-1}{u}_{n}^{2}{u}_{n+1}+{u}_{n}{u}_{n-1}^{2}{u}_{n-2})\\ +{u}_{n}{u}_{n-1}[{\left({u}_{n}+{u}_{n-2}\right)}^{2}{u}_{n-1}^{2}\\ +(2{u}_{n}^{2}{u}_{n+1}+{u}_{n}{u}_{n-2}{u}_{n+1}\\ +{u}_{n-3}{u}_{n-2}^{2}){u}_{n-1}+{u}_{n}{u}_{n+1}^{2}({u}_{n}+{u}_{n+2})].\end{array}\end{eqnarray*}$ |
When we continue to take the value of m, we will get a series of new discrete equations. These equations may also have some good properties worthy of our further study, such as integrability, Hamiltonian structure and conservation law as done in [5, 28, 29], which are not our main goal in this paper. Our next main task is to study various exact solutions of equation (5 ) via the discrete generalized (m, 2N − m)-fold DT [28, 29] proposed by one of the authors in this paper and analyze their physical properties. Before doing this, we first use the continuous limit technique to correspond the first three equations of this hierarchy to the continuous KdV equation with physical significance.
3. Continuous limit
As we all know, the continuous limit of discrete systems is one of the remarkably important research areas in soliton theory [9–12, 24, 25]. Since the continuity limit of equation (5 ), as a special case of equation (2 ), has been discussed [24, 25], next we mainly discuss the continuous limits of equations (12 ) and (14 ). Using the same approach, equation (12 ) is mapped into
$\begin{eqnarray}({u}_{\tau }+6{{uu}}_{x}+{u}_{{xxx}}){\delta }^{3}+O({\delta }^{5})=0,\end{eqnarray}$
under the transformation $\begin{eqnarray}\begin{array}{rcl}{u}_{n} & = & 1+\displaystyle \frac{{\delta }^{2}}{2}u[(n+12t)\delta ,-4{\delta }^{3}t]\\ & & +O({\delta }^{3})\triangleq 1+\displaystyle \frac{{\delta }^{2}}{2}u(x,\tau )+O({\delta }^{3}),\end{array}\end{eqnarray}$
which happens to be the famous KdV equation if O(δ5) is ignored and τ is changed into t. Here δ is an arbitrary small parameter.When the function un meets the following condition14 ) is also transformed into (16 ). If we neglect its O(δ5), equation (14 ) is also continuous to the KdV equation. Unfortunately, we can not correspond the higher-order discrete equations in this hierarchy to the higher-order KdV equation. Since equations (12 ) and (14 ) also continuously correspond to the KdV equation, we also call them the discrete KdV equations.
$\begin{eqnarray}\begin{array}{rcl}{u}_{n} & = & 1+\displaystyle \frac{{\delta }^{2}}{2}u[(n+60t)\delta ,-30{\delta }^{3}t]\\ & & +O({\delta }^{3})\triangleq 1+\displaystyle \frac{{\delta }^{2}}{2}u(x,\tau )+O({\delta }^{3}).\end{array}\end{eqnarray}$
Equation (Through the above process, we find that the first few discrete equations in this hierarchy can continuously correspond to the continuous KdV equation, so it is appropriate to call the hierarchy in section 2 the discrete KdV hierarchy. The discrete equations in this hierarchy only can correspond to the continuous KdV equation, not correspond to the continuous mKdV equation, although equation (5 ) is a special case of discrete mKdV equation (2 ), in fact, it only can correspond to the continuous KdV equation. Therefore, that is another reason why we call this hierarchy the discrete KdV hierarchy. Continuing this process, we can get more discrete KdV equations from the above hierarchy. Although they are all discrete KdV equations, we know that equations (12 ) and (14 ) have more nonlinear terms than equation (5 ), so it is possible that these higher-order equations are more accurate and effective as discrete numerical schemes of continuous KdV equations in some aspects. Whether these equations have new properties is worth further discussion. As far as we know, when α ≠ 0, equation (2 ) has been investigated via DT [27], and when α = 0, the DT of equation (2 ) can not be reduced to the DT of equation (5 ), so when α = 0, the special equation (5 ) need further investigation, which is also our main task below.
4. Discrete generalized (m, 2N − m)-fold DT
As mentioned above, equation (5 ) is a special case of (2 ) when α = 0. For this special case, its generalized (m, 2N − m)-fold DT has not been studied, and the generalized (m, 2N − m)-fold DT of newly obtained equations (12 ) and (14 ) have not been studied. Therefore, in this section, we take equation (5 ) as an example to discuss its generalized (m, 2N − m)-fold DT through the Taylor expansion and a limit procedure [28, 29]. First, we consider a gauge transformation as
$\begin{eqnarray}{\tilde{\phi }}_{n}={T}_{n}{\phi }_{n},\end{eqnarray}$
which can yield the following equations $\begin{eqnarray}\begin{array}{l}{\tilde{\phi }}_{n+1}={\tilde{U}}_{n}{\tilde{\phi }}_{n}={T}_{n+1}{U}_{n}{T}_{n}^{-1}{\tilde{\phi }}_{n},\\ {\tilde{\phi }}_{n,t}={\tilde{V}}_{n}{\tilde{\phi }}_{n}=({T}_{n,t}+{V}_{n}{T}_{n}){T}_{n}^{-1}{\tilde{\phi }}_{n}.\end{array}\end{eqnarray}$
The forms of ${\tilde{U}}_{n}$, ${\tilde{V}}_{n}$ are the same as Un, Vn, except that the new solution ${\tilde{u}}_{n}$ is used to replace the old solution un. Then we define a special Darboux matrix Tn as $\begin{eqnarray}\begin{array}{l}{T}_{n}=\left(\begin{array}{cc}{a}_{n} & {b}_{n}\\ {c}_{n} & {d}_{n}\end{array}\right)\\ \quad =\left(\begin{array}{cc}{\lambda }^{2N}+\sum _{j=1}^{N}{a}_{n}^{(2j-2)}{\lambda }^{2j-2} & \sum _{j=1}^{N}{b}_{n}^{(2j-1)}{\lambda }^{2j-1}\\ \sum _{j=1}^{N}{c}_{n}^{(2j-1)}{\lambda }^{2j-1} & \sum _{j=1}^{N}{d}_{n}^{(2j)}{\lambda }^{2j}+{a}_{n}^{(0)}\end{array}\right),\end{array}\end{eqnarray}$
where N is a positive integer, and ${a}_{n}^{(0)}$, ${a}_{n}^{(2j-2)}$, ${b}_{n}^{(2j-1)}$, ${c}_{n}^{(2j-1)}$ and ${d}_{n}^{(2j)}(j=1,2,...,N)$ are 4N functions of the variables n and t which are determined later. Let $T({\lambda }_{i}+\varepsilon )\,={T}_{n}^{(0)}+{T}_{n}^{(1)}\varepsilon +\cdots +{T}_{n}^{({m}_{i})}{\varepsilon }^{{m}_{i}}$, ${\phi }_{n}({\lambda }_{i}+\varepsilon )={\phi }_{n}^{(0)}({\lambda }_{i})+{\phi }_{n}^{(1)}({\lambda }_{i})+{\phi }_{n}^{(2)}({\lambda }_{i}){\varepsilon }^{2}+{\phi }_{n}^{(3)}({\lambda }_{i}){\varepsilon }^{3}+\cdots $, then we have $\begin{eqnarray}\begin{array}{l}T({\lambda }_{i}+\varepsilon ){\phi }_{n}({\lambda }_{i}+\varepsilon )\\ \quad =\sum _{k=0}^{+\infty }\sum _{j=0}^{k}{T}^{(j)}({\lambda }_{i}){\phi }_{n}^{(k-j)}({\lambda }_{i}){\varepsilon }^{k},\end{array}\end{eqnarray}$
in which ϵ is an arbitrary small parameter. In order to find the solutions of the 4N unknowns of Tn when m(m < 2N) spectral parameters λi (i = 1, 2,…,m) are taken, we only need to solve the following 4N equations $\begin{eqnarray*}\left\{\begin{array}{l}{T}_{n}^{(0)}({\lambda }_{i}){\phi }_{n}^{(0)}({\lambda }_{i})=0,\\ {T}_{n}^{(0)}({\lambda }_{i}){\phi }_{n}^{(1)}({\lambda }_{i})+{T}_{n}^{(1)}({\lambda }_{i}){\phi }_{n}^{(0)}({\lambda }_{i})=0,\\ {T}_{n}^{(0)}({\lambda }_{i}){\phi }_{n}^{(2)}({\lambda }_{i})+{T}_{n}^{(1)}({\lambda }_{i}){\phi }_{n}^{(1)}({\lambda }_{i})+{T}_{n}^{(2)}({\lambda }_{i}){\phi }_{n}^{(0)}({\lambda }_{i})=0,\\ \cdots \cdots ,\\ \sum _{j=0}^{{m}_{i}}{T}_{n}^{(j)}({\lambda }_{i}){\phi }_{n}^{({m}_{i}-j)}({\lambda }_{i})=0,\end{array}\right.\end{eqnarray*}$
where nonnegative integers m, mi and N satisfy the relation $2N=m+{\sum }_{i=1}^{m}{m}_{i}$. According to the above analysis, we can give the following discrete generalized (m, 2N − m)-fold DT theorem:Let ${\phi }_{n}({\lambda }_{i})={\left({\varphi }_{n}({\lambda }_{i}),{\psi }_{n}({\lambda }_{i})\right)}^{{\rm{T}}}$ be m column vector solutions of Lax pair (
$\begin{eqnarray}{\tilde{u}}_{n}={d}_{n+1}^{(2N)}{u}_{n}+{c}_{n+1}^{(2N-1)},\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{a}_{n}^{(0)}=\displaystyle \frac{{\rm{\Delta }}{a}_{n}^{(0)}}{{{\rm{\Delta }}}_{1}},{c}_{n}^{(2N-1)}=\displaystyle \frac{{\rm{\Delta }}{c}_{n}^{(2N-1)}}{{{\rm{\Delta }}}_{2}},\\ {d}_{n}^{(2N)}=\displaystyle \frac{{\rm{\Delta }}{d}_{n}^{(2N)}}{{{\rm{\Delta }}}_{2}},\end{array}\end{eqnarray}$
where ${{\rm{\Delta }}}_{1}=\det (\left[{{\rm{\Delta }}}_{1}^{(1)}\right.$, ${\left.{{\rm{\Delta }}}_{1}^{(2)},...,{{\rm{\Delta }}}_{1}^{(m)}\right]}^{{\rm{T}}})$, ${{\rm{\Delta }}}_{2}=\det (\left[{{\rm{\Delta }}}_{2}^{(1)}\right.$, ${\left.{{\rm{\Delta }}}_{2}^{(2)},...,{{\rm{\Delta }}}_{2}^{(m)}\right]}^{{\rm{T}}})$, ${{\rm{\Delta }}}_{1}^{(i)}={\left({{\rm{\Delta }}}_{1,j,s}^{(i)}\right)}_{({m}_{i}+1)\times 2N}$, ${{\rm{\Delta }}}_{2}^{(i)}={\left({{\rm{\Delta }}}_{2,j,s}^{(i)}\right)}_{({m}_{i}+1)\times 2N}$ in which ${{\rm{\Delta }}}_{1,j,s}^{(i)}$, ${{\rm{\Delta }}}_{2,j,s}^{(i)}$ $(1\leqslant j\leqslant {m}_{i}+1,1\leqslant s\leqslant 2N,1\leqslant i\leqslant m)$ are given by following formulae: $\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1,j,s}^{(i)}=\left\{\begin{array}{l}\sum _{k=0}^{j-1}{C}_{2N-2s}^{k}{\lambda }_{i}^{2N-2s-k}{\varphi }_{n}^{(j-1-k)}({\lambda }_{i})\quad \quad \quad \quad \mathrm{for}\quad 1\leqslant j\leqslant {m}_{i}+1,1\leqslant s\leqslant N,\\ \sum _{k=0}^{j-1}{C}_{4N-2s+1}^{k}{\lambda }_{i}^{4N-2s-k+1}{\psi }_{n}^{(j-1-k)}({\lambda }_{i})\quad \quad \mathrm{for}\quad 1\leqslant j\leqslant {m}_{i}+1,N+1\leqslant s\leqslant 2N,\end{array}\right.\\ {{\rm{\Delta }}}_{2,j,s}^{(i)}=\left\{\begin{array}{l}\sum _{k=0}^{j-1}{C}_{2N-2s+1}^{k}{\lambda }_{i}^{2N-2s-k+1}{\varphi }_{n}^{(j-1-k)}({\lambda }_{i})\quad \quad \mathrm{for}\quad 1\leqslant j\leqslant {m}_{i}+1,1\leqslant s\leqslant N,\\ \sum _{k=0}^{j-1}{C}_{4N-2s+2}^{k}{\lambda }_{i}^{4N-2s-k+2}{\psi }_{n}^{(j-1-k)}({\lambda }_{i})\quad \quad \mathrm{for}\quad 1\leqslant j\leqslant {m}_{i}+1,N+1\leqslant s\leqslant 2N,\end{array}\right.\end{array}\end{eqnarray*}$
and ${\rm{\Delta }}{a}_{n}^{(0)}$ is derived by replacing the Nth column of determinant ${{\rm{\Delta }}}_{1}$ with ${a}_{j}^{(i)}={\sum }_{k=0}^{j-1}$ ${C}_{2N-2s}^{k}{\lambda }_{i}^{2N-2s-k}$ ${\varphi }_{n}^{(j-1-k)}({\lambda }_{i})$ $(1\leqslant j\leqslant {m}_{i}+1,1\leqslant i\leqslant m)$. Similarly, ${\rm{\Delta }}{c}_{n}^{(2N-1)}$ and ${\rm{\Delta }}{d}_{n}^{(2N)}$ are obtained by replacing first and $(N+1)$th column of determinant ${{\rm{\Delta }}}_{2}$ with ${c}_{j}^{(i)}=-{a}_{n}^{(0)}{\psi }_{n}^{(j-1)}({\lambda }_{i})$ and ${d}_{j}^{(i)}=-{a}_{n}^{(0)}$ ${\psi }_{n}^{(j-1)}({\lambda }_{i})$ $(1\leqslant j\leqslant {m}_{i}+1,1\leqslant i\leqslant m)$, respectively.We need to note that m represents the number of spectral parameters and $(2N-m)$ represents the number of equations that we need to get from (
5. Diverse exact solutions and their asymptotic analysis
Asymptotic analysis is a very important method in integrable systems, which can be used to study physical phenomena and asymptotic behaviors. Recently, the asymptotic behaviors of soliton solutions and rational solutions of some special equations have been obtained by an improved asymptotic analysis method which relies on the balance between different algebraic terms up to the subdominant level [31–33], such as the Gerdjikov-Ivanov equation [31], Hirota equation [32] and defocusing nonlocal nonlinearSchrödinger equation [33]. In this part, we will use the generalized (m, 2N − m)-fold DT to give diverse exact solutions on non-zero seed background of equation (5 ), and then discuss their asymptotic behaviors by making an asymptotic analysis technique.
5.1. Soliton solutions and their physical properties and asymptotic analysis
Let us rewrite the solution of Lax pair (6 ) and (7 ) as φn = (φn, ψn), in which ${\varphi }_{n}={\left({\varphi }_{1,n},{\varphi }_{2,n}\right)}^{{\rm{T}}}$, ${\psi }_{n}={\left({\psi }_{1,n},{\psi }_{2,n}\right)}^{{\rm{T}}}$. Taking the trivial seed solution un = 1 and substituting it into the Lax pair (6 ) and (7 ), its basic solution are given by5 ) from (23 ) by the usual 2N-fold DT. In order to understand them more vividly, we draw their structures when N = 1 as shown in figure 1.
$\begin{eqnarray}\begin{array}{l}{\varphi }_{n}=\left(\begin{array}{c}{\tau }_{1}^{n}{{\rm{e}}}^{{\rho }_{1}t}\\ {\tau }_{1}^{n-1}{{\rm{e}}}^{{\rho }_{1}t}\end{array}\right),\\ {\psi }_{n}=\left(\begin{array}{c}{\tau }_{2}^{n}{{\rm{e}}}^{{\rho }_{2}t}\\ {\tau }_{2}^{n-1}{{\rm{e}}}^{{\rho }_{2}t}\end{array}\right),\end{array}\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{rcl}{\tau }_{1} & = & \displaystyle \frac{\lambda }{2}+\displaystyle \frac{\sqrt{{\lambda }^{2}-4}}{2},\quad {\tau }_{2}=\displaystyle \frac{\lambda }{2}-\displaystyle \frac{\sqrt{{\lambda }^{2}-4}}{2},\\ {\rho }_{1} & = & 1-\displaystyle \frac{{\lambda }^{2}}{2}+\displaystyle \frac{\lambda \sqrt{{\lambda }^{2}-4}}{2},\quad {\rho }_{2}=1-\displaystyle \frac{{\lambda }^{2}}{2}-\displaystyle \frac{\lambda \sqrt{{\lambda }^{2}-4}}{2},\end{array}\end{eqnarray*}$
where ∣λ∣ > 2. Furthermore, we define ${\delta }_{i,n}=\tfrac{{\varphi }_{2,n}({\lambda }_{i}){\lambda }_{i}-{r}_{i}{\psi }_{2,n}({\lambda }_{i})}{{\varphi }_{1,n}({\lambda }_{i}){\lambda }_{i}-{r}_{i}{\psi }_{1,n}({\lambda }_{i})}$ (i = 1, 2,…,N), where ri (i = 1, 2,…,N) are arbitrary constants, then we can obtain multi-soliton solutions of equations (
Figure 1. (a1) Bell-shaped one-soliton structure with parameters λ1 = 4, λ2 = 2, r1 = − r2 = 1; (b1) The propagation processes for one-soliton solution at t = −2 (dashdot line), t = 0 (longdash line) and t = 2 (solid line). (a2) Bell-shaped two-soliton structure with parameters ${\lambda }_{1}=\tfrac{5}{2},{\lambda }_{2}=3,{r}_{1}=-{r}_{2}=-1$; (b2) The propagation processes for two-soliton solution at t = −6 (dashdot line), t = 0 (longdash line) and t = 6 (solid line). |
When N = 1, λ = λi (i = 1, 2), we can obtain the following exact solutions as26 ), we can get one-soliton and two-soliton solutions if the parameters are selected properly.
$\begin{eqnarray}{\tilde{u}}_{n}={d}_{n+1}^{(2)}{u}_{n}+{c}_{n+1}^{(1)},\end{eqnarray}$
where ${a}_{n}^{(0)}=\tfrac{{\rm{\Delta }}{a}_{n}^{(0)}}{{{\rm{\Delta }}}_{1}}$, ${c}_{n}^{(1)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{c}_{n}^{(1)}}{{{\rm{\Delta }}}_{2}}$ and ${d}_{n}^{(2)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{d}_{n}^{(2)}}{{{\rm{\Delta }}}_{2}}$, in which $\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1}=\left|\begin{array}{cc}1 & {\lambda }_{1}{\delta }_{1,n}\\ 1 & {\lambda }_{2}{\delta }_{2,n}\end{array}\right|,\,{{\rm{\Delta }}}_{2}=\left|\begin{array}{cc}{\lambda }_{1} & {\lambda }_{1}^{2}{\delta }_{1,n}\\ {\lambda }_{2} & {\lambda }_{2}^{2}{\delta }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{a}_{n}^{(0)}=\left|\begin{array}{cc}-{\lambda }_{1}^{2} & {\lambda }_{1}{\delta }_{1,n}\\ -{\lambda }_{2}^{2} & {\lambda }_{2}{\delta }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{c}_{n}^{(1)}=\left|\begin{array}{cc}-{\delta }_{1,n} & {\lambda }_{1}^{2}{\delta }_{1,n}\\ -{\delta }_{2,n} & {\lambda }_{2}^{2}{\delta }_{2,n}\end{array}\right|,\\ {\rm{\Delta }}{d}_{n}^{(2)}=\left|\begin{array}{cc}{\lambda }_{1} & -{\delta }_{1,n}\\ {\lambda }_{2} & -{\delta }_{2,n}\end{array}\right|,\end{array}\end{eqnarray*}$
where ${a}_{n+1}^{(0)}$, ${c}_{n+1}^{(1)}$ and ${d}_{n+1}^{(2)}$ are obtained from ${a}_{n}^{(0)}$, ${c}_{n}^{(1)}$ and ${d}_{n}^{(2)}$ by changing n into n + 1. According to the expression (| I | (I)If one of the parameters λ1 and λ2 is equal to 2, without losing generality, let λ2 = 2, we can get the one-soliton solution by the expression ( $\begin{eqnarray}{\tilde{u}}_{n}=1+\displaystyle \frac{{\lambda }_{1}^{2}-4}{4}{{\rm{sech}} }^{2}\xi ,\end{eqnarray}$ with $\begin{eqnarray*}\xi =n\mathrm{ln}\left(\displaystyle \frac{{\lambda }_{1}}{2}+\displaystyle \frac{\sqrt{{\lambda }_{1}^{2}-4}}{2}\right)+\displaystyle \frac{{\lambda }_{1}\sqrt{{\lambda }_{1}^{2}-4}}{2}t,\end{eqnarray*}$ where ∣λ1∣ > 2. From (In order to analyze the propagation stability of the one-soliton solution, we numerically simulate it by using the finite difference method [34], which is given by figure 2. Figure 2(a) shows the exact one-soliton solution, which is consistent with figure 1(a1). Figure 2(b) is the numerical solution without any noise, which is almost the same as the exact solution in figure 2(a). Figure 2(c) is the time evolution of the numerical solution after increasing a 2% noise to an exact solution. We can find that it has little change compared with the numerical solution in figure 2(b). When we increase a 5% noise, the numerical solution will change more obviously, but its overall shape will not change greatly, as is shown in figure 2(d). The numerical simulation result further illustrates the correctness of the one-soliton solution, and the soliton propagation resists small noise and keeps stability in a relatively short time. |
| II | (II)If neither λ1 nor λ2 are equal to 2, we can derive the two-soliton solution from ( $\begin{eqnarray}{\tilde{u}}_{n}=1-\displaystyle \frac{2({\lambda }_{1}^{2}-{\lambda }_{2}^{2})[({\lambda }_{2}^{2}-4)\cosh (2{\xi }_{1})+({\lambda }_{1}^{2}-4)\cosh (2{\xi }_{2})-{\lambda }_{1}^{2}+{\lambda }_{2}^{2}]}{{\left[({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}-{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4})\cosh ({\xi }_{1}+{\xi }_{2})+({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}+{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4})\cosh ({\xi }_{1}-{\xi }_{2})\right]}^{2}},\end{eqnarray}$ where ${\xi }_{i}=n\mathrm{ln}$ $\left(\tfrac{{\lambda }_{i}}{2}+\tfrac{\sqrt{{\lambda }_{i}^{2}-4}}{2}\right)$ + $\tfrac{{\lambda }_{i}\sqrt{{\lambda }_{i}^{2}-4}}{2}t$, ∣λi∣ > 2 (i = 1, 2). Using the asymptotic analysis technique (see [30–33] and references therein), we can give the following four kinds of asymptotic states of two-soliton solution ( |
Figure 2. One-soliton solution ( |
Table 1. Physical characteristics of one-soliton solution. |
| Soliton | Amplitude | Width | Velocity | Wave number | Primary phase | Energy |
|---|---|---|---|---|---|---|
| un | $\tfrac{{\lambda}_{1}^{2}}{4}$ | $\tfrac{1}{\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $-\tfrac{{\lambda}_{1}\sqrt{{\lambda}_{1}^{2}-4}}{2\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)$ | 0 | $\tfrac{{\left({\lambda}_{1}^{2}-4\right)}^{2}}{12\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ |
Before the interaction (t → − ∞ ):28 ) before they interact.
$\begin{eqnarray}\begin{array}{l}{\tilde{u}}_{n}\to {u}_{1}^{-}=1+\displaystyle \frac{{\lambda }_{1}^{2}-4}{4}\\ {\times \,{\rm{sech}} }^{2}\left({\xi }_{1}-\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\left({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}-{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4}\right)}^{2}}{4({\lambda }_{2}^{2}-{\lambda }_{1}^{2})}\right),\\ ({\xi }_{1}\sim 0,{\xi }_{2}\to -\infty ),\\ {\tilde{u}}_{n}\to {u}_{2}^{-}=1+\displaystyle \frac{{\lambda }_{2}^{2}-4}{4}\\ {\times \,{\rm{sech}} }^{2}\left({\xi }_{2}-\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\left({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}-{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4}\right)}^{2}}{4({\lambda }_{2}^{2}-{\lambda }_{1}^{2})}\right),\\ ({\xi }_{2}\sim 0,{\xi }_{1}\to -\infty ),\end{array}\end{eqnarray}$
in which ${u}_{1}^{-},{u}_{2}^{-}$ stand for the limit states of equation (After the interaction (t → + ∞ ):28 ) after they interact.
$\begin{eqnarray}\begin{array}{l}{\tilde{u}}_{n}\to {u}_{1}^{+}=1+\displaystyle \frac{{\lambda }_{1}^{2}-4}{4}\\ {\times \,{\rm{sech}} }^{2}\left({\xi }_{1}+\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\left({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}-{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4}\right)}^{2}}{4({\lambda }_{2}^{2}-{\lambda }_{1}^{2})}\right),\\ ({\xi }_{1}\sim 0,{\xi }_{2}\to +\infty ),\\ {\tilde{u}}_{n}\to {u}_{2}^{+}=1+\displaystyle \frac{{\lambda }_{2}^{2}-4}{4}\\ {\times \,{\rm{sech}} }^{2}\left({\xi }_{2}+\displaystyle \frac{1}{2}\mathrm{ln}\displaystyle \frac{{\left({\lambda }_{1}\sqrt{{\lambda }_{2}^{2}-4}-{\lambda }_{2}\sqrt{{\lambda }_{1}^{2}-4}\right)}^{2}}{4({\lambda }_{2}^{2}-{\lambda }_{1}^{2})}\right),\\ ({\xi }_{2}\sim 0,{\xi }_{1}\to +\infty ),\end{array}\end{eqnarray}$
where ${u}_{1}^{+},{u}_{2}^{+}$ stand for the limit states of equation (From the above analysis, we can expose the physical properties of a two-soliton solution (28 ), which are listed in table 2, from which we know that the amplitude, velocity and energy of ${\tilde{u}}_{n}$ do not change before and after the interactions, but its phases have changed, so the collision interaction of two solitons in (28 ) is elastic.
Table 2. Physical characteristics of the two-soliton solution. |
| Solitons | Amplitudes | Widths | Velocities | Wave numbers | Primary phases | Energies |
|---|---|---|---|---|---|---|
| ${u}_{1}^{-}$ | $\tfrac{{\lambda}_{1}^{2}}{4}$ | $\tfrac{1}{\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $-\tfrac{{\lambda}_{1}\sqrt{{\lambda}_{1}^{2}-4}}{2\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)$ | $-\tfrac{1}{2}\mathrm{ln}\tfrac{{\left({\lambda}_{1}\sqrt{{\lambda}_{2}^{2}-4}-{\lambda}_{2}\sqrt{{\lambda}_{1}^{2}-4}\right)}^{2}}{4({\lambda}_{2}^{2}-{\lambda}_{1}^{2})}$ | $\tfrac{{\left({\lambda}_{1}^{2}-4\right)}^{2}}{12\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ |
| ${u}_{2}^{-}$ | $\tfrac{{\lambda}_{2}^{2}}{4}$ | $\tfrac{1}{\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ | $-\tfrac{{\lambda}_{2}\sqrt{{\lambda}_{2}^{2}-4}}{2\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ | $\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)$ | $-\tfrac{1}{2}\mathrm{ln}\tfrac{{\left({\lambda}_{1}\sqrt{{\lambda}_{2}^{2}-4}-{\lambda}_{2}\sqrt{{\lambda}_{1}^{2}-4}\right)}^{2}}{4({\lambda}_{2}^{2}-{\lambda}_{1}^{2})}$ | $\tfrac{{\left({\lambda}_{2}^{2}-4\right)}^{2}}{12\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ |
| ${u}_{1}^{+}$ | $\tfrac{{\lambda}_{1}^{2}}{4}$ | $\tfrac{1}{\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $-\tfrac{{\lambda}_{1}\sqrt{{\lambda}_{1}^{2}-4}}{2\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ | $\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)$ | $\tfrac{1}{2}\mathrm{ln}\tfrac{{\left({\lambda}_{1}\sqrt{{\lambda}_{2}^{2}-4}-{\lambda}_{2}\sqrt{{\lambda}_{1}^{2}-4}\right)}^{2}}{4({\lambda}_{2}^{2}-{\lambda}_{1}^{2})}$ | $\tfrac{{\left({\lambda}_{1}^{2}-4\right)}^{2}}{12\mathrm{ln}\left(\tfrac{{\lambda}_{1}}{2}+\tfrac{\sqrt{{\lambda}_{1}^{2}-4}}{2}\right)}$ |
| ${u}_{2}^{+}$ | $\tfrac{{\lambda}_{2}^{2}}{4}$ | $\tfrac{1}{\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ | $-\tfrac{{\lambda}_{2}\sqrt{{\lambda}_{2}^{2}-4}}{2\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ | $\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)$ | $\tfrac{1}{2}\mathrm{ln}\tfrac{{\left({\lambda}_{1}\sqrt{{\lambda}_{2}^{2}-4}-{\lambda}_{2}\sqrt{{\lambda}_{1}^{2}-4}\right)}^{2}}{4({\lambda}_{2}^{2}-{\lambda}_{1}^{2})}$ | $\tfrac{{\left({\lambda}_{2}^{2}-4\right)}^{2}}{12\mathrm{ln}\left(\tfrac{{\lambda}_{2}}{2}+\tfrac{\sqrt{{\lambda}_{2}^{2}-4}}{2}\right)}$ |
Using the same method as the numerical simulation of one-soliton, we numerically simulate the two-soliton solution (28 ). The relevant evolution result is shown in figure 3, figure 3(a) represents the exact two-soliton solution, figure 3(b) shows the numerical evolution of the two-soliton solution without any noise. Figures 3(c) and (d) exhibit the numerical evolutions by adding 2% and 5% noises, respectively. From figure 3, we know that the numerical evolution of the two-soliton solution is robust against a small noise in a relatively short time.
Figure 3. Two-soliton solution ( |
5.2. Rational solutions and asymptotic analysis
In this subsection, we will give the rational solutions of equation (5 ) with the initial seed solution un = 1 by using the generalized (1, 2N − 1)-fold DT. For the convenience of study, we rewrite solution (25 ) as5 ).
36 ) as37 ), we can get its limit states with the aid of symbolic computation Maple. If taking ${\zeta }_{1}=n+2t-{\left(10+6\sqrt{5}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$, ${\zeta }_{2}=n+2t-{\left(10-6\sqrt{5}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$, $b={\left(10+6\sqrt{5}\right)}^{\tfrac{1}{3}}-{\left(10-6\sqrt{5}\right)}^{\tfrac{1}{3}}\gt 0$, it will yield four kinds of asymptotic states as the following shows.
$\begin{eqnarray}\begin{array}{l}{\phi }_{n}=\left(\begin{array}{c}{\varphi }_{n}\\ {\psi }_{n}\end{array}\right)\\ =\left(\begin{array}{c}{C}_{1}{\tau }_{1}^{n}{{\rm{e}}}^{{\rho }_{1}t+\delta (\varepsilon )}+{C}_{2}{\tau }_{2}^{n}{{\rm{e}}}^{{\rho }_{2}t-\delta (\varepsilon )}\\ {C}_{1}{\tau }_{1}^{n-1}{{\rm{e}}}^{{\rho }_{1}t+\delta (\varepsilon )}+{C}_{2}{\tau }_{2}^{n-1}{{\rm{e}}}^{{\rho }_{2}t-\delta (\varepsilon )}\end{array}\right),\end{array}\end{eqnarray}$
where $\delta (\varepsilon )=\sqrt{{\lambda }_{1}^{2}-4}{{\rm{\Sigma }}}_{j=0}^{2N-1}{e}_{j}{\varepsilon }^{2j},$ ej(j = 0, 1,…,2N − 1) are arbitrary parameters, ϵ is a small parameter, τ1, τ2, ρ1, ρ2 are the same as the previous definitions, and C1, C2 are arbitrary real constants. Let λ = λ1 + ϵ2, λ1 = 2, C1 = C2 = 1, then expand φn by Taylor series at ϵ = 0, we have $\begin{eqnarray}\begin{array}{rcl}{\phi }_{n}({\varepsilon }^{2}) & = & {\phi }_{n}^{(0)}+{\phi }_{n}^{(1)}{\varepsilon }^{2}+{\phi }_{n}^{(2)}{\varepsilon }^{4}+{\phi }_{n}^{(3)}{\varepsilon }^{6}\\ & & +{\phi }_{n}^{(4)}{\varepsilon }^{8}+{\phi }_{n}^{(5)}{\varepsilon }^{10}+\cdots ,\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{\phi }_{n}^{(0)}=\left(\begin{array}{c}{\varphi }_{n}^{(0)}\\ {\psi }_{n}^{(0)}\end{array}\right)\,=\,\left(\begin{array}{c}2{{\rm{e}}}^{-t}\\ 2{{\rm{e}}}^{-t}\end{array}\right),\\ {\phi }_{n}^{(1)}=\left(\begin{array}{c}{\varphi }_{n}^{(1)}\\ {\psi }_{n}^{(1)}\end{array}\right)\\ =\left(\begin{array}{c}{{\rm{e}}}^{-t}({\xi }^{2}+4{e}_{0}\xi +4{e}_{0}^{2}-4t)\\ {{\rm{e}}}^{-t}({\xi }^{2}+4\xi {e}_{0}+4{e}_{0}^{2}-4t-2\xi -4{e}_{0}+1)\end{array}\right),\\ {\phi }_{n}^{(2)}=\left(\begin{array}{c}{\varphi }_{n}^{(2)}\\ {\psi }_{n}^{(2)}\end{array}\right),\end{array}\end{eqnarray}$
in which $\begin{eqnarray*}\begin{array}{l}{\varphi }_{n}^{(2)}=\displaystyle \frac{1}{12}[{{\rm{e}}}^{-t}({\xi }^{4}+8{\xi }^{3}{e}_{0}+24{\xi }^{2}{e}_{0}^{2}\\ \quad +32\xi {e}_{0}^{3}+16{e}_{0}^{4}-24t{\xi }^{2}\\ \quad -96t\xi {e}_{0}-96{{te}}_{0}^{2}+48{t}^{2}+32t\xi +64{{te}}_{0}-{\xi }^{2}\\ \quad +4\xi {e}_{0}+12{e}_{0}^{2}-12t)],\\ {\psi }_{n}^{(2)}=\displaystyle \frac{1}{12}[{{\rm{e}}}^{-t}({\xi }^{4}+8{\xi }^{3}{e}_{0}+24{\xi }^{2}{e}_{0}^{2}\\ \quad +32\xi {e}_{0}^{3}+16{e}_{0}^{4}-24t{\xi }^{2}\\ -96t\xi {e}_{0}-96{{te}}_{0}^{2}-4{\xi }^{3}-24{\xi }^{2}{e}_{0}-48\xi {e}_{0}^{2}-32{e}_{0}^{3}\\ \quad +48{t}^{2}+80t\xi +160{{te}}_{0}+5{\xi }^{2}\\ \quad +28\xi {e}_{0}+36{e}_{0}^{2}-68t-2\xi -12{e}_{0})],\end{array}\end{eqnarray*}$
where ξ = n + 2t, and the rest of ${\phi }_{n}^{(j)}(j=3,4,5,\cdots )$ are left out here. Next, taking N = 1, 2, we give the rational solutions of equation (| I | (I)When N = 1, we can get the first-order rational solution by the generalized (1, 1)-fold DT as follows: $\begin{eqnarray}{\tilde{u}}_{n}={d}_{n+1}^{(2)}{u}_{n}+{c}_{n+1}^{(1)},\end{eqnarray}$ where ${a}_{n}^{(0)}=\tfrac{{\rm{\Delta }}{a}_{n}^{(0)}}{{{\rm{\Delta }}}_{1}}$, ${c}_{n}^{(1)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{c}_{n}^{(1)}}{{{\rm{\Delta }}}_{2}}$ and ${d}_{n}^{(2)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{d}_{n}^{(2)}}{{{\rm{\Delta }}}_{2}}$, in which $\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1}=\left|\begin{array}{cc}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(0)}\\ {\varphi }_{n}^{(1)} & {\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)}\end{array}\right|,\\ {{\rm{\Delta }}}_{2}=\left|\begin{array}{cc}{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(1)}+2{\lambda }_{1}{\psi }_{n}^{(0)}\end{array}\right|,\\ {\rm{\Delta }}{c}_{n}^{(1)}=\left|\begin{array}{cc}-{\psi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ -{\psi }_{n}^{(1)} & {\lambda }_{1}^{2}{\psi }_{n}^{(1)}+2{\lambda }_{1}{\psi }_{n}^{(0)}\end{array}\right|,\\ {\rm{\Delta }}{a}_{n}^{(0)}=\left|\begin{array}{cc}-{\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(0)}\\ -{\lambda }_{1}^{2}{\varphi }_{n}^{(1)}-2{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)}\end{array}\right|,\\ {\rm{\Delta }}{d}_{n}^{(2)}=\left|\begin{array}{cc}{\lambda }_{1}{\varphi }_{n}^{(0)} & -{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & -{\psi }_{n}^{(1)}\end{array}\right|,\end{array}\end{eqnarray*}$ where ${a}_{n+1}^{(0)}$, ${c}_{n+1}^{(1)}$ and ${d}_{n+1}^{(2)}$ are obtained from ${a}_{n}^{(0)}$, ${c}_{n}^{(1)}$ and ${d}_{n}^{(2)}$ by changing n into n + 1. From the above formulas, we can calculate the expression of equation ( $\begin{eqnarray}{\tilde{u}}_{n}=1-\displaystyle \frac{1}{{\left(n+2t+2{e}_{0}\right)}^{2}}=1-\displaystyle \frac{1}{{\left(\xi +2{e}_{0}\right)}^{2}},\end{eqnarray}$ from which we can see that solution ( |
| II | (II)When N = 2, we can get the second-order rational solution by the generalized (1, 3)-fold DT as $\begin{eqnarray}{\tilde{u}}_{n}={d}_{n+1}^{(4)}{u}_{n}+{c}_{n+1}^{(3)},\end{eqnarray}$ where ${a}_{n}^{(0)}=\tfrac{{\rm{\Delta }}{a}_{n}^{(0)}}{{{\rm{\Delta }}}_{1}}$, ${c}_{n}^{(3)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{c}_{n}^{(3)}}{{{\rm{\Delta }}}_{2}}$ and ${d}_{n}^{(4)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{d}_{n}^{(4)}}{{{\rm{\Delta }}}_{2}}$, in which |
$\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1}=\left|\begin{array}{cccc}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(0)} & {\lambda }_{1}^{3}{\psi }_{n}^{(0)}\\ {\varphi }_{n}^{(1)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(1)}+2{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)} & {\lambda }_{1}^{3}{\psi }_{n}^{(1)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ {\varphi }_{n}^{(2)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(2)}+2{\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(2)}+{\psi }_{n}^{(1)} & {\lambda }_{1}^{3}{\psi }_{n}^{(2)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(1)}+3{\lambda }_{1}{\psi }_{n}^{(0)}\\ {\varphi }_{n}^{(3)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(3)}+2{\lambda }_{1}{\varphi }_{n}^{(2)}+{\varphi }_{n}^{(1)} & {\lambda }_{1}{\psi }_{n}^{(3)}+{\psi }_{n}^{(2)} & {\lambda }_{1}^{3}{\psi }_{n}^{(3)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(2)}+3{\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)}\end{array}\right|,\\ {\rm{\Delta }}{d}_{n}^{(4)}=\left|\begin{array}{cccc}{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(0)} & -{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(1)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(1)}+2{\lambda }_{1}{\psi }_{n}^{(0)} & -{\psi }_{n}^{(1)}\\ {\lambda }_{1}{\varphi }_{n}^{(2)}+{\varphi }_{n}^{(1)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(2)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(1)}+3{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(2)}+2{\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)} & -{\psi }_{n}^{(2)}\\ {\lambda }_{1}{\varphi }_{n}^{(3)}+{\varphi }_{n}^{(2)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(3)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(2)}+3{\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(3)}+2{\lambda }_{1}{\psi }_{n}^{(2)}+{\psi }_{n}^{(1)} & -{\psi }_{n}^{(3)}\end{array}\right|,\\ {\rm{\Delta }}{c}_{n}^{(3)}=\left|\begin{array}{cccc}{\lambda }_{1}{\varphi }_{n}^{(0)} & -{\psi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & -{\psi }_{n}^{(1)} & {\lambda }_{1}^{2}{\psi }_{n}^{(1)}+2{\lambda }_{1}{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(1)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(2)}+{\varphi }_{n}^{(1)} & -{\psi }_{n}^{(2)} & {\lambda }_{1}^{2}{\psi }_{n}^{(2)}+2{\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(2)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(1)}+6{\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(3)}+{\varphi }_{n}^{(2)} & -{\psi }_{n}^{(3)} & {\lambda }_{1}^{2}{\psi }_{n}^{(3)}+2{\lambda }_{1}{\psi }_{n}^{(2)}+{\psi }_{n}^{(1)} & {\lambda }_{1}^{4}{\psi }_{n}^{(3)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(2)}+6{\lambda }_{1}^{2}{\psi }_{n}^{(1)}+4{\lambda }_{1}{\psi }_{n}^{(0)}\end{array}\right|,\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{2}=\left|\begin{array}{cccc}{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(1)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(1)}+2{\lambda }_{1}{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(1)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(2)}+{\varphi }_{n}^{(1)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(2)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(1)}+3{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(2)}+2{\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)} & {\lambda }_{1}^{4}{\psi }_{n}^{(2)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(1)}+6{\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ {\lambda }_{1}{\varphi }_{n}^{(3)}+{\varphi }_{n}^{(2)} & {\lambda }_{1}^{3}{\varphi }_{n}^{(3)}+3{\lambda }_{1}^{2}{\varphi }_{n}^{(2)}+3{\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\psi }_{n}^{(3)}+2{\lambda }_{1}{\psi }_{n}^{(2)}+{\psi }_{n}^{(1)} & {\lambda }_{1}^{4}{\psi }_{n}^{(3)}+4{\lambda }_{1}^{3}{\psi }_{n}^{(2)}+6{\lambda }_{1}^{2}{\psi }_{n}^{(1)}+4{\lambda }_{1}{\psi }_{n}^{(0)}\end{array}\right|,\\ {\rm{\Delta }}{a}_{n}^{(0)}=\left|\begin{array}{cccc}-{\lambda }_{1}^{4}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(0)} & {\lambda }_{1}^{3}{\psi }_{n}^{(0)}\\ -{\lambda }_{1}^{4}{\varphi }_{n}^{(1)}-4{\lambda }_{1}^{3}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(1)}+2{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)} & {\lambda }_{1}^{3}{\psi }_{n}^{(1)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(0)}\\ -{\lambda }_{1}^{4}{\varphi }_{n}^{(2)}-4{\lambda }_{1}^{3}{\varphi }_{n}^{(1)}-6{\lambda }_{1}^{2}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(2)}+2{\lambda }_{1}{\varphi }_{n}^{(1)}+{\varphi }_{n}^{(0)} & {\lambda }_{1}{\psi }_{n}^{(2)}+{\psi }_{n}^{(1)} & {\lambda }_{1}^{3}{\psi }_{n}^{(2)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(1)}+3{\lambda }_{1}{\psi }_{n}^{(0)}\\ -{\lambda }_{1}^{4}{\varphi }_{n}^{(3)}-4{\lambda }_{1}^{3}{\varphi }_{n}^{(2)}-6{\lambda }_{1}^{2}{\varphi }_{n}^{(1)}-4{\lambda }_{1}{\varphi }_{n}^{(0)} & {\lambda }_{1}^{2}{\varphi }_{n}^{(3)}+2{\lambda }_{1}{\varphi }_{n}^{(2)}+{\varphi }_{n}^{(1)} & {\lambda }_{1}{\psi }_{n}^{(3)}+{\psi }_{n}^{(2)} & {\lambda }_{1}^{3}{\psi }_{n}^{(3)}+3{\lambda }_{1}^{2}{\psi }_{n}^{(2)}+3{\lambda }_{1}{\psi }_{n}^{(1)}+{\psi }_{n}^{(0)}\end{array}\right|,\end{array}\end{eqnarray*}$
where ${a}_{n+1}^{(0)}$, ${c}_{n+1}^{(3)}$ and ${d}_{n+1}^{(4)}$ are obtained from ${a}_{n}^{(0)}$, ${c}_{n}^{(3)}$ and ${d}_{n}^{(4)}$ by changing n into n + 1. From the above formulas, we can calculate the analytical expression of equation ( $\begin{eqnarray}{\tilde{u}}_{n}=\displaystyle \frac{{EF}}{G},\end{eqnarray}$
with $\begin{eqnarray*}\begin{array}{l}E=64{e}_{0}^{6}+(192n+384t-192){e}_{0}^{5}+(240{n}^{2}+(960t-480)n\\ \quad +960{t}^{2}-960t+120){e}_{0}^{4}+(160{n}^{3}+(960t-480){n}^{2}\\ \quad +(1920{t}^{2}-1920t+260)n+1280{t}^{3}-1920{t}^{2}+360t\\ \quad -240{e}_{1}+60){e}_{0}^{3}+(60{n}^{4}+(480t-240){n}^{3}+(1440{t}^{2}\\ \quad -1440t+210){n}^{2}+(1920{t}^{3}-2880{t}^{2}\\ \quad +600t-360{e}_{1}+60)n\\ \quad +960{t}^{4}-1920{t}^{3}+360{t}^{2}+(-720{e}_{1}+360)t\\ \quad +360{e}_{1}-\displaystyle \frac{135}{2}){e}_{0}^{2}+(12{n}^{5}+(120t-60){n}^{4}\\ \quad +(480{t}^{2}-480t+75){n}^{3}+(960{t}^{3}-1440{t}^{2}\\ \quad +330t-180{e}_{1}+15){n}^{2}\\ \quad +\left(960{t}^{4}-1920{t}^{3}+420{t}^{2}+(-720{e}_{1}+300)t+360{e}_{1}\right.\\ \quad \left.-\displaystyle \frac{213}{4}\right)n+384{t}^{5}-960{t}^{4}+120{t}^{3}+(-720{e}_{1}+540){t}^{2}\\ \quad +(720{e}_{1}-\displaystyle \frac{405}{2})t-135{e}_{1}+\displaystyle \frac{45}{4}){e}_{0}+{n}^{6}+(12t-6){n}^{5}\\ \quad +(60{t}^{2}-60t+10){n}^{4}+(160{t}^{3}-240{t}^{2}+60t-30{e}_{1}){n}^{3}\\ \quad +(240{t}^{4}-480{t}^{3}+120{t}^{2}+(-180{e}_{1}+60)t+90{e}_{1}-11){n}^{2}\\ \quad +(192{t}^{5}-480{t}^{4}+80{t}^{3}+(-360{e}_{1}+240){t}^{2}+(-72\\ \quad +360{e}_{1})t+6-\displaystyle \frac{75}{2}{e}_{1})n+64{t}^{6}-192{t}^{5}+(-240{e}_{1}+240){t}^{3}\\ \quad +(360{e}_{1}-180){t}^{2}-315{{te}}_{1}-\displaystyle \frac{45}{2}{e}_{1}-180{e}_{1}^{2},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}F=64{e}_{0}^{6}+(192n+384t+192){e}_{0}^{5}+(240{n}^{2}+(960t+480)n\\ \quad +960{t}^{2}+960t+120){e}_{0}^{4}+(160{n}^{3}+(960t+480){n}^{2}\\ \quad +(1920{t}^{2}+1920t+260)n+1280{t}^{3}+1920{t}^{2}\\ \quad +360t-240{e}_{1}-60){e}_{0}^{3}+(60{n}^{4}+(480t+240){n}^{3}+(1440{t}^{2}\\ \quad +1440t+210){n}^{2}+(1920{t}^{3}+2880{t}^{2}+600t-360{e}_{1}-60)n\\ \quad +960{t}^{4}+1920{t}^{3}+360{t}^{2}+(-720{e}_{1}-360)t-360{e}_{1}\\ \quad -\displaystyle \frac{135}{2}){e}_{0}^{2}+(12{n}^{5}+(120t+60){n}^{4}+(480{t}^{2}+480t+75){n}^{3}\\ \quad +(960{t}^{3}+1440{t}^{2}+330t-180{e}_{1}-15){n}^{2}+(960{t}^{4}\\ \quad +1920{t}^{3}+420{t}^{2}+(-720{e}_{1}-300)t-360{e}_{1}-\displaystyle \frac{213}{4})n+384{t}^{5}\\ \quad +960{t}^{4}+120{t}^{3}+(-720{e}_{1}-540){t}^{2}+(-720{e}_{1}\\ \quad -\displaystyle \frac{405}{2})t-135{e}_{1}-\displaystyle \frac{45}{4}){e}_{0}+{n}^{6}\\ \quad +(12t+6){n}^{5}+(60{t}^{2}+60t+10){n}^{4}\\ \quad +(160{t}^{3}+240{t}^{2}+60t-30{e}_{1}){n}^{3}+(240{t}^{4}\\ \quad +480{t}^{3}+120{t}^{2}+(-180{e}_{1}-60)t-90{e}_{1}-11){n}^{2}+(192{t}^{5}+480{t}^{4}\\ \quad +80{t}^{3}+(-360{e}_{1}-240){t}^{2}+(-72-360{e}_{1})t\\ \quad -6-\displaystyle \frac{75}{2}{e}_{1})n+64{t}^{6}+192{t}^{5}+(-240{e}_{1}-240){t}^{3}\\ \quad +(-360{e}_{1}-180){t}^{2}-315{{te}}_{1}+\displaystyle \frac{45}{2}{e}_{1}-180{e}_{1}^{2},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{l}G=\left[64{e}_{0}^{6}+(192n+384t){e}_{0}^{5}+(240{n}^{2}+960{nt}+960{t}^{2}-120){e}_{0}^{4}\right.\\ \quad +(160{n}^{3}+960{n}^{2}t+(1920{t}^{2}-220)n+1280{t}^{3}\\ \quad -600t-240{e}_{1}){e}_{0}^{3}+(60{n}^{4}+480{n}^{3}t+(1440{t}^{2}-150){n}^{2}\\ \quad +(1920{t}^{3}-840t-360{e}_{1})n-720{{te}}_{1}-1080{t}^{2}+960{t}^{4}\\ \quad +\displaystyle \frac{45}{2}){e}_{0}^{2}+(12{n}^{5}+120{n}^{4}t+(480{t}^{2}-45){n}^{3}\\ \quad +(960{t}^{3}-390t-180{e}_{1}){n}^{2}+(-720{{te}}_{1}\\ \quad +960{t}^{4}-1020{t}^{2}+\displaystyle \frac{87}{4})n\\ \quad -720{t}^{2}{e}_{1}+\displaystyle \frac{135}{2}t+45{e}_{1}-840{t}^{3}\\ \quad +384{t}^{5}){e}_{0}+{n}^{6}+12{n}^{5}t\\ \quad +(60{t}^{2}-5){n}^{4}+(160{t}^{3}-60t-30{e}_{1}){n}^{3}+(240{t}^{4}\\ \quad -240{t}^{2}-180{{te}}_{1}+4){n}^{2}+(-360{t}^{2}{e}_{1}+48t\\ \quad +\displaystyle \frac{105}{2}{e}_{1}-400{t}^{3}\\ \quad {\left.+192{t}^{5})n+64{t}^{6}-240{t}^{4}-240{t}^{3}{e}_{1}-135{{te}}_{1}-180{e}_{1}^{2}\right]}^{2},\end{array}\end{eqnarray*}$
where e0, e1 are arbitrary real constants. In order to understand the properties of solution (| i | (i)If ${\zeta }_{1}=n+2t-{\left(10+6\sqrt{5}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$ is regarded as a fixed constant, we can obtain ${\zeta }_{2}={\zeta }_{1}+{{bt}}^{\tfrac{1}{3}}$ and ζ2 → ± ∞ when t → ± ∞ , from which we can get that the first two limit states are $\begin{eqnarray}{\tilde{u}}_{n}\to {u}_{1}^{\pm }=1-\displaystyle \frac{1}{{\left({\zeta }_{1}+2{e}_{0}\right)}^{2}}.\end{eqnarray}$ |
| ii | (ii)If ${\zeta }_{2}=n+2t-{\left(10-6\sqrt{5}\right)}^{\tfrac{1}{3}}{t}^{\tfrac{1}{3}}$ is regarded as a fixed constant, we can obtain ${\zeta }_{1}={\zeta }_{2}-{{bt}}^{\tfrac{1}{3}}$ and ζ1 → ∓ ∞ when t → ± ∞ , from which we can get that the last two limit states are $\begin{eqnarray}{\tilde{u}}_{n}\to {u}_{2}^{\pm }=1-\displaystyle \frac{1}{{\left({\zeta }_{2}+2{e}_{0}\right)}^{2}}.\end{eqnarray}$ |
From the above analysis, we can find that two curves ζ1 + 2e0 = 0 and ζ2 + 2e0 = 0 are the central trajectories of solution ${\tilde{u}}_{n}$, which are also its two singularity lines. The limit states of the second-order rational solution (37 ) are similar to the first-order rational solution, and the difference is that the first-order rational solution's singularity line is a straight line, while the second-order rational solution's singularity line is a curve. Similarly, we can change the position of the singularity line by the constant e0. Using the same method as above, we can analyze the asymptotic behaviors of more complex higher-order rational solutions, which are omitted to discuss here. Next we only give some simple mathematical features of rational solutions of equation (5 ) listed in table 3, which also reflects their good mathematical laws.
Table 3. Main mathematical features of rational solution ${\tilde{u}}_{n}$ of order N. |
| N | Background of ${\tilde{u}}_{n}$ | Highest powers in the numerator | Highest powers in the denominator |
|---|---|---|---|
| 1 | 1 | 2 | 2 |
| 2 | 1 | 12 | 12 |
| 3 | 1 | 30 | 30 |
| ... | ... | ... | ... |
| N | 1 | 2N(2N − 1) | 2N(2N − 1) |
In table 3, the first column represents the order number of rational solution ${\tilde{u}}_{n}$, the second column is its initial background, and the third and fourth columns are the highest powers of the numerator and denominator of it respectively. With the increase of the order number, the background remains unchanged, and the highest powers in the numerator and denominator of the rational solution increase accordingly, but they are always equal.
It should be noted here that for discrete KdV equation (
5.3. Mixed solution and asymptotic state analysis
In this section, we will discuss the mixed solution of equation (5 ) by using the generalized (2, 2N − 2)-fold DT and analyze its asymptotic behaviors. Next, we only discuss the case of N = 1.
Different from the previous steps for solving soliton and rational solutions, here we choose two spectral parameters λ1 and λ2, and take ${C}_{1}=-{C}_{2}=\tfrac{1}{\varepsilon }$, λ1 = λ1 + ϵ2, λ2 = 4 with λ1 = 2. For λ1, we make the Taylor expansion as (32 ) by expanding φn of (31 ) at ϵ = 0, while for λ2, we do not make Taylor expansion, then the mixed solution of equation (5 ) is written as40 ) as
Before the interaction (t → − ∞ ):
$\begin{eqnarray}{\tilde{u}}_{n}={d}_{n+1}^{(2)}{u}_{n}+{c}_{n+1}^{(1)},\end{eqnarray}$
with ${a}_{n}^{(0)}=\tfrac{{\rm{\Delta }}{a}_{n}^{(0)}}{{{\rm{\Delta }}}_{1}}$, ${c}_{n}^{(1)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{c}_{n}^{(1)}}{{{\rm{\Delta }}}_{2}}$ and ${d}_{n}^{(2)}={a}_{n}^{(0)}\tfrac{{\rm{\Delta }}{d}_{n}^{(2)}}{{{\rm{\Delta }}}_{2}}$, in which $\begin{eqnarray*}\begin{array}{l}{{\rm{\Delta }}}_{1}=\left|\begin{array}{cc}{\varphi }_{n}^{(0)}({\lambda }_{1}) & {\lambda }_{1}{\psi }_{n}^{(0)}({\lambda }_{1})\\ {\varphi }_{n}({\lambda }_{2}) & {\lambda }_{2}{\psi }_{n}({\lambda }_{2})\end{array}\right|,\\ {{\rm{\Delta }}}_{2}=\left|\begin{array}{cc}{\lambda }_{1}{\varphi }_{n}^{(0)}({\lambda }_{1}) & {\lambda }_{1}^{2}{\psi }_{n}^{(0)}({\lambda }_{1})\\ {\lambda }_{2}{\varphi }_{n}({\lambda }_{2}) & {\lambda }_{2}^{2}{\psi }_{n}({\lambda }_{2})\end{array}\right|,\\ {\rm{\Delta }}{a}_{n}^{(0)}=\left|\begin{array}{cc}-{\lambda }_{1}^{2}{\varphi }_{n}^{(0)}({\lambda }_{1}) & {\lambda }_{1}{\psi }_{n}^{(0)}({\lambda }_{1})\\ -{\lambda }_{2}^{2}{\varphi }_{n}({\lambda }_{2}) & {\lambda }_{2}{\psi }_{n}({\lambda }_{2})\end{array}\right|,\\ {\rm{\Delta }}{c}_{n}^{(1)}=\left|\begin{array}{cc}-{\psi }_{n}^{(0)}({\lambda }_{1}) & {\lambda }_{1}^{2}{\psi }_{n}^{(0)}({\lambda }_{1})\\ -{\psi }_{n}({\lambda }_{2}) & {\lambda }_{2}^{2}{\psi }_{n}({\lambda }_{2})\end{array}\right|,\\ {\rm{\Delta }}{d}_{n}^{(2)}=\left|\begin{array}{cc}{\lambda }_{1}{\varphi }_{n}^{(0)}({\lambda }_{1}) & -{\psi }_{n}^{(0)}({\lambda }_{1})\\ {\lambda }_{2}{\varphi }_{n}({\lambda }_{2}) & -{\psi }_{n}({\lambda }_{2})\end{array}\right|,\end{array}\end{eqnarray*}$
where ${a}_{n+1}^{(0)}$, ${c}_{n+1}^{(1)}$ and ${d}_{n+1}^{(2)}$ are obtained from ${a}_{n}^{(0)}$, ${c}_{n}^{(1)}$ and ${d}_{n}^{(2)}$ by changing n into n + 1. According to the above formulas, we can calculate the analytical expression of solution ( $\begin{eqnarray}{\tilde{u}}_{n}=1-\displaystyle \frac{3[\cosh (2{\xi }_{1})+6{\left({\xi }_{2}+{e}_{0}\right)}^{2}+1]}{2{\left[2\cosh {\xi }_{1}-\sqrt{3}({\xi }_{2}+2{e}_{0})\sinh {\xi }_{1}\right]}^{2}},\end{eqnarray}$
in which ${\xi }_{1}=n\mathrm{ln}(2+\sqrt{3})+4\sqrt{3}t$, ξ2 = n + 2t. Next, we analyze the limit states of mixed solution before and after the interactions.| i | (i)If ξ1 is equal to a fixed constant as t → ± ∞ , we can get ξ2 → ± ∞ , and two asymptotic states of solution ( $\begin{eqnarray}{\tilde{u}}_{n}\to {u}_{1}^{\pm }=1-3{\mathrm{csch}}^{2}{\xi }_{1},\end{eqnarray}$ where ${u}_{1}^{-},{u}_{1}^{+}$ stand for the limit states of solution ( |
| ii | (ii)If ξ2 is equal to a fixed constant as t → ± ∞ , we can get ξ1 → ± ∞ , and two asymptotic states of solution ( |
$\begin{eqnarray}{\tilde{u}}_{n}\to {u}_{2}^{-}=1-\displaystyle \frac{3}{{\left(\sqrt{3}{\xi }_{2}+2\sqrt{3}{e}_{0}+2\right)}^{2}}.\end{eqnarray}$
After the interaction (t → + ∞ ):41 ) before and after the interactions, respectively.
$\begin{eqnarray}{\tilde{u}}_{n}\to {u}_{2}^{+}=1-\displaystyle \frac{3}{{\left(\sqrt{3}{\xi }_{2}+2\sqrt{3}{e}_{0}-2\right)}^{2}},\end{eqnarray}$
where ${u}_{2}^{-},{u}_{2}^{+}$ stand for the limit states of solution (It should be noted here that for this case, we should get mixed superposition solutions of usual soliton and rational solution in theory. However, from the above analysis, we know that when ${\xi }_{1}$ is fixed and ${\xi }_{2}\to \pm \infty $, the asymptotic states of the mixed solution (
6. Conclusions
In this paper, we have studied some properties of a discrete KdV equation (5 ) and obtained its exact solutions such as soliton solutions, rational solutions, and mixed solutions on a non-zero seed background, which reflects the good physical significance of the discrete KdV equation. The main achievements of this paper are as follows: First, we have constructed the hierarchy of the discrete KdV equation and obtained two new discrete KdV equations, (12 ) and (14 ). Second, we have mapped the discrete equations (12 ) and (14 ) to the KdV equation by using the continuous limit. Third, we have constructed the discrete generalized (m, 2N − m)-fold DT of equation (5 ) in theorem 1, which is different from the usual N-fold DT, to give soliton, rational and mixed solutions. The structures of one-soliton and two-soliton solutions are demonstrated in figure 1, and their physical properties are listed in tables 1 and 2, respectively. Moreover, we also numerically simulate one-soliton and two-soliton solutions in order to better understand their dynamic behaviors as shown in figures 2 and 3. At the same time, the asymptotic states of various exact solutions have also been analyzed. These results given in this paper might help us understand some physical phenomena described by KdV equations.