# Extended Mixed Function Method and Its Application for Solving Two Classic Toda Lattice Equations

Authors: Weiguo Rui; Xinsong Yang
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

The mixed function method is extended from the(1+1)-dimensional space
to the (2+1)-dimensional one, even those forms of exact solution do not exist in (1+1)-dimensional NDDEs. By using this extended method, the Toda lattice and (2+1)-dimensional
Toda lattice equations are studied. Some new solutions such as discrete solitary wave solutions,
discrete kink and antikink wave solutions, and discrete breather soliton solutions are
obtained, and their dynamic properties are discussed.


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## Body

## 1. Introduction

In [1], motivated by the structures of the exact solutions obtained from Darboux transformation, the authors constructed the mixed function method in (1 + 1)-dimensional space. Further, they studied the generalized Hybrid lattice equation and the two-component Volterra lattice equation by using this method, then they obtained some new exact solutions such as discrete solitary wave solutions and kink wave solutions. By this token, the mixed function method is as powerful as the exp method [2–7] and tanh method [8–10]. In this paper, we will extend this method from the (1 + 1)-dimensional space to the (2 + 1)-dimensional one and present some new forms on establishing different kinds of exact solutions, then this extended method will be applied to many higher-order nonlinear differential-difference equations (NDDEs). Moreover, we provide a simplified way to solve large numbers of high-power algebraic equations using a computer. Of course, these high-power algebraic equations mentioned above are all derived from high-order or high-power NDDEs. Therefore, the procedure of computation can be greatly simplified, and time is saved. We notice that this extended method is as powerful as the other symbolic computation methods such as tanh-function method, sine-cosine function method, exp-function method, Jacobian elliptic function method, (G′/G)-expansion method, and Adomian decomposition method, and these methods (please refer to [2–18] and references cited therein) are popular tools in the field of the nonlinear differential-difference equations.We consider the following (2 + 1)-dimensional Toda lattice equation [19](1.1)∂2un∂x∂t=(∂un∂t+1)(un-1-2un+un+1),
and the Toda lattice equation [20](1.2)d2undt2=(dundt+1)(un-1-2un+un+1).
It is well known that there is another form for the (2 + 1)-dimensional Toda lattice equation(1.3)∂2yn∂x∂t=exp(yn-1-yn)-exp(yn-yn+1),
and there are another two forms for the Toda lattice equation(1.4)d2yndt2=exp(yn-1-yn)-exp(yn-yn+1),(1.5)u̇n=un(vn-1-vn),v̇n=un-un+1.
Under the transformation ∂un/∂t=exp(yn-1-yn)-1,   (1.3) and (1.4) can be reduced to   (1.1) and (1.2), respectively. For   (1.1) and (1.2), there are important physical models and biological models which have been studied by many authors, see [7, 10, 21–39] and references cited therein. For example, the Toda lattice equation was used as DNA models [36–38], this equation can also be used as a model to describe the pressure pulse wave in aorta.Recently, by using the exp-function method,   Zhu [7] studied (2 + 1)-dimensional Toda lattice equation and obtained some exact solutions of exp-function type. By using tanh method,  S. Zhang and H.-Q. Zhang  [10] studied the same equation, they obtained some exact solutions of hyperbolic function type. By using the symbolic computation of hyperbolic tangent solutions, Baldwin et al. [11] studied both   (1.1) and (1.2), and they obtained some exact solutions of tanh-function type. By using the modified hyperbolic function method, Zhi et al. [34] studied   (1.2), they obtained some discrete soliton solutions of hyperbolic function type. In this paper, using the extended mixed function method, we discuss   (1.1) and (1.2), and new solutions which are different from the results in   [7, 10, 11, 34] are obtained.The rest of this paper is organized as follows. In Section2, we introduce the extended mixed function method. In Section 3, by using the extended method, we obtain new exact solutions of (1.1), and discuss their dynamic properties. In Section 4, we will obtain new exact solutions of   (1.2), and also discuss their dynamic properties.


## 2. The Extended Mixed Function Method

In this section, we begin formally introducing the extended mixed function method motivated by   [1].First, we extend the mixed function method from (1 + 1)-dimensional space to (2 + 1)-dimensional one, and later, we present some new forms of solutions on establishing different kinds of exact solutions of NDDEs. Without loss of generality, we suppose that a polynomial (2 + 1)-dimensional NDDEs and (1 + 1)-dimensional NDDEs have the following form:(2.1)Ψ(…,un-2,un-1,un,un+1,un+2,…,∂un∂t,∂un∂x,∂2un∂x∂t,…)=0,(2.2)Ψ(…,un-2,un-1,un,un+1,un+2,…,∂un∂t,∂2un∂t2,…)=0,
respectively, where un=u(t,n,x) in   (2.1) and un=u(t,n) in   (2.2), n is a discrete variable, and x and t are two continuous variable. The general framework of the extended method for NDDEs is shown in the following.Step 1.
Corresponding to   (2.1), as in [1], we suppose that its exact solutions has the following three expansions:
(2.3)un=∑p=0map[ωneαx-βt+γ0-Be-(αx-βt+γ0)ωneαx-βt+γ0+Ce-(αx-βt+γ0)]p,(2.4)un=∑p=0map[wnw2neαx-βt+γ0+Ce-(αx-βt+γ0)]p,(2.5)un=∑p=0map[wnexp(αx-βt+γ0)]p∑p=0mbp[wnexp(αx-βt+γ0)]p,
where m is a positive integer which can be given by the homogeneous balance principle or the character of the idiographic NDDEs, the parameters B,C,w,α,β,γ0,ap,bp(p=0,1,…,m) are constants which need to be determined later, and k∈N,B≠-C,a0/b0≠a2/b2≠⋯≠am/bm.
Besides the above forms of solutions, we will add three new expansions. If   (2.1) has no term ∂un/∂x, then we suppose their expansions as follows:
(2.6)un=Anf(x)+∑p=0mbp[ωneβt+γ0-Be-(βt+γ0)ωneβt+γ0+Ce-(βt+γ0)]p,(2.7)un=Anf(x)+∑p=0mbp[wnw2neβt+γ0+Ce-(βt+γ0)]p,b2k+1=0,(2.8)un=Anf(x)+∑p=0mcp(wn)p(eβt+γ0)p+B(wn)p,c2k+1=0,
where f(x) is an arbitrary function which can be chosen as hyperbolic functions and trigonometric functions. In  [1], we supposed that   (2.2)'s exact solution is one of the following two kinds of expansion forms:
(2.9)un=∑i=0mai[wneΩt-Be-ΩtwneΩt+Ce-Ωt]i,(2.10)un=∑i=0mai[wnw2neΩt+Ce-Ωt]i.
We add two new forms of solutions for   (2.2) here
(2.11)un=∑p=0map(wneΩt)p∑p=0mbp(wneΩt)p,a2k+1=0,b2k+1=0,(2.12)un=∑p=0map(wn)p(eΩt)p+B(wn)p,a2k+1=0,
where m is a positive integer which can be given by the homogeneous balance principle or the character of the idiographic NDDEs, the parameters B,C,w,Ω,ap,bp(p=0,1,…,m) are constants which need to be determined later, and k∈N,B≠-C,a0/b0≠a2/b2≠⋯≠am/bm.Step 2.
By using the homogeneous balance principle or according to character of the idiographic NDDEs, we determine the value ofm then substitute it in the expressions (2.3)–(2.12) of Step 1. Sometimes, we can directly assume that m=1 or m=2. In [8], by using the homogeneous balance method, the balance number of the generalized hybrid equation has been obtained successfully.Step 3.
Substituting the presupposed solutions determined by Step2 in the original equation (2.1) or (2.2), then setting the coefficients of all independent terms in ekξ,e-kξ(k=0,1,2,…,N,ξ=αx-βt+γ0 or ξ=Ωt) to zero, and we get a series of algebraic equations from which the corresponding undetermined constants are explicitly solved by the use of mathematical software Maple or Mathematica.Step 4.
Substituting the values of these constantsB,C,w,Ω,α,β,γ0,ap,bp(p=0,1,…,m) given by Step 3 in the solutions presupposed by Step 2, thus, the exact solutions of the original equation  (2.1) or (2.2) are obtained finally.


## 3. New Exact Solutions of (2 + 1)-Dimensional Toda Lattice Equation and Their Dynamic Properties

In this section, by using the extended method shown in the Section2, we discuss exact solutions of (2 + 1)-dimensional Toda lattice equation (1.1) and their dynamic properties.


### 3.1. Exact Solutions of the Form (2.3)

Suppose that (1.1) has exact solutions of the form (2.3). By using the balance procedure, we easily obtain m=1. Thus, we let(3.1)un=a0+a1[wnexp(αx-βt+γ0)-Bexp(-(αx-βt+γ0))wnexp(αx-βt+γ0)+Cexp(-(αx-βt+γ0))],
where a0, a1, w,α, β,γ0,B, C are constants to be determined later, and B≠-C. After substituting  (3.1) into   (1.1), multiplying both sides by the common denominator (ωneαx-βt+γ0+Ce-(αx-βt+γ0))3(ωn-1eαx-βt+γ0+Ce-(αx-βt+γ0))(ωn+1eαx-βt+γ0+Ce-(αx-βt+γ0)) and dividing both sides by the common factor a1(B+C), it follows:(3.2)A-3e-3(αx-βt+γ0)-A-1e-(αx-βt+γ0)+A1eαx-βt+γ0+A3e3(αx-βt+γ0)=0,
where (3.3)A-3=-C3[wn+1+(4αβ-2)wn+wn-1],A-1=C[(-4αβC+2a1βB-C+2a1βC)w2n+1+(-4a1βB+2C-4a1βC+4αβC)w2n+(-4αβC+2a1βB-C+2a1βC)w2n-1],A1=(-4αβC+2a1βB-C+2a1βC)w3n+1-(-4a1βB+2C-4a1βC+4αβC)w3n-(-4αβC+2a1βB-C+2a1βC)w3n-1,A3=w4n+1+(4αβ-2)w4n+w4n-1.In (3.2), setting the coefficients of all independent terms e-3(αx-βt+γ0),e-(αx-βt+γ0), eαx-βt+γ0,e3(αx-βt+γ0) to zero, we get a series of algebraic equations as follows:(3.4)A-3=0,A-1=0,A1=0,A3=0.
Equation (3.4) is a group of high-power algebraic equations, so the computational load may be heavy when we use computer to solve it. Notice that A-3=-C3wn-1[w2+(4αβ-2)w+1], we let (3.5)M-3=A-3(-C3wn-1)=w2+(4αβ-2)w+1.
As a result, the equation   A-3=0 becomes M-3=0, that is, w2+(4αβ-2)w+1=0, which is a low-power equation. Similarly, we obtain (3.6)M-1=A-1(Cw2n-1)=(-4αβC+2a1βB-C+2a1βC)w2+(-4a1βB+2C-4a1βC+4αβC)w+(-4αβC+2a1βB-C+2a1βC),M1=A1(-w3n-1)=(-4αβC+2a1βB-C+2a1βC)w2+(-4a1βB+2C-4a1βC+4αβC)w+(-4αβC+2a1βB-C+2a1βC),M3=A3w4n-1=w2+(4αβ-2)w+1.
Notice that M3=M-3,M1=M-1, thus,  (3.4) can be reduced to a group of low-power algebraic equations as follows:(3.7)M1=0,M3=0.
Solving   (3.7) yields(3.8)B=C(2α-a1)a1,(3.9)w1,2=1-2αβ±2αβ(αβ-1),
and C,α,β,γ0,a0,a1 are arbitrary nonzero constants. The process of computation can be greatly simplified by the above settings, and time is saved.From   (3.1), (3.8), and (3.9), we obtain two exact solutions of   (1.1) as follows:(3.10)un=a0+a1[w1nexp(αx-βt+γ0)-(C(2α-a1)/a1)exp(-αx+βt-γ0)w1nexp(αx-βt+γ0)+Cexp(-αx+βt-γ0)],(3.11)un=a0+a1[w2nexp(αx-βt+γ0)-(C(2α-a1)/a1)exp(-αx+βt-γ0)w2nexp(αx-βt+γ0)+Cexp(-αx+βt-γ0)],
where w1,2 are given by   (3.9).In fact,   (3.10) and (3.11) are solutions of local discretization because the n is a discrete variable while the x,t are continuous variables. Here we still call them discrete exact solutions. In order to describe the dynamic properties of these two discrete soliton solutions intuitively, we plot their profile figures for some fixed parametric values, as shown in Figure 1. Setting a0=2,a1=0.3,α=-1.5,β=2,γ0=-1,t=1, when C=4>0, the solution (3.10) shows a shape of discrete kink soliton, see Figure 1(a); when C=-4<0, the solution un (3.10) shows another shape of discrete kink soliton, see Figure 1(b). The shape shown by Figure 1(b) has jumping phenomenon, but this is not the case shown by Figure 1(a). Setting a0=2,a1=0.3,α=0.8,β=1.5,γ0=3,t=1, when C=1.4>0, the solution (3.11) shows a shape of discrete antikink soliton, see Figure 1(c); when C=-1.4<0, the solution un (3.11) shows another shape of discrete antikink soliton, see Figure 1(d). From Figures 1(c) to 1(d), the waveforms have both discrete and continuous character, and the soliton is discrete along the n-axes and is continuous along the x-axes.The solutionsun in   (3.10) and (3.11) show four shapes of discrete kink and  antikink solitons for fixed parametric values of a0,a1,α,β,γ0,t and different kinds of C values: (a) C=4>0; (b)  C=-4<0; (c) C=1.4>0; (d)  C=-1.4<0.


### 3.2. Exact Solutions of the Form (2.5)

Suppose that   (1.1) has exact solution of the form (2.5). By using the balance procedure, we obtain m=2. For simplicity, here we only consider the case of a1=b1=0. Suppose that(3.12)un=a0+a2w2nexp[2(αx-βt+γ0)]b0+b2w2nexp[2(αx-βt+γ0)],
where a0, a2, b0, b2, w, α, β are constants to be determined later, and a0/b0≠a2/b2, the γ0 is an arbitrary constant. As in  Section 3.1, after substituting   (3.12) into  (1.1), multiplying both sides of the result by the common denominator (b0+b2w2ne2(αx-βt+γ0))3(b0+b2w2(n-1)e2(αx-βt+γ0))(b0+b2w2(n+1)e2(αx-βt+γ0)), it follows:(3.13)A2e2(αx-βt+γ0)+A4e4(αx-βt+γ0)+A6e6(αx-βt+γ0)+A8e8(αx-βt+γ0)=0,
where (3.14)A2=(b03w2n-2)[(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0],A4=(b0w4n-2)[(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4],A6=(-b2w6n-2)[(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4],A8=(-b23w8n-2)[(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0].
In  (3.13), setting the coefficients of all independent exp-function terms to zero, it follows(3.15)A2=0,A4=0,A6=0,A8=0.
Let M2=A2/(b03w2n-2),M4=A4/(b0w4n-2),M6=A6/(-b2w6n-2),M8=A8/(-b23w8n-2). Then (3.16)M2=M8=(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0,M4=M6=(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)].
Thus,   (3.15) can be reduced to the following equations: (3.17)M2=0,M4=0.
Solving  (3.17), we obtain(3.18)α=b0a2-a0b22b0b2,(3.19)w1,2=±b0b2(b0b2-b0a2β+a0b2β+β(b2a0-a2b0)(a0b2β+2b0b2-a2b0β))b0b2,(3.20)w3,4=±b0b2(b0b2-b0a2β+a0b2β-β(b2a0-a2b0)(a0b2β+2b0b2-a2b0β))b0b2,
and a0,a2,b0,b2,β,γ0 are arbitrary nonzero constants with a0/b0≠a2/b2.Based on  (3.12), (3.18), (3.19), and (3.20), we obtain four exact solutions of   (1.1) as follows:(3.21)un=a0+a2(w1)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w1)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.22)un=a0+a2(w2)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w2)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.23)un=a0+a2(w3)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w3)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.24)un=a0+a2(w4)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w4)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),
where w1,w2,w3,w4 are given by  (3.19), (3.20), and w1,2,3,4≠1,b0a2≠a0b2.The dynamic properties of solutions (3.21), (3.22), (3.23), and (3.24) are similar to those of (3.10) and (3.11). As an example, we plot two profile figures of the solutions (3.21) and (3.22), see Figure 2.The solutionsun in   (3.21) and (3.22) show two shapes of discrete antikink  and kink solitons for fixed parametric values: a0=2, b0=3, a2=-1.5, b2=2.5, β=0.8, γ0=5, t=2.


### 3.3. Exact Solutions of the Form (2.6)

Suppose that (1.1) has exact solution of the form (2.6). By using the balance procedure, we obtain m=1. So, we suppose that   (1.1)'s solution has the following form:(3.25)un=Anf(x)+b0+b1(wneβt+γ0-Be-βt-γ0wneβt+γ0+Ce-βt-γ0),
where A, B, C, b0, b1, w, β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in Section 3.1, after substituting (3.25) into   (1.1), multiplying both sides of the result by the common denominator (wneβt+γ0+Ce-βt-γ0)3(wn-1eβt+γ0+Ce-βt-γ0)(wn+1eβt+γ0+Ce-βt-γ0), it follows:(3.26)A-3e-3(βt+γ0)+A-1e-(βt+γ0)+A1eβt+γ0+A3e3(βt+γ0)=0,
where (3.27)A-3=-a1C3(B+C)wn-1(w-1)2,A-1=-a1C(B+C)(2a1βB+C+2a1βC)w2n-1(w-1)2,A1=a1(B+C)(2a1βB+C+2a1βC)w3n-1(w-1)2,A3=a1(B+C)w4n-1(w-1)2.In (3.26), setting the coefficients of all independent exp-function terms to zero, we obtain(3.28)A-3=0,A-1=0,A1=0,A3=0.
Directly solving  (3.28), we obtain w=1 and A,B,C,β,b0,b1 are arbitrary constants. Thus, we obtain a class of exact solutions of  (1.1) as follows:(3.29)un=Anf(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),
where f(x) is an arbitrary function, for which let us set f(x) to be tanh(x), sech(x), sin(x), cos(x), tanh(x)+sin(x), sech(x)+cos(x); respectively, we obtain a series of soliton solutions and breather solutions as follows:(3.30)un=Antanh(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.31)un=Ansech(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.32)un=Ansin(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.33)un=Ancos(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.34)un=An[tanh(x)+sin(x)]+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.35)un=An[sech(x)+cos(x)]+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0).In order to describe the dynamic properties of the above solutions intuitively, as examples, we draw the profile figures of the solutions (3.31), (3.34), and (3.35) in Figure 3. Setting A=0.5,B=3,C=5,b0=2,b1=-2,β=1,γ0=-2,t=1,n∈[-15,15],x∈[-15,15], the solution (3.31) shows a shape of discrete soliton, see Figure 3(a); the solutions (3.34) and (3.35) show two shapes of discrete breather oscillations, see Figures 3(c) and 3(e). Setting A=0.5,B=3,C=5,b0=2,b1=-2,β=1,γ0=-2,n=2,t∈[-15,15],x∈[-15,15], the solution (3.31) shows a shape of continuous double soliton, see Figure 3(b); the solutions (3.34) and (3.35) show two shapes of continuous breather oscillations, see Figures 3(d) and 3(f). From Figure 3, the solutions (3.30)–(3.35) have diplex dynamic characters, that is, the discrete character and the continuous character coexist. If the parameter t is fixed, the profiles show discrete dynamic character, see Figures 3(a), 3(c), and 3(e); while the parameter n is fixed, the profiles show continuous dynamic character, see Figures 3(b), 3(d), and 3(f).The solutionsun in   (3.31), (3.34), and (3.35) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
(a)
Discrete soliton(b)
Continuous double soliton(c)
Discrete breather oscillation(d)
Continuous breather oscillation(e)
Discrete breather oscillation(f)
Continuous breather oscillation


### 3.4. Exact Solutions of the Form (2.7)

Suppose that   (1.1) has exact solution of the form (2.7). By using the balance procedure, we obtain m=2. So, we suppose that   (1.1) has exact solution as the following form:(3.36)un=Anf(x)+b0+b2(wnw2neβt+γ0+Ce-βt-γ0)2,
where A, C, b0, b2, w, β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in  Section 3.3, by using  (3.36) and (1.1), we can obtain the exact solution of   (1.1); here, we omit those processes and directly give the result as follows:(3.37)un=Anf(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2.
Similarly, setting f(x) to be tanh2(x),sech2(x),sin2(x),cos2(x),tanh2(x)+sin2(x),sech2(x)+cos2(x), respectively, we obtain a series of soliton solutions and breather solutions as follows:(3.38)un=Antanh2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.39)un=Ansech2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.40)un=Ansin2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.41)un=Ancos2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.42)un=An[tanh2(x)+sin2(x)]+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.43)un=An[sech2(x)+cos2(x)]+b0+b2(1eβt+γ0+Ce-βt-γ0)2.The dynamic properties of solutions (3.38)–(3.43) are similar to those of solutions (3.30)–(3.35). As examples, we plot six profile figures of the solutions (3.39), (3.42), and (3.43), see Figure 4. Setting A=0.2,C=3,b0=4,b2=200,β=0.8,γ0=-2,t=1,n∈[-20,20],x∈[-20,20], the solution (3.39) shows a shape of discrete soliton, see Figure 4(a); the solutions (3.42) and (3.43) show two shapes of discrete breather oscillations, see Figures 4(c) and 4(e). Setting A=0.2,C=3,b0=4,b2=2,β=0.8,γ0=-2,n=1,t∈[-5,8],x∈[-6,6],   (3.39) shows a shape of continuous double soliton, see Figure 4(b); the solutions (3.42) and (3.43) show two shapes of continuous breather oscillations, see Figures 4(d) and 4(f).The solutionsun in   (3.39), (3.42), and (3.43) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
(a)
Discrete soliton(b)
Continuous double soliton(c)
Discrete breather oscillation(d)
Continuous breather oscillation(e)
Discrete breather oscillation(f)
Continuous breather oscillation


### 3.5. Exact Solutions of the Form (2.8)

Suppose that  (1.1) has exact solution of the form (2.8). By using the balance procedure, we obtain m=2. So, we suppose that   (1.1) has exact solution as the following form:(3.44)un=Anf(x)+c01+B+c2w2ne2(βt+γ0)+Bw2n,
where A, B,c0,c2, w,β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in Section 3.3, by using   (3.44) and (1.1) and the expatiatory computation, we obtain w=±1. So we obtain a family of exact solutions with arbitrary function as follows:(3.45)un=Anf(x)+c01+B+c2e2(βt+γ0)+B.
Similarly, assume f(x) the same functions as in Section 3.3, we obtain a series of soliton solutions and breather solutions as follows:(3.46)un=Antanh(x)+c01+B+c2e2(βt+γ0)+B,(3.47)un=Ansech(x)+c01+B+c2e2(βt+γ0)+B,(3.48)un=Ansin(x)+c01+B+c2e2(βt+γ0)+B,(3.49)un=Ancos(x)+c01+B+c2e2(βt+γ0)+B,(3.50)un=An[tanh(x)+sin(x)]+c01+B+c2e2(βt+γ0)+B,(3.51)un=An[sech(x)+cos(x)]+c01+B+c2e2(βt+γ0)+B.The dynamic properties of solutions (3.46)–(3.51) are similar to those of solutions (3.30)–(3.35). As examples, we plot six profile figures of the solutions (3.46), (3.47), and (3.48), see Figure 5. Setting A=0.2,C=3,b0=4,b2=200,β=0.8,γ0=-2,t=1,n∈[-20,20],x∈[-20,20], the solution (3.46) shows a shape of discrete kink soliton, see Figure 5(a); the solution (3.47) shows a shape of discrete soliton, see Figure 5(c); the solution (3.48) shows a shape of discrete breather oscillations, see Figure 5(e). Setting A=0.2,C=3,b0=4,b2=2,β=0.8,γ0=-2,n=1,t∈[-5,8],x∈[-6,6], the solution (3.46) shows a shape of continuous double kink soliton, see Figure 5(b); the solution (3.47) shows a shape of continuous double soliton, see Figure 5(c); the solution (3.48) shows a shape of continuous breather oscillations, see Figure 5(e).The solutionsun in   (3.46), (3.47) and (3.48) show six shapes of discrete  and continuous solitons, and discrete and continuous breather oscillations.
(a)
Discrete kink soliton(b)
Continuous double kink soliton(c)
Discrete soliton(d)
Continuous double soliton(e)
Discrete breather oscillation(f)
Continuous breather oscillation


## 3.1. Exact Solutions of the Form (2.3)

Suppose that (1.1) has exact solutions of the form (2.3). By using the balance procedure, we easily obtain m=1. Thus, we let(3.1)un=a0+a1[wnexp(αx-βt+γ0)-Bexp(-(αx-βt+γ0))wnexp(αx-βt+γ0)+Cexp(-(αx-βt+γ0))],
where a0, a1, w,α, β,γ0,B, C are constants to be determined later, and B≠-C. After substituting  (3.1) into   (1.1), multiplying both sides by the common denominator (ωneαx-βt+γ0+Ce-(αx-βt+γ0))3(ωn-1eαx-βt+γ0+Ce-(αx-βt+γ0))(ωn+1eαx-βt+γ0+Ce-(αx-βt+γ0)) and dividing both sides by the common factor a1(B+C), it follows:(3.2)A-3e-3(αx-βt+γ0)-A-1e-(αx-βt+γ0)+A1eαx-βt+γ0+A3e3(αx-βt+γ0)=0,
where (3.3)A-3=-C3[wn+1+(4αβ-2)wn+wn-1],A-1=C[(-4αβC+2a1βB-C+2a1βC)w2n+1+(-4a1βB+2C-4a1βC+4αβC)w2n+(-4αβC+2a1βB-C+2a1βC)w2n-1],A1=(-4αβC+2a1βB-C+2a1βC)w3n+1-(-4a1βB+2C-4a1βC+4αβC)w3n-(-4αβC+2a1βB-C+2a1βC)w3n-1,A3=w4n+1+(4αβ-2)w4n+w4n-1.In (3.2), setting the coefficients of all independent terms e-3(αx-βt+γ0),e-(αx-βt+γ0), eαx-βt+γ0,e3(αx-βt+γ0) to zero, we get a series of algebraic equations as follows:(3.4)A-3=0,A-1=0,A1=0,A3=0.
Equation (3.4) is a group of high-power algebraic equations, so the computational load may be heavy when we use computer to solve it. Notice that A-3=-C3wn-1[w2+(4αβ-2)w+1], we let (3.5)M-3=A-3(-C3wn-1)=w2+(4αβ-2)w+1.
As a result, the equation   A-3=0 becomes M-3=0, that is, w2+(4αβ-2)w+1=0, which is a low-power equation. Similarly, we obtain (3.6)M-1=A-1(Cw2n-1)=(-4αβC+2a1βB-C+2a1βC)w2+(-4a1βB+2C-4a1βC+4αβC)w+(-4αβC+2a1βB-C+2a1βC),M1=A1(-w3n-1)=(-4αβC+2a1βB-C+2a1βC)w2+(-4a1βB+2C-4a1βC+4αβC)w+(-4αβC+2a1βB-C+2a1βC),M3=A3w4n-1=w2+(4αβ-2)w+1.
Notice that M3=M-3,M1=M-1, thus,  (3.4) can be reduced to a group of low-power algebraic equations as follows:(3.7)M1=0,M3=0.
Solving   (3.7) yields(3.8)B=C(2α-a1)a1,(3.9)w1,2=1-2αβ±2αβ(αβ-1),
and C,α,β,γ0,a0,a1 are arbitrary nonzero constants. The process of computation can be greatly simplified by the above settings, and time is saved.From   (3.1), (3.8), and (3.9), we obtain two exact solutions of   (1.1) as follows:(3.10)un=a0+a1[w1nexp(αx-βt+γ0)-(C(2α-a1)/a1)exp(-αx+βt-γ0)w1nexp(αx-βt+γ0)+Cexp(-αx+βt-γ0)],(3.11)un=a0+a1[w2nexp(αx-βt+γ0)-(C(2α-a1)/a1)exp(-αx+βt-γ0)w2nexp(αx-βt+γ0)+Cexp(-αx+βt-γ0)],
where w1,2 are given by   (3.9).In fact,   (3.10) and (3.11) are solutions of local discretization because the n is a discrete variable while the x,t are continuous variables. Here we still call them discrete exact solutions. In order to describe the dynamic properties of these two discrete soliton solutions intuitively, we plot their profile figures for some fixed parametric values, as shown in Figure 1. Setting a0=2,a1=0.3,α=-1.5,β=2,γ0=-1,t=1, when C=4>0, the solution (3.10) shows a shape of discrete kink soliton, see Figure 1(a); when C=-4<0, the solution un (3.10) shows another shape of discrete kink soliton, see Figure 1(b). The shape shown by Figure 1(b) has jumping phenomenon, but this is not the case shown by Figure 1(a). Setting a0=2,a1=0.3,α=0.8,β=1.5,γ0=3,t=1, when C=1.4>0, the solution (3.11) shows a shape of discrete antikink soliton, see Figure 1(c); when C=-1.4<0, the solution un (3.11) shows another shape of discrete antikink soliton, see Figure 1(d). From Figures 1(c) to 1(d), the waveforms have both discrete and continuous character, and the soliton is discrete along the n-axes and is continuous along the x-axes.The solutionsun in   (3.10) and (3.11) show four shapes of discrete kink and  antikink solitons for fixed parametric values of a0,a1,α,β,γ0,t and different kinds of C values: (a) C=4>0; (b)  C=-4<0; (c) C=1.4>0; (d)  C=-1.4<0.


## 3.2. Exact Solutions of the Form (2.5)

Suppose that   (1.1) has exact solution of the form (2.5). By using the balance procedure, we obtain m=2. For simplicity, here we only consider the case of a1=b1=0. Suppose that(3.12)un=a0+a2w2nexp[2(αx-βt+γ0)]b0+b2w2nexp[2(αx-βt+γ0)],
where a0, a2, b0, b2, w, α, β are constants to be determined later, and a0/b0≠a2/b2, the γ0 is an arbitrary constant. As in  Section 3.1, after substituting   (3.12) into  (1.1), multiplying both sides of the result by the common denominator (b0+b2w2ne2(αx-βt+γ0))3(b0+b2w2(n-1)e2(αx-βt+γ0))(b0+b2w2(n+1)e2(αx-βt+γ0)), it follows:(3.13)A2e2(αx-βt+γ0)+A4e4(αx-βt+γ0)+A6e6(αx-βt+γ0)+A8e8(αx-βt+γ0)=0,
where (3.14)A2=(b03w2n-2)[(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0],A4=(b0w4n-2)[(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4],A6=(-b2w6n-2)[(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4],A8=(-b23w8n-2)[(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0].
In  (3.13), setting the coefficients of all independent exp-function terms to zero, it follows(3.15)A2=0,A4=0,A6=0,A8=0.
Let M2=A2/(b03w2n-2),M4=A4/(b0w4n-2),M6=A6/(-b2w6n-2),M8=A8/(-b23w8n-2). Then (3.16)M2=M8=(b2a0-a2b0)w4+2(b2a0-a2b0)(2αβ-1)w2+b2a0-a2b0,M4=M6=(b2a0-a2b0)(2b2a0β+4b2b0αβ+b0b2-2a2b0β)w4-2(b2a0-a2b0)(2b2a0β+2b2b0αβ+b0b2-2a2b0β)w2+(b2a0-a2b0)×(2b2a0β+4b2b0αβ+b0b2-2a2b0β)].
Thus,   (3.15) can be reduced to the following equations: (3.17)M2=0,M4=0.
Solving  (3.17), we obtain(3.18)α=b0a2-a0b22b0b2,(3.19)w1,2=±b0b2(b0b2-b0a2β+a0b2β+β(b2a0-a2b0)(a0b2β+2b0b2-a2b0β))b0b2,(3.20)w3,4=±b0b2(b0b2-b0a2β+a0b2β-β(b2a0-a2b0)(a0b2β+2b0b2-a2b0β))b0b2,
and a0,a2,b0,b2,β,γ0 are arbitrary nonzero constants with a0/b0≠a2/b2.Based on  (3.12), (3.18), (3.19), and (3.20), we obtain four exact solutions of   (1.1) as follows:(3.21)un=a0+a2(w1)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w1)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.22)un=a0+a2(w2)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w2)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.23)un=a0+a2(w3)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w3)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),(3.24)un=a0+a2(w4)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0)b0+b2(w4)2nexp(((b0a2-a0b2)/b0b2)x-2βt+2γ0),
where w1,w2,w3,w4 are given by  (3.19), (3.20), and w1,2,3,4≠1,b0a2≠a0b2.The dynamic properties of solutions (3.21), (3.22), (3.23), and (3.24) are similar to those of (3.10) and (3.11). As an example, we plot two profile figures of the solutions (3.21) and (3.22), see Figure 2.The solutionsun in   (3.21) and (3.22) show two shapes of discrete antikink  and kink solitons for fixed parametric values: a0=2, b0=3, a2=-1.5, b2=2.5, β=0.8, γ0=5, t=2.


## 3.3. Exact Solutions of the Form (2.6)

Suppose that (1.1) has exact solution of the form (2.6). By using the balance procedure, we obtain m=1. So, we suppose that   (1.1)'s solution has the following form:(3.25)un=Anf(x)+b0+b1(wneβt+γ0-Be-βt-γ0wneβt+γ0+Ce-βt-γ0),
where A, B, C, b0, b1, w, β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in Section 3.1, after substituting (3.25) into   (1.1), multiplying both sides of the result by the common denominator (wneβt+γ0+Ce-βt-γ0)3(wn-1eβt+γ0+Ce-βt-γ0)(wn+1eβt+γ0+Ce-βt-γ0), it follows:(3.26)A-3e-3(βt+γ0)+A-1e-(βt+γ0)+A1eβt+γ0+A3e3(βt+γ0)=0,
where (3.27)A-3=-a1C3(B+C)wn-1(w-1)2,A-1=-a1C(B+C)(2a1βB+C+2a1βC)w2n-1(w-1)2,A1=a1(B+C)(2a1βB+C+2a1βC)w3n-1(w-1)2,A3=a1(B+C)w4n-1(w-1)2.In (3.26), setting the coefficients of all independent exp-function terms to zero, we obtain(3.28)A-3=0,A-1=0,A1=0,A3=0.
Directly solving  (3.28), we obtain w=1 and A,B,C,β,b0,b1 are arbitrary constants. Thus, we obtain a class of exact solutions of  (1.1) as follows:(3.29)un=Anf(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),
where f(x) is an arbitrary function, for which let us set f(x) to be tanh(x), sech(x), sin(x), cos(x), tanh(x)+sin(x), sech(x)+cos(x); respectively, we obtain a series of soliton solutions and breather solutions as follows:(3.30)un=Antanh(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.31)un=Ansech(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.32)un=Ansin(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.33)un=Ancos(x)+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.34)un=An[tanh(x)+sin(x)]+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0),(3.35)un=An[sech(x)+cos(x)]+b0+b1(eβt+γ0-Be-βt-γ0eβt+γ0+Ce-βt-γ0).In order to describe the dynamic properties of the above solutions intuitively, as examples, we draw the profile figures of the solutions (3.31), (3.34), and (3.35) in Figure 3. Setting A=0.5,B=3,C=5,b0=2,b1=-2,β=1,γ0=-2,t=1,n∈[-15,15],x∈[-15,15], the solution (3.31) shows a shape of discrete soliton, see Figure 3(a); the solutions (3.34) and (3.35) show two shapes of discrete breather oscillations, see Figures 3(c) and 3(e). Setting A=0.5,B=3,C=5,b0=2,b1=-2,β=1,γ0=-2,n=2,t∈[-15,15],x∈[-15,15], the solution (3.31) shows a shape of continuous double soliton, see Figure 3(b); the solutions (3.34) and (3.35) show two shapes of continuous breather oscillations, see Figures 3(d) and 3(f). From Figure 3, the solutions (3.30)–(3.35) have diplex dynamic characters, that is, the discrete character and the continuous character coexist. If the parameter t is fixed, the profiles show discrete dynamic character, see Figures 3(a), 3(c), and 3(e); while the parameter n is fixed, the profiles show continuous dynamic character, see Figures 3(b), 3(d), and 3(f).The solutionsun in   (3.31), (3.34), and (3.35) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
(a)
Discrete soliton(b)
Continuous double soliton(c)
Discrete breather oscillation(d)
Continuous breather oscillation(e)
Discrete breather oscillation(f)
Continuous breather oscillation


## 3.4. Exact Solutions of the Form (2.7)

Suppose that   (1.1) has exact solution of the form (2.7). By using the balance procedure, we obtain m=2. So, we suppose that   (1.1) has exact solution as the following form:(3.36)un=Anf(x)+b0+b2(wnw2neβt+γ0+Ce-βt-γ0)2,
where A, C, b0, b2, w, β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in  Section 3.3, by using  (3.36) and (1.1), we can obtain the exact solution of   (1.1); here, we omit those processes and directly give the result as follows:(3.37)un=Anf(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2.
Similarly, setting f(x) to be tanh2(x),sech2(x),sin2(x),cos2(x),tanh2(x)+sin2(x),sech2(x)+cos2(x), respectively, we obtain a series of soliton solutions and breather solutions as follows:(3.38)un=Antanh2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.39)un=Ansech2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.40)un=Ansin2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.41)un=Ancos2(x)+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.42)un=An[tanh2(x)+sin2(x)]+b0+b2(1eβt+γ0+Ce-βt-γ0)2,(3.43)un=An[sech2(x)+cos2(x)]+b0+b2(1eβt+γ0+Ce-βt-γ0)2.The dynamic properties of solutions (3.38)–(3.43) are similar to those of solutions (3.30)–(3.35). As examples, we plot six profile figures of the solutions (3.39), (3.42), and (3.43), see Figure 4. Setting A=0.2,C=3,b0=4,b2=200,β=0.8,γ0=-2,t=1,n∈[-20,20],x∈[-20,20], the solution (3.39) shows a shape of discrete soliton, see Figure 4(a); the solutions (3.42) and (3.43) show two shapes of discrete breather oscillations, see Figures 4(c) and 4(e). Setting A=0.2,C=3,b0=4,b2=2,β=0.8,γ0=-2,n=1,t∈[-5,8],x∈[-6,6],   (3.39) shows a shape of continuous double soliton, see Figure 4(b); the solutions (3.42) and (3.43) show two shapes of continuous breather oscillations, see Figures 4(d) and 4(f).The solutionsun in   (3.39), (3.42), and (3.43) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
(a)
Discrete soliton(b)
Continuous double soliton(c)
Discrete breather oscillation(d)
Continuous breather oscillation(e)
Discrete breather oscillation(f)
Continuous breather oscillation


## 3.5. Exact Solutions of the Form (2.8)

Suppose that  (1.1) has exact solution of the form (2.8). By using the balance procedure, we obtain m=2. So, we suppose that   (1.1) has exact solution as the following form:(3.44)un=Anf(x)+c01+B+c2w2ne2(βt+γ0)+Bw2n,
where A, B,c0,c2, w,β are constants to be determined later, γ0 is an arbitrary constant, and f(x) is an arbitrary function. As in Section 3.3, by using   (3.44) and (1.1) and the expatiatory computation, we obtain w=±1. So we obtain a family of exact solutions with arbitrary function as follows:(3.45)un=Anf(x)+c01+B+c2e2(βt+γ0)+B.
Similarly, assume f(x) the same functions as in Section 3.3, we obtain a series of soliton solutions and breather solutions as follows:(3.46)un=Antanh(x)+c01+B+c2e2(βt+γ0)+B,(3.47)un=Ansech(x)+c01+B+c2e2(βt+γ0)+B,(3.48)un=Ansin(x)+c01+B+c2e2(βt+γ0)+B,(3.49)un=Ancos(x)+c01+B+c2e2(βt+γ0)+B,(3.50)un=An[tanh(x)+sin(x)]+c01+B+c2e2(βt+γ0)+B,(3.51)un=An[sech(x)+cos(x)]+c01+B+c2e2(βt+γ0)+B.The dynamic properties of solutions (3.46)–(3.51) are similar to those of solutions (3.30)–(3.35). As examples, we plot six profile figures of the solutions (3.46), (3.47), and (3.48), see Figure 5. Setting A=0.2,C=3,b0=4,b2=200,β=0.8,γ0=-2,t=1,n∈[-20,20],x∈[-20,20], the solution (3.46) shows a shape of discrete kink soliton, see Figure 5(a); the solution (3.47) shows a shape of discrete soliton, see Figure 5(c); the solution (3.48) shows a shape of discrete breather oscillations, see Figure 5(e). Setting A=0.2,C=3,b0=4,b2=2,β=0.8,γ0=-2,n=1,t∈[-5,8],x∈[-6,6], the solution (3.46) shows a shape of continuous double kink soliton, see Figure 5(b); the solution (3.47) shows a shape of continuous double soliton, see Figure 5(c); the solution (3.48) shows a shape of continuous breather oscillations, see Figure 5(e).The solutionsun in   (3.46), (3.47) and (3.48) show six shapes of discrete  and continuous solitons, and discrete and continuous breather oscillations.
(a)
Discrete kink soliton(b)
Continuous double kink soliton(c)
Discrete soliton(d)
Continuous double soliton(e)
Discrete breather oscillation(f)
Continuous breather oscillation


## 4. New Exact Solutions of Toda Lattice Equation and Its Dynamic Properties

In this section, using the extended method offered in Section2, we discuss the exact solutions of the Toda lattice equation (1.2).


### 4.1. Exact Solutions of the Form (2.9)

Suppose that   (1.2) has exact solution of the form (2.9) as follows:(4.1)un=a0+a1(wneΩt-Be-ΩtwneΩt+Ce-Ωt).
After substituting (4.1) in   (1.2), multiplying both sides by the common denominator (wneΩt+Ce-Ωt)3(wn-1eΩt+Ce-Ωt)(wn+1eΩt+Ce-Ωt), it follows:(4.2)A-3e-3Ωt+A-1e-Ωt+A1eΩt+A3e3Ωt=0,
where (4.3)A-3=-a1C3(B+C)wn-1[w2-(4Ω+2)w+1],A-1=-a1C(B+C)w2n-1[(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2],A1=a1(B+C)w3n-1[(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2],A3=a1(B+C)w4n-1[w2-(4Ω+2)w+1].
In (4.2), setting the coefficients of all independent exp-function terms to zero, we obtain(4.4)A-3=0,A-1=0,A1=0,A3=0.
Let M-3=A-3/[-a1C3(B+C)wn-1],
M-1=A-1/[-a1C(B+C)w2n-1],M1=A1/[a1(B+C)w3n-1],M3=A3/[a1(B+C)w4n-1], then (4.5)M3=M-3=w2-(4Ω+2)w+1,M1=M-1=(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2].
Thus,  (4.4) can be reduced to(4.6)M1=0,M3=0.
Solving  (4.6) yields(4.7)B=C(2Ω-a1)a1,w1,2=2Ω2+1±2ΩΩ2+1
and C,a0,a1,Ω are arbitrary constants. From   (4.1), (4.7), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.8)un=a0+a1[(2Ω2+1+2ΩΩ2+1)neΩt-(C(2Ω-a1)/a1)e-ΩtwneΩt+Ce-Ωt],un=a0+a1[(2Ω2+1-2ΩΩ2+1)neΩt-(C(2Ω-a1)/a1)e-ΩtwneΩt+Ce-Ωt].
Similarly, by using the same method, we can obtain exact solutions as the form of   (2.10); here, we omit them.


### 4.2. Exact Solutions of the Form (2.11)

Suppose that   (1.2) has exact solution of the form (2.11) as follows:(4.9)un=a0+a2w2ne2Ωt+a4w4ne4Ωtb0+b2w2ne2Ωt+b4w4ne4Ωt,
where a0, a2,a4, b0, b2, b4 are constants to be determined later and a0/b0≠a2/b2≠a4/b4. After substituting   (4.9) in   (1.2), multiplying both sides by the common denominator (b0+b2w2ne2Ωt+b4w4ne4Ωt)3(b0+b2w2(n-1)e2Ωt+b4w4(n-1)e4Ωt)(b0+b2w2(n+1)e2Ωt+b4w4(n+1)e4Ωt), it follows:(4.10)A18e18Ωt+A16e16Ωt+A14e14Ωt+A12e12Ωt+A10e10Ωt+A8e8Ωt+A6e6Ωt+A4e4Ωt+A2e2Ωt=0,
whereA18=-b43(a2b4-a4b2)w18n-2[w4-(4Ω2+2)w2+1],A16=b4w16n-4[(a4b0b42-a0b43)w8+(a2b4-a4b2)(2Ωb4a2-b2b4+4b4Ω2b2-2Ω4b2)w6+(-2a4b0b42+16Ω2a0b43-4Ωa22b42-2a4b22b4-4Ωa42b22+8Ωa2b4a4b2+2a2b42b2+2a0b43-16Ω2a4b0b42-4Ω2a2b42b2+4Ω2a4b22b4)w4+(a2b4-a4b2)(2Ωb4a2-b2b4+4b4Ω2b2-2Ω4b2)w2+a4b0b42-a0b43],A14=w14n-4[(-3a0b2b43+2a2b0b43+4Ω2a2b0b43+2Ωa2b43a0-2Ωa4b2a0b42+a4b0b2b42+2Ωa42b0b2b4-2Ωa2b42a4b0-4Ω2a4b0b2b42)w8+(4Ωa2b43a0+a0b2b43+4Ω2a4b23b4-a4b23b4-2Ωa42b23-2Ωa22b42b2+a2b22b42-4a2b0b43+4Ωa2b4a4b22-16Ω2a4b0b2b42+16Ω2a0b2b43+4Ωa42b0b2b4-4Ω2a2b42b22+3a4b0b2b42-4Ωa4b2a0b42-4Ωa2b42a4b0)w6+(4a0b2b43+4a2b0b43-12Ωa2b43a0-12Ωa42b0b2b4+12Ωa4b2a0b42-2a2b22b42+4Ωa42b23-8a4b0b2b42+2a4b23b4-24Ω2a2b0b43+12Ωa2b42a4b0+4Ω2a2b42b22+12Ω2a4b0b2b42+12Ω2a0b2b43+4Ωa22b42b2-4Ω2a4b23b4-8Ωa2b4a4b22)w4+(4Ωa2b43a0+a0b2b43+4Ω2a4b23b4-a4b23b4-2Ωa42b23-2Ωa22b42b2+a2b22b42-4a2b0b43+4Ωa2b4a4b22-16Ω2a4b0b2b42+16Ω2a0b2b43+4Ωa42b0b2b4-4Ω2a2b42b22+3a4b0b2b42-4Ωa4b2a0b42-4Ωa2b42a4b0)w2-3a0b2b43+2a2b0b43+4Ω2a2b0b43+2Ωa2b43a0-2Ωa4b2a0b42+a4b0b2b42+2Ωa42b0b2b4-2Ωa2b42a4b0-4Ω2a4b0b2b42],A12=w12n-4[(-3a0b22b42+a4b42b02-a0b43b0+4Ωa02b43+4a2b0b2b42-a4b0b4b22-16Ω2a4b02b42+16Ω2a0b43b0-4Ωa22b0b42-2Ωa42b0b22+4Ωa42b02b4+2Ωa0b2a2b42+6Ωa2b0a4b2b4-8Ωa4b0a0b42-2Ωa0b22a4b4+4Ω2a4b0b4b22-4Ω2a2b0b2b42)w8+(-4Ωa0b2a2b42+12Ω2a0b22b42+8Ω2a4b0b4b22+a0b22b42+2a4b0b4b22+4Ωa0b22a4b4-3a2b0b2b42-20Ω2a2b0b2b42+2Ωa22b0b42-2Ωa42b0b22+a2b23b4-a4b24)w6+(4a0b22b42+2a0b43b0-4Ω2a2b23b4+2a4b24+20Ω2a0b22b42+4Ω2a4b24-2a4b0b4b22-16Ω2a0b43b0+4Ωa22b0b42-4Ω2a2b0b2b42-2a2b23b4-2a2b0b2b42+8Ωa42b0b22+16Ω2a4b02b42-2a4b42b02-4Ωa0b22a4b4-8Ωa42b02b4-16Ω2a4b0b4b22-12Ωa2b0a4b2b4+16Ωa4b0a0b42+4Ωa0b2a2b42-8Ωa02b43)w4-(4Ωa0b2a2b42-12Ω2a0b22b42-8Ω2a4b0b4b22-a0b22b42-2a4b0b4b22-4Ωa0b22a4b4+3a2b0b2b42+20Ω2a2b0b2b42-2Ωa22b0b42+2Ωa42b0b22-a2b23b4+a4b24)w2-3a0b22b42+a4b42b02-a0b43b0+4Ωa02b43+4a2b0b2b42-a4b0b4b22-16Ω2a4b02b42+16Ω2a0b43b0-4Ωa22b0b42-2Ωa42b0b22+4Ωa42b02b4+2Ωa0b2a2b42+6Ωa2b0a4b2b4-8Ωa4b0a0b42-2Ωa0b22a4b4+4Ω2a4b0b4b22-4Ω2a2b0b2b42],A10=w10n-4[(-a4b0b23-2a4b2b4b02-a0b23b4+4a2b42b02+12Ω2a0b2b0b42-2a0b2b0b42-24Ω2a2b42b02-6Ωa42b2b02+2a2b0b4b22+12Ω2a4b2b4b02+6Ωa02b2b42+12Ωa2b02a4b4-12Ωa2b0a0b42)w8-(4Ωa02b2b42-2a2b0b4b22+6a2b42b02-4Ω2a0b23b4+a0b23b4+8Ω2a2b0b4b22+a4b0b23-3a4b2b4b02-4Ω2a4b0b23-3a0b2b0b42-8Ωa2b0a0b42-4Ωa42b2b02+8Ωa2b02a4b4)w6+(4a2b42b02-8a2b0b4b22+4a0b23b4+4a4b0b23-4Ω2a4b2b4b02-4Ω2a0b2b0b42+8Ω2a2b42b02-2a4b2b4b02-8Ωa2b02a4b4+12Ω2a4b0b23-2a0b2b0b42+8Ωa2b0a0b42+4Ωa42b2b02-4Ωa02b2b42-24Ω2a2b0b4b22+12Ω2a0b23b4)w4-(4Ωa02b2b42-2a2b0b4b22+6a2b42b02-4Ω2a0b23b4+a0b23b4+8Ω2a2b0b4b22+a4b0b23-3a4b2b4b02-4Ω2a4b0b23-3a0b2b0b42-8Ωa2b0a0b42-4Ωa42b2b02+8Ωa2b02a4b4)w2-a4b0b23-2a4b2b4b02-a0b23b4+4a2b42b02+12Ω2a0b2b0b42-2a0b2b0b42-24Ω2a2b42b02-6Ωa42b2b02+2a2b0b4b22+12Ω2a4b2b4b02+6Ωa02b2b42+12Ωa2b02a4b4-12Ωa2b0a0b42],A8=w8n-4[-(4Ω2a2b2b4b02+4Ωa42b03+3a4b22b02-b02a0b42+b03a4b4-2Ωa4b0a0b22+2Ωa4b02a2b2-4a2b2b4b02-4Ωa22b4b02+a0b4b0b22-2Ωa02b4b22+4Ωa02b42b0-16Ω2a4b03b4-4Ω2a0b4b0b22+16Ω2a0b42b02-8Ωa4b02a0b4+6Ωa2b4a0b2b0)w8+(a2b0b23-a0b24+2Ωa02b4b22+2a0b4b0b22-2Ωa22b4b02+a4b22b02+8Ω2a0b4b0b22-20Ω2a2b2b4b02+12Ω2a4b22b02-3a2b2b4b02+4Ωa4b02a2b2-4Ωa4b0a0b22)w6-(8Ωa02b4b22+2b02a0b42+4Ωa22b4b02-4a4b22b02-16Ω2a0b42b02-2b03a4b4+4Ω2a2b2b4b02+16Ω2a0b4b0b22+2a2b2b4b02-8Ωa42b03+4Ωa4b02a2b2+2a0b4b0b22+16Ωa4b02a0b4-8Ωa02b42b0-4Ωa4b0a0b22+16Ω2a4b03b4+2a2b0b23-12Ωa2b4a0b2b0-4Ω2a0b24-2a0b24+4Ω2a2b0b23-20Ω2a4b22b02)w4+(a2b0b23-a0b24+2Ωa02b4b22+2a0b4b0b22-2Ωa22b4b02+a4b22b02+8Ω2a0b4b0b22-20Ω2a2b2b4b02+12Ω2a4b22b02-3a2b2b4b02+4Ωa4b02a2b2-4Ωa4b0a0b22)w2-4Ω2a2b2b4b02-4Ωa42b03-3a4b22b02+b02a0b42-b03a4b4+2Ωa4b0a0b22-2Ωa4b02a2b2+4a2b2b4b02+4Ωa22b4b02-a0b4b0b22+2Ωa02b4b22-4Ωa02b42b0+16Ω2a4b03b4+4Ω2a0b4b0b22-16Ω2a0b42b02+8Ωa4b02a0b4-6Ωa2b4a0b2b0],A6=w6n-4[-(2Ωa2b03a4-4Ω2a2b03b4-a0b2b4b02-2Ωa2b02a0b4+2Ωa02b2b4b0-2Ωa4b02a0b2-2a2b03b4+3a4b03b2+4Ω2a0b2b4b02)w8+(2Ωa22b02b2-4Ω2a2b02b22-a0b23b0+16Ω2a4b03b2+4Ω2a0b23b0+3a0b2b4b02-4a2b03b4+a2b22b02+a4b03b2+2Ωa02b23-16Ω2a0b2b4b02-4Ωa2b0a0b22+4Ωa4b02a0b2+4Ωa2b02a0b4-4Ωa02b2b4b0-4Ωa2b03a4)w6-(24Ω2a2b03b4-12Ωa2b03a4+8a0b2b4b02+12Ωa4b02a0b2-12Ωa02b2b4b0-4a2b03b4-12Ω2a0b2b4b02-4a4b03b2+12Ωa2b02a0b4+2a2b22b02-2a0b23b0+4Ωa02b23+4Ω2a0b23b0+4Ωa22b02b2-12Ω2a4b03b2-4Ω2a2b02b22-8Ωa2b0a0b22)w4+(2Ωa22b02b2-4Ω2a2b02b22-a0b23b0+16Ω2a4b03b2+4Ω2a0b23b0+3a0b2b4b02-4a2b03b4+a2b22b02+a4b03b2+2Ωa02b23-16Ω2a0b2b4b02-4Ωa2b0a0b22+4Ωa4b02a0b2+4Ωa2b02a0b4-4Ωa02b2b4b0-4Ωa2b03a4)w2-2Ωa2b03a4+4Ω2a2b03b4+a0b2b4b02+2Ωa2b02a0b4-2Ωa02b2b4b0+2Ωa4b02a0b2+2a2b03b4-3a4b03b2-4Ω2a0b2b4b02],A4=-b0w4n-4[(a4b03-a0b4b02)w8+(2Ωa22b02+a2b02b2-4Ω2a2b02b2+2Ωa02b22-4Ωa2b0a0b2+4Ω2a0b22b0-a0b22b0)w6-(4Ω2a0b22b0-2a0b4b02+2a2b02b2-4Ω2a2b02b2+16Ω2a4b03+4Ωa22b02+4Ωa02b22-8Ωa2b0a0b2+2a4b03-2a0b22b0-16Ω2a0b4b02)w4+(2Ωa22b02a2b02b2-4Ω2a2b02b2+2Ωa02b22-4Ωa2b0a0b2+4Ω2a0b22b0-a0b22b0)w2+a4b03-a0b4b02],A2=b03(a0b2-a2b0)w2n-2[w4-(4Ω2+2)w2+1].In   (4.10), we let(4.11)A18=0,A16=0,A14=0,A12=0,A10=0,A8=0,A6=0,A4=0,A2=0.
Solving the group of   (4.11), we obtain three group of parametric conditions which satisfy (4.10).Case 1.
We have the following:(4.12)a0=a2b0b2±w2-1w,a4=b4=0,Ω=±w2-12w,
where a2, b0, b2, w are arbitrary nonzero constants with a2≠b0≠b2,w≠±1.Case 2.
We have the following:(4.13)a0=-p4w4±p3w3+p2w2±p1w+p0b42(w3-w2-w+1)(w+1),b0=±q2w2+q1w±q0b43(w-1)2(w+1)2,Ω=±w2-12w,
where p4=-a2b43b2+a4b22b42,
p3=2a42b22b4-3a2b42a4b2+a22b43,
p2=-2a42b4b2a2-2a4b22b42+2a2b43b2+a4b42a22+a43b22,
p1=3a2b42a4b2-2a42b22b4-a22b43,
p0=-a2b43b2+a4b22b42,
q2=a4b22b4-a2b42b2,
q1=-2b4b2a4a2+b42a22+b22a42,
q0=a2b42b2-a4b22b4 and a2,a4,b2,b4,w are arbitrary nonzero constants with a2≠a4≠b2≠b4,w≠±1.Under Case1, substituting   (4.12) in   (4.9), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.14)un=a2b0±(w-(1/w))+a2b2w2nexp[±(w-(1/w))t]b0b2+b22w2nexp[±(w-(1/w))t],
where a2,b0,b2,w are arbitrary nonzero constants with a2≠b0≠b2,w≠±1.Under Case2, substituting  (4.13)   into   (4.9), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.15)un=𝔏+a2w2nexp[±(w-(1/w))t]+a4w4nexp[±(2w-(2/w))t]𝔖+b2w2nexp[±(w-(1/w))t]+b4w4nexp[±(2w-(2/w))t],
where 𝔏 denotes -((p4w4±p3w3+p2w2±p1w+p0)/(w3-w2-w+1)) and 𝔖 denotes (±q2w2+q1w±q0)/(b4(w+1)(w-1)2),  and p4,p3,p2,p1,p0,q2,q1,q0 have been given above, and a2,a4,b2,b4,w are arbitrary nonzero constants with a2≠a4≠b2≠b4,w≠±1.In order to describe the dynamic properties of the above solutions intuitively, as an example, we draw the profile figures of the solution (4.15), see Figure 6. Setting a2=0.8,a4=2,b2=-0.5<0,b4=1.5,w=4,t=1,n∈[-20,25], the solution (4.15) shows a shape of discrete antikink wave, see Figure 6(a). Setting a2=0.8,a4=2,b2=0.5>0,b4=1.5,w=4,t=1,n∈[-20,25], the solution (4.15) shows a shape of discrete kink wave, see Figure 6(b).The solutionun in   (4.15) shows shapes of discrete antikink and kink solitons.
(a)
Discrete antikink wave(b)
Discrete kink wave


### 4.3. Exact Solutions of the Form (2.12)

Suppose that   (1.2) has exact solution of the form (2.12) as follows:(4.16)un=ã0+a2w2ne2Ωt+b2w2n,
where ã0=a0/(1+b2) and a0, a2, b2 are constants to be determined later. After substituting   (4.16) in   (1.2), multiplying both sides by the common denominator (e2Ωt+b2w2n)3(e2Ωt+b2w2(n-1))(e2Ωt+b2w2(n+1)), it follows(4.17)a2w2n-2M8e8Ωt+a2w4n-2M6e6Ωt-b2a2w6n-2M4e4Ωt+b23a2w8n-2M2e2Ωt=0,
where M8=-1-w4+(4Ω2+2)w2,
M6=(-b2+2a2Ω+4Ω2b2)w4+(2b2-4Ω2b2-4a2Ω)w2-b2+2a2Ω+4Ω2b2,
M4=(-b2+2a2Ω+4Ω2b2)w4+(2b2-4Ω2b2-4a2Ω)w2-b2+2a2Ω+4Ω2b2,M2=1+w4+(-4Ω2-2)w2.In   (4.17), we let(4.18)M8=0,M6=0,M4=0,M2=0.
Solving the group of   (4.18), we obtain two groups of parametric conditions which satisfy  (4.17).Case 1.
We have the following:(4.19)a2=-2b2Ω,w=-Ω±Ω2+1,
where b2,Ω are two arbitrary nonzero constants.Case 2.
We have the following:(4.20)a2=-2b2Ω,w=Ω±Ω2+1,
where b2,Ω are also two arbitrary nonzero constants.Under Case1, substituting   (4.19) in  (4.16), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.21)un=a01+b2-2b2Ω(-Ω±Ω2+1)2ne2Ωt+b2(-Ω±Ω2+1)2n,
where a0,b2,Ω are arbitrary nonzero constants.Under Case2, substituting   (4.20) in   (4.16), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.22)un=a01+b2-2b2Ω(Ω±Ω2+1)2ne2Ωt+b2(Ω±Ω2+1)2n,
where a0,b2,Ω are arbitrary nonzero constants.The dynamic properties of solutions (4.21) and (4.22) are similar to those of solutions (4.14) and (4.15). So we omit their profile figures here.


## 4.1. Exact Solutions of the Form (2.9)

Suppose that   (1.2) has exact solution of the form (2.9) as follows:(4.1)un=a0+a1(wneΩt-Be-ΩtwneΩt+Ce-Ωt).
After substituting (4.1) in   (1.2), multiplying both sides by the common denominator (wneΩt+Ce-Ωt)3(wn-1eΩt+Ce-Ωt)(wn+1eΩt+Ce-Ωt), it follows:(4.2)A-3e-3Ωt+A-1e-Ωt+A1eΩt+A3e3Ωt=0,
where (4.3)A-3=-a1C3(B+C)wn-1[w2-(4Ω+2)w+1],A-1=-a1C(B+C)w2n-1[(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2],A1=a1(B+C)w3n-1[(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2],A3=a1(B+C)w4n-1[w2-(4Ω+2)w+1].
In (4.2), setting the coefficients of all independent exp-function terms to zero, we obtain(4.4)A-3=0,A-1=0,A1=0,A3=0.
Let M-3=A-3/[-a1C3(B+C)wn-1],
M-1=A-1/[-a1C(B+C)w2n-1],M1=A1/[a1(B+C)w3n-1],M3=A3/[a1(B+C)w4n-1], then (4.5)M3=M-3=w2-(4Ω+2)w+1,M1=M-1=(2a1BΩ+2a1CΩ+C-4CΩ2)w2+(4CΩ2-4a1BΩ-4a1CΩ-2C)w+2a1BΩ+2a1CΩ+C-4CΩ2].
Thus,  (4.4) can be reduced to(4.6)M1=0,M3=0.
Solving  (4.6) yields(4.7)B=C(2Ω-a1)a1,w1,2=2Ω2+1±2ΩΩ2+1
and C,a0,a1,Ω are arbitrary constants. From   (4.1), (4.7), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.8)un=a0+a1[(2Ω2+1+2ΩΩ2+1)neΩt-(C(2Ω-a1)/a1)e-ΩtwneΩt+Ce-Ωt],un=a0+a1[(2Ω2+1-2ΩΩ2+1)neΩt-(C(2Ω-a1)/a1)e-ΩtwneΩt+Ce-Ωt].
Similarly, by using the same method, we can obtain exact solutions as the form of   (2.10); here, we omit them.


## 4.2. Exact Solutions of the Form (2.11)

Suppose that   (1.2) has exact solution of the form (2.11) as follows:(4.9)un=a0+a2w2ne2Ωt+a4w4ne4Ωtb0+b2w2ne2Ωt+b4w4ne4Ωt,
where a0, a2,a4, b0, b2, b4 are constants to be determined later and a0/b0≠a2/b2≠a4/b4. After substituting   (4.9) in   (1.2), multiplying both sides by the common denominator (b0+b2w2ne2Ωt+b4w4ne4Ωt)3(b0+b2w2(n-1)e2Ωt+b4w4(n-1)e4Ωt)(b0+b2w2(n+1)e2Ωt+b4w4(n+1)e4Ωt), it follows:(4.10)A18e18Ωt+A16e16Ωt+A14e14Ωt+A12e12Ωt+A10e10Ωt+A8e8Ωt+A6e6Ωt+A4e4Ωt+A2e2Ωt=0,
whereA18=-b43(a2b4-a4b2)w18n-2[w4-(4Ω2+2)w2+1],A16=b4w16n-4[(a4b0b42-a0b43)w8+(a2b4-a4b2)(2Ωb4a2-b2b4+4b4Ω2b2-2Ω4b2)w6+(-2a4b0b42+16Ω2a0b43-4Ωa22b42-2a4b22b4-4Ωa42b22+8Ωa2b4a4b2+2a2b42b2+2a0b43-16Ω2a4b0b42-4Ω2a2b42b2+4Ω2a4b22b4)w4+(a2b4-a4b2)(2Ωb4a2-b2b4+4b4Ω2b2-2Ω4b2)w2+a4b0b42-a0b43],A14=w14n-4[(-3a0b2b43+2a2b0b43+4Ω2a2b0b43+2Ωa2b43a0-2Ωa4b2a0b42+a4b0b2b42+2Ωa42b0b2b4-2Ωa2b42a4b0-4Ω2a4b0b2b42)w8+(4Ωa2b43a0+a0b2b43+4Ω2a4b23b4-a4b23b4-2Ωa42b23-2Ωa22b42b2+a2b22b42-4a2b0b43+4Ωa2b4a4b22-16Ω2a4b0b2b42+16Ω2a0b2b43+4Ωa42b0b2b4-4Ω2a2b42b22+3a4b0b2b42-4Ωa4b2a0b42-4Ωa2b42a4b0)w6+(4a0b2b43+4a2b0b43-12Ωa2b43a0-12Ωa42b0b2b4+12Ωa4b2a0b42-2a2b22b42+4Ωa42b23-8a4b0b2b42+2a4b23b4-24Ω2a2b0b43+12Ωa2b42a4b0+4Ω2a2b42b22+12Ω2a4b0b2b42+12Ω2a0b2b43+4Ωa22b42b2-4Ω2a4b23b4-8Ωa2b4a4b22)w4+(4Ωa2b43a0+a0b2b43+4Ω2a4b23b4-a4b23b4-2Ωa42b23-2Ωa22b42b2+a2b22b42-4a2b0b43+4Ωa2b4a4b22-16Ω2a4b0b2b42+16Ω2a0b2b43+4Ωa42b0b2b4-4Ω2a2b42b22+3a4b0b2b42-4Ωa4b2a0b42-4Ωa2b42a4b0)w2-3a0b2b43+2a2b0b43+4Ω2a2b0b43+2Ωa2b43a0-2Ωa4b2a0b42+a4b0b2b42+2Ωa42b0b2b4-2Ωa2b42a4b0-4Ω2a4b0b2b42],A12=w12n-4[(-3a0b22b42+a4b42b02-a0b43b0+4Ωa02b43+4a2b0b2b42-a4b0b4b22-16Ω2a4b02b42+16Ω2a0b43b0-4Ωa22b0b42-2Ωa42b0b22+4Ωa42b02b4+2Ωa0b2a2b42+6Ωa2b0a4b2b4-8Ωa4b0a0b42-2Ωa0b22a4b4+4Ω2a4b0b4b22-4Ω2a2b0b2b42)w8+(-4Ωa0b2a2b42+12Ω2a0b22b42+8Ω2a4b0b4b22+a0b22b42+2a4b0b4b22+4Ωa0b22a4b4-3a2b0b2b42-20Ω2a2b0b2b42+2Ωa22b0b42-2Ωa42b0b22+a2b23b4-a4b24)w6+(4a0b22b42+2a0b43b0-4Ω2a2b23b4+2a4b24+20Ω2a0b22b42+4Ω2a4b24-2a4b0b4b22-16Ω2a0b43b0+4Ωa22b0b42-4Ω2a2b0b2b42-2a2b23b4-2a2b0b2b42+8Ωa42b0b22+16Ω2a4b02b42-2a4b42b02-4Ωa0b22a4b4-8Ωa42b02b4-16Ω2a4b0b4b22-12Ωa2b0a4b2b4+16Ωa4b0a0b42+4Ωa0b2a2b42-8Ωa02b43)w4-(4Ωa0b2a2b42-12Ω2a0b22b42-8Ω2a4b0b4b22-a0b22b42-2a4b0b4b22-4Ωa0b22a4b4+3a2b0b2b42+20Ω2a2b0b2b42-2Ωa22b0b42+2Ωa42b0b22-a2b23b4+a4b24)w2-3a0b22b42+a4b42b02-a0b43b0+4Ωa02b43+4a2b0b2b42-a4b0b4b22-16Ω2a4b02b42+16Ω2a0b43b0-4Ωa22b0b42-2Ωa42b0b22+4Ωa42b02b4+2Ωa0b2a2b42+6Ωa2b0a4b2b4-8Ωa4b0a0b42-2Ωa0b22a4b4+4Ω2a4b0b4b22-4Ω2a2b0b2b42],A10=w10n-4[(-a4b0b23-2a4b2b4b02-a0b23b4+4a2b42b02+12Ω2a0b2b0b42-2a0b2b0b42-24Ω2a2b42b02-6Ωa42b2b02+2a2b0b4b22+12Ω2a4b2b4b02+6Ωa02b2b42+12Ωa2b02a4b4-12Ωa2b0a0b42)w8-(4Ωa02b2b42-2a2b0b4b22+6a2b42b02-4Ω2a0b23b4+a0b23b4+8Ω2a2b0b4b22+a4b0b23-3a4b2b4b02-4Ω2a4b0b23-3a0b2b0b42-8Ωa2b0a0b42-4Ωa42b2b02+8Ωa2b02a4b4)w6+(4a2b42b02-8a2b0b4b22+4a0b23b4+4a4b0b23-4Ω2a4b2b4b02-4Ω2a0b2b0b42+8Ω2a2b42b02-2a4b2b4b02-8Ωa2b02a4b4+12Ω2a4b0b23-2a0b2b0b42+8Ωa2b0a0b42+4Ωa42b2b02-4Ωa02b2b42-24Ω2a2b0b4b22+12Ω2a0b23b4)w4-(4Ωa02b2b42-2a2b0b4b22+6a2b42b02-4Ω2a0b23b4+a0b23b4+8Ω2a2b0b4b22+a4b0b23-3a4b2b4b02-4Ω2a4b0b23-3a0b2b0b42-8Ωa2b0a0b42-4Ωa42b2b02+8Ωa2b02a4b4)w2-a4b0b23-2a4b2b4b02-a0b23b4+4a2b42b02+12Ω2a0b2b0b42-2a0b2b0b42-24Ω2a2b42b02-6Ωa42b2b02+2a2b0b4b22+12Ω2a4b2b4b02+6Ωa02b2b42+12Ωa2b02a4b4-12Ωa2b0a0b42],A8=w8n-4[-(4Ω2a2b2b4b02+4Ωa42b03+3a4b22b02-b02a0b42+b03a4b4-2Ωa4b0a0b22+2Ωa4b02a2b2-4a2b2b4b02-4Ωa22b4b02+a0b4b0b22-2Ωa02b4b22+4Ωa02b42b0-16Ω2a4b03b4-4Ω2a0b4b0b22+16Ω2a0b42b02-8Ωa4b02a0b4+6Ωa2b4a0b2b0)w8+(a2b0b23-a0b24+2Ωa02b4b22+2a0b4b0b22-2Ωa22b4b02+a4b22b02+8Ω2a0b4b0b22-20Ω2a2b2b4b02+12Ω2a4b22b02-3a2b2b4b02+4Ωa4b02a2b2-4Ωa4b0a0b22)w6-(8Ωa02b4b22+2b02a0b42+4Ωa22b4b02-4a4b22b02-16Ω2a0b42b02-2b03a4b4+4Ω2a2b2b4b02+16Ω2a0b4b0b22+2a2b2b4b02-8Ωa42b03+4Ωa4b02a2b2+2a0b4b0b22+16Ωa4b02a0b4-8Ωa02b42b0-4Ωa4b0a0b22+16Ω2a4b03b4+2a2b0b23-12Ωa2b4a0b2b0-4Ω2a0b24-2a0b24+4Ω2a2b0b23-20Ω2a4b22b02)w4+(a2b0b23-a0b24+2Ωa02b4b22+2a0b4b0b22-2Ωa22b4b02+a4b22b02+8Ω2a0b4b0b22-20Ω2a2b2b4b02+12Ω2a4b22b02-3a2b2b4b02+4Ωa4b02a2b2-4Ωa4b0a0b22)w2-4Ω2a2b2b4b02-4Ωa42b03-3a4b22b02+b02a0b42-b03a4b4+2Ωa4b0a0b22-2Ωa4b02a2b2+4a2b2b4b02+4Ωa22b4b02-a0b4b0b22+2Ωa02b4b22-4Ωa02b42b0+16Ω2a4b03b4+4Ω2a0b4b0b22-16Ω2a0b42b02+8Ωa4b02a0b4-6Ωa2b4a0b2b0],A6=w6n-4[-(2Ωa2b03a4-4Ω2a2b03b4-a0b2b4b02-2Ωa2b02a0b4+2Ωa02b2b4b0-2Ωa4b02a0b2-2a2b03b4+3a4b03b2+4Ω2a0b2b4b02)w8+(2Ωa22b02b2-4Ω2a2b02b22-a0b23b0+16Ω2a4b03b2+4Ω2a0b23b0+3a0b2b4b02-4a2b03b4+a2b22b02+a4b03b2+2Ωa02b23-16Ω2a0b2b4b02-4Ωa2b0a0b22+4Ωa4b02a0b2+4Ωa2b02a0b4-4Ωa02b2b4b0-4Ωa2b03a4)w6-(24Ω2a2b03b4-12Ωa2b03a4+8a0b2b4b02+12Ωa4b02a0b2-12Ωa02b2b4b0-4a2b03b4-12Ω2a0b2b4b02-4a4b03b2+12Ωa2b02a0b4+2a2b22b02-2a0b23b0+4Ωa02b23+4Ω2a0b23b0+4Ωa22b02b2-12Ω2a4b03b2-4Ω2a2b02b22-8Ωa2b0a0b22)w4+(2Ωa22b02b2-4Ω2a2b02b22-a0b23b0+16Ω2a4b03b2+4Ω2a0b23b0+3a0b2b4b02-4a2b03b4+a2b22b02+a4b03b2+2Ωa02b23-16Ω2a0b2b4b02-4Ωa2b0a0b22+4Ωa4b02a0b2+4Ωa2b02a0b4-4Ωa02b2b4b0-4Ωa2b03a4)w2-2Ωa2b03a4+4Ω2a2b03b4+a0b2b4b02+2Ωa2b02a0b4-2Ωa02b2b4b0+2Ωa4b02a0b2+2a2b03b4-3a4b03b2-4Ω2a0b2b4b02],A4=-b0w4n-4[(a4b03-a0b4b02)w8+(2Ωa22b02+a2b02b2-4Ω2a2b02b2+2Ωa02b22-4Ωa2b0a0b2+4Ω2a0b22b0-a0b22b0)w6-(4Ω2a0b22b0-2a0b4b02+2a2b02b2-4Ω2a2b02b2+16Ω2a4b03+4Ωa22b02+4Ωa02b22-8Ωa2b0a0b2+2a4b03-2a0b22b0-16Ω2a0b4b02)w4+(2Ωa22b02a2b02b2-4Ω2a2b02b2+2Ωa02b22-4Ωa2b0a0b2+4Ω2a0b22b0-a0b22b0)w2+a4b03-a0b4b02],A2=b03(a0b2-a2b0)w2n-2[w4-(4Ω2+2)w2+1].In   (4.10), we let(4.11)A18=0,A16=0,A14=0,A12=0,A10=0,A8=0,A6=0,A4=0,A2=0.
Solving the group of   (4.11), we obtain three group of parametric conditions which satisfy (4.10).Case 1.
We have the following:(4.12)a0=a2b0b2±w2-1w,a4=b4=0,Ω=±w2-12w,
where a2, b0, b2, w are arbitrary nonzero constants with a2≠b0≠b2,w≠±1.Case 2.
We have the following:(4.13)a0=-p4w4±p3w3+p2w2±p1w+p0b42(w3-w2-w+1)(w+1),b0=±q2w2+q1w±q0b43(w-1)2(w+1)2,Ω=±w2-12w,
where p4=-a2b43b2+a4b22b42,
p3=2a42b22b4-3a2b42a4b2+a22b43,
p2=-2a42b4b2a2-2a4b22b42+2a2b43b2+a4b42a22+a43b22,
p1=3a2b42a4b2-2a42b22b4-a22b43,
p0=-a2b43b2+a4b22b42,
q2=a4b22b4-a2b42b2,
q1=-2b4b2a4a2+b42a22+b22a42,
q0=a2b42b2-a4b22b4 and a2,a4,b2,b4,w are arbitrary nonzero constants with a2≠a4≠b2≠b4,w≠±1.Under Case1, substituting   (4.12) in   (4.9), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.14)un=a2b0±(w-(1/w))+a2b2w2nexp[±(w-(1/w))t]b0b2+b22w2nexp[±(w-(1/w))t],
where a2,b0,b2,w are arbitrary nonzero constants with a2≠b0≠b2,w≠±1.Under Case2, substituting  (4.13)   into   (4.9), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.15)un=𝔏+a2w2nexp[±(w-(1/w))t]+a4w4nexp[±(2w-(2/w))t]𝔖+b2w2nexp[±(w-(1/w))t]+b4w4nexp[±(2w-(2/w))t],
where 𝔏 denotes -((p4w4±p3w3+p2w2±p1w+p0)/(w3-w2-w+1)) and 𝔖 denotes (±q2w2+q1w±q0)/(b4(w+1)(w-1)2),  and p4,p3,p2,p1,p0,q2,q1,q0 have been given above, and a2,a4,b2,b4,w are arbitrary nonzero constants with a2≠a4≠b2≠b4,w≠±1.In order to describe the dynamic properties of the above solutions intuitively, as an example, we draw the profile figures of the solution (4.15), see Figure 6. Setting a2=0.8,a4=2,b2=-0.5<0,b4=1.5,w=4,t=1,n∈[-20,25], the solution (4.15) shows a shape of discrete antikink wave, see Figure 6(a). Setting a2=0.8,a4=2,b2=0.5>0,b4=1.5,w=4,t=1,n∈[-20,25], the solution (4.15) shows a shape of discrete kink wave, see Figure 6(b).The solutionun in   (4.15) shows shapes of discrete antikink and kink solitons.
(a)
Discrete antikink wave(b)
Discrete kink wave


## 4.3. Exact Solutions of the Form (2.12)

Suppose that   (1.2) has exact solution of the form (2.12) as follows:(4.16)un=ã0+a2w2ne2Ωt+b2w2n,
where ã0=a0/(1+b2) and a0, a2, b2 are constants to be determined later. After substituting   (4.16) in   (1.2), multiplying both sides by the common denominator (e2Ωt+b2w2n)3(e2Ωt+b2w2(n-1))(e2Ωt+b2w2(n+1)), it follows(4.17)a2w2n-2M8e8Ωt+a2w4n-2M6e6Ωt-b2a2w6n-2M4e4Ωt+b23a2w8n-2M2e2Ωt=0,
where M8=-1-w4+(4Ω2+2)w2,
M6=(-b2+2a2Ω+4Ω2b2)w4+(2b2-4Ω2b2-4a2Ω)w2-b2+2a2Ω+4Ω2b2,
M4=(-b2+2a2Ω+4Ω2b2)w4+(2b2-4Ω2b2-4a2Ω)w2-b2+2a2Ω+4Ω2b2,M2=1+w4+(-4Ω2-2)w2.In   (4.17), we let(4.18)M8=0,M6=0,M4=0,M2=0.
Solving the group of   (4.18), we obtain two groups of parametric conditions which satisfy  (4.17).Case 1.
We have the following:(4.19)a2=-2b2Ω,w=-Ω±Ω2+1,
where b2,Ω are two arbitrary nonzero constants.Case 2.
We have the following:(4.20)a2=-2b2Ω,w=Ω±Ω2+1,
where b2,Ω are also two arbitrary nonzero constants.Under Case1, substituting   (4.19) in  (4.16), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.21)un=a01+b2-2b2Ω(-Ω±Ω2+1)2ne2Ωt+b2(-Ω±Ω2+1)2n,
where a0,b2,Ω are arbitrary nonzero constants.Under Case2, substituting   (4.20) in   (4.16), we obtain discrete kink or antikink soliton solution of   (1.2) as follows:(4.22)un=a01+b2-2b2Ω(Ω±Ω2+1)2ne2Ωt+b2(Ω±Ω2+1)2n,
where a0,b2,Ω are arbitrary nonzero constants.The dynamic properties of solutions (4.21) and (4.22) are similar to those of solutions (4.14) and (4.15). So we omit their profile figures here.


## 5. Conclusion

In this work, we introduced an extended method based on the mixed function method. Using this extended method, we studied the Toda lattice equation and (2 + 1)-dimensional Toda lattice equation. We obtained some new exact solutions of discrete type for these two classic Toda lattice equations. As for the (2 + 1)-dimensional Toda lattice equation, its exact solutions which we obtained contain the discrete soliton solutions (3.31), (3.38), (3.39), and (3.47), the discrete kink and antikink wave solutions (3.10)-(3.11), (3.21)–(3.23) and (3.46), and the discrete breather soliton solutions (3.32)–(3.35), (3.40)–(3.43) and (3.48)–(3.51). These exact solutions have both discrete and continuous dynamic properties. In other words, their waveforms have discrete character along the n-axes and have continuous character along the x-axes.Among the above discrete soliton solutions and kink (or antikink) wave solutions, the waveforms of the solutions (3.10), (3.21), (3.31), (3.39), (3.46), and (3.47) are similar to those smooth traveling waves which appear in continuous systems except their discrete characters, see Figures 1(a), 1(c), 2(a), 3(a), 4(a), 5(a), and 5(c); the waveforms of the solutions (3.11) and (3.22) are similar to those nonsmooth traveling waves such as peakon,  and cusp wave which appear in continuous systems except their discrete characters, see Figures 1(b), 1(d), and 2(b). It is worthy to regard that Li et al. [40–42] explained the causes of the smooth traveling waves, nonsmooth traveling waves, peakons and cusp waves by the bifurcation theory. In addition, the waveforms with continuous characters of the breather soliton solutions and double kink wave solutions obtained in this paper are very similar to those in   [43] though the studied problems (model equations) are different. This shows that the waveforms of these exact solutions obtained by us are partly similar to some of traveling waves appeared in continuous systems though the obtained solutions and the studied problems are different.



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*Source: 293607-2011-10-03.xml*