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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 293607, 24 pages
http://dx.doi.org/10.1155/2011/293607
Research Article

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

Center for Nonlinear Science Research, College of Mathematics, Honghe University, Yunnan, Mengzi 661100, China

Received 14 April 2011; Revised 26 May 2011; Accepted 15 June 2011

Academic Editor: Zhengqiu Zhang

Copyright © 2011 Weiguo Rui and Xinsong Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [27] and tanh method [810]. 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 [218] 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]𝜕2𝑢𝑛=𝜕𝑥𝜕𝑡𝜕𝑢𝑛𝑢𝜕𝑡+1𝑛12𝑢𝑛+𝑢𝑛+1,(1.1) and the Toda lattice equation [20]𝑑2𝑢𝑛𝑑𝑡2=𝑑𝑢𝑛𝑢𝑑𝑡+1𝑛12𝑢𝑛+𝑢𝑛+1.(1.2) It is well known that there is another form for the (2 + 1)-dimensional Toda lattice equation𝜕2𝑦𝑛𝑦𝜕𝑥𝜕𝑡=exp𝑛1𝑦𝑛𝑦exp𝑛𝑦𝑛+1,(1.3) and there are another two forms for the Toda lattice equation𝑑2𝑦𝑛𝑑𝑡2𝑦=exp𝑛1𝑦𝑛𝑦exp𝑛𝑦𝑛+1,(1.4)̇𝑢𝑛=𝑢𝑛𝑣𝑛1𝑣𝑛,̇𝑣𝑛=𝑢𝑛𝑢𝑛+1.(1.5) Under the transformation 𝜕𝑢𝑛/𝜕𝑡=exp(𝑦𝑛1𝑦𝑛)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, 2139] and references cited therein. For example, the Toda lattice equation was used as DNA models [3638], 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 Section 2, 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:Ψ,𝑢𝑛2,𝑢𝑛1,𝑢𝑛,𝑢𝑛+1,𝑢𝑛+2,,𝜕𝑢𝑛,𝜕𝑡𝜕𝑢𝑛,𝜕𝜕𝑥2𝑢𝑛Ψ𝜕𝑥𝜕𝑡,=0,(2.1),𝑢𝑛2,𝑢𝑛1,𝑢𝑛,𝑢𝑛+1,𝑢𝑛+2,,𝜕𝑢𝑛,𝜕𝜕𝑡2𝑢𝑛𝜕𝑡2,=0,(2.2) 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: 𝑢𝑛=𝑚𝑝=0𝑎𝑝𝜔𝑛𝑒𝛼𝑥𝛽𝑡+𝛾0𝐵𝑒(𝛼𝑥𝛽𝑡+𝛾0)𝜔𝑛𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐶𝑒(𝛼𝑥𝛽𝑡+𝛾0)𝑝,𝑢(2.3)𝑛=𝑚𝑝=0𝑎𝑝𝑤𝑛𝑤2𝑛𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐶𝑒(𝛼𝑥𝛽𝑡+𝛾0)𝑝𝑢,(2.4)𝑛=𝑚𝑝=0𝑎𝑝𝑤𝑛exp𝛼𝑥𝛽𝑡+𝛾0𝑝𝑚𝑝=0𝑏𝑝𝑤𝑛exp𝛼𝑥𝛽𝑡+𝛾0𝑝,(2.5) 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 𝑘𝑁,𝐵𝐶,𝑎0/𝑏0𝑎2/𝑏2𝑎𝑚/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: 𝑢𝑛=𝐴𝑛𝑓(𝑥)+𝑚𝑝=0𝑏𝑝𝜔𝑛𝑒𝛽𝑡+𝛾0𝐵𝑒(𝛽𝑡+𝛾0)𝜔𝑛𝑒𝛽𝑡+𝛾0+𝐶𝑒(𝛽𝑡+𝛾0)𝑝,𝑢(2.6)𝑛=𝐴𝑛𝑓(𝑥)+𝑚𝑝=0𝑏𝑝𝑤𝑛𝑤2𝑛𝑒𝛽𝑡+𝛾0+𝐶𝑒(𝛽𝑡+𝛾0)𝑝,𝑏2𝑘+1𝑢=0,(2.7)𝑛=𝐴𝑛𝑓(𝑥)+𝑚𝑝=0𝑐𝑝(𝑤𝑛)𝑝𝑒𝛽𝑡+𝛾0𝑝+𝐵(𝑤𝑛)𝑝,𝑐2𝑘+1=0,(2.8) 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: 𝑢𝑛=𝑚𝑖=0𝑎𝑖𝑤𝑛𝑒Ω𝑡𝐵𝑒Ω𝑡𝑤𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡𝑖,𝑢(2.9)𝑛=𝑚𝑖=0𝑎𝑖𝑤𝑛𝑤2𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡𝑖.(2.10) We add two new forms of solutions for (2.2) here 𝑢𝑛=𝑚𝑝=0𝑎𝑝𝑤𝑛𝑒Ω𝑡𝑝𝑚𝑝=0𝑏𝑝𝑤𝑛𝑒Ω𝑡𝑝,𝑎2𝑘+1=0,𝑏2𝑘+1𝑢=0,(2.11)𝑛=𝑚𝑝=0𝑎𝑝(𝑤𝑛)𝑝𝑒Ω𝑡𝑝+𝐵(𝑤𝑛)𝑝,𝑎2𝑘+1=0,(2.12) 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 𝑘𝑁,𝐵𝐶,𝑎0/𝑏0𝑎2/𝑏2𝑎𝑚/𝑏𝑚.

Step 2. By using the homogeneous balance principle or according to character of the idiographic NDDEs, we determine the value of 𝑚 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 Step 2 in the original equation (2.1) or (2.2), then setting the coefficients of all independent terms in 𝑒𝑘𝜉,𝑒𝑘𝜉(𝑘=0,1,2,,𝑁,𝜉=𝛼𝑥𝛽𝑡+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 constants 𝐵,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 Section 2, 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 let𝑢𝑛=𝑎0+𝑎1𝑤𝑛exp𝛼𝑥𝛽𝑡+𝛾0𝐵exp𝛼𝑥𝛽𝑡+𝛾0𝑤𝑛exp𝛼𝑥𝛽𝑡+𝛾0+𝐶exp𝛼𝑥𝛽𝑡+𝛾0,(3.1) where 𝑎0, 𝑎1, 𝑤,𝛼, 𝛽,𝛾0,𝐵, 𝐶 are constants to be determined later, and 𝐵𝐶. After substituting (3.1) into (1.1), multiplying both sides by the common denominator (𝜔𝑛𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐶𝑒(𝛼𝑥𝛽𝑡+𝛾0))3(𝜔𝑛1𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐶𝑒(𝛼𝑥𝛽𝑡+𝛾0))(𝜔𝑛+1𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐶𝑒(𝛼𝑥𝛽𝑡+𝛾0)) and dividing both sides by the common factor 𝑎1(𝐵+𝐶), it follows:𝐴3𝑒3(𝛼𝑥𝛽𝑡+𝛾0)𝐴1𝑒(𝛼𝑥𝛽𝑡+𝛾0)+𝐴1𝑒𝛼𝑥𝛽𝑡+𝛾0+𝐴3𝑒3(𝛼𝑥𝛽𝑡+𝛾0)=0,(3.2) where 𝐴3=𝐶3𝑤𝑛+1+(4𝛼𝛽2)𝑤𝑛+𝑤𝑛1,𝐴1=𝐶4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶2𝑛+1+4𝑎1𝛽𝐵+2𝐶4𝑎1𝑤𝛽𝐶+4𝛼𝛽𝐶2𝑛+4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶2𝑛1,𝐴1=4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶3𝑛+14𝑎1𝛽𝐵+2𝐶4𝑎1𝑤𝛽𝐶+4𝛼𝛽𝐶3𝑛4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶3𝑛1,𝐴3=𝑤4𝑛+1+(4𝛼𝛽2)𝑤4𝑛+𝑤4𝑛1.(3.3)

In (3.2), setting the coefficients of all independent terms 𝑒3(𝛼𝑥𝛽𝑡+𝛾0),𝑒(𝛼𝑥𝛽𝑡+𝛾0), 𝑒𝛼𝑥𝛽𝑡+𝛾0,𝑒3(𝛼𝑥𝛽𝑡+𝛾0) to zero, we get a series of algebraic equations as follows:𝐴3=0,𝐴1=0,𝐴1=0,𝐴3=0.(3.4) 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 𝐴3=𝐶3𝑤𝑛1[𝑤2+(4𝛼𝛽2)𝑤+1], we let 𝑀3=𝐴3𝐶3𝑤𝑛1=𝑤2+(4𝛼𝛽2)𝑤+1.(3.5) As a result, the equation 𝐴3=0 becomes 𝑀3=0, that is, 𝑤2+(4𝛼𝛽2)𝑤+1=0, which is a low-power equation. Similarly, we obtain 𝑀1=𝐴1𝐶𝑤2𝑛1=4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶2+4𝑎1𝛽𝐵+2𝐶4𝑎1𝑤+𝛽𝐶+4𝛼𝛽𝐶4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1,𝑀𝛽𝐶1=𝐴1𝑤3𝑛1=4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1𝑤𝛽𝐶2+4𝑎1𝛽𝐵+2𝐶4𝑎1𝑤+𝛽𝐶+4𝛼𝛽𝐶4𝛼𝛽𝐶+2𝑎1𝛽𝐵𝐶+2𝑎1,𝑀𝛽𝐶3=𝐴3𝑤4𝑛1=𝑤2+(4𝛼𝛽2)𝑤+1.(3.6) Notice that 𝑀3=𝑀3,𝑀1=𝑀1, thus, (3.4) can be reduced to a group of low-power algebraic equations as follows:𝑀1=0,𝑀3=0.(3.7) Solving (3.7) yields𝐶𝐵=2𝛼𝑎1𝑎1,𝑤(3.8)1,2=12𝛼𝛽±2𝛼𝛽(𝛼𝛽1),(3.9) 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:𝑢𝑛=𝑎0+𝑎1𝑤𝑛1exp𝛼𝑥𝛽𝑡+𝛾0𝐶2𝛼𝑎1/𝑎1exp𝛼𝑥+𝛽𝑡𝛾0𝑤𝑛1exp𝛼𝑥𝛽𝑡+𝛾0+𝐶exp𝛼𝑥+𝛽𝑡𝛾0𝑢,(3.10)𝑛=𝑎0+𝑎1𝑤𝑛2exp𝛼𝑥𝛽𝑡+𝛾0𝐶2𝛼𝑎1/𝑎1exp𝛼𝑥+𝛽𝑡𝛾0𝑤𝑛2exp𝛼𝑥𝛽𝑡+𝛾0+𝐶exp𝛼𝑥+𝛽𝑡𝛾0,(3.11) 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.

fig1
Figure 1: The solutions 𝑢𝑛 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 that𝑢𝑛=𝑎0+𝑎2𝑤2𝑛2exp𝛼𝑥𝛽𝑡+𝛾0𝑏0+𝑏2𝑤2𝑛2exp𝛼𝑥𝛽𝑡+𝛾0,(3.12) where 𝑎0, 𝑎2, 𝑏0, 𝑏2, 𝑤, 𝛼, 𝛽 are constants to be determined later, and 𝑎0/𝑏0𝑎2/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(𝛼𝑥𝛽𝑡+𝛾0))3(𝑏0+𝑏2𝑤2(𝑛1)𝑒2(𝛼𝑥𝛽𝑡+𝛾0))(𝑏0+𝑏2𝑤2(𝑛+1)𝑒2(𝛼𝑥𝛽𝑡+𝛾0)), it follows:𝐴2𝑒2(𝛼𝑥𝛽𝑡+𝛾0)+𝐴4𝑒4(𝛼𝑥𝛽𝑡+𝛾0)+𝐴6𝑒6(𝛼𝑥𝛽𝑡+𝛾0)+𝐴8𝑒8(𝛼𝑥𝛽𝑡+𝛾0)=0,(3.13) where 𝐴2=𝑏30𝑤2𝑛2𝑏2𝑎0𝑎2𝑏0𝑤4𝑏+22𝑎0𝑎2𝑏0(2𝛼𝛽1)𝑤2+𝑏2𝑎0𝑎2𝑏0,𝐴4=𝑏0𝑤4𝑛2𝑏2𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤4𝑏22𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+2𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤2+𝑏2𝑎0𝑎2𝑏0×2𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽,𝐴6=𝑏2𝑤6𝑛2𝑏2𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤4𝑏22𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+2𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤2+𝑏2𝑎0𝑎2𝑏0×2𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽,𝐴8=𝑏32𝑤8𝑛2𝑏2𝑎0𝑎2𝑏0𝑤4𝑏+22𝑎0𝑎2𝑏0(2𝛼𝛽1)𝑤2+𝑏2𝑎0𝑎2𝑏0.(3.14) In (3.13), setting the coefficients of all independent exp-function terms to zero, it follows𝐴2=0,𝐴4=0,𝐴6=0,𝐴8=0.(3.15) Let 𝑀2=𝐴2/(𝑏30𝑤2𝑛2),𝑀4=𝐴4/(𝑏0𝑤4𝑛2),𝑀6=𝐴6/(𝑏2𝑤6𝑛2),𝑀8=𝐴8/(𝑏32𝑤8𝑛2). Then 𝑀2=𝑀8=𝑏2𝑎0𝑎2𝑏0𝑤4𝑏+22𝑎0𝑎2𝑏0(2𝛼𝛽1)𝑤2+𝑏2𝑎0𝑎2𝑏0,𝑀4=𝑀6=𝑏2𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤4𝑏22𝑎0𝑎2𝑏02𝑏2𝑎0𝛽+2𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽𝑤2+𝑏2𝑎0𝑎2𝑏0×2𝑏2𝑎0𝛽+4𝑏2𝑏0𝛼𝛽+𝑏0𝑏22𝑎2𝑏0𝛽.(3.16) Thus, (3.15) can be reduced to the following equations: 𝑀2=0,𝑀4=0.(3.17) Solving (3.17), we obtain𝑏𝛼=0𝑎2𝑎0𝑏22𝑏0𝑏2,𝑤(3.18)1,2=±𝑏0𝑏2𝑏0𝑏2𝑏0𝑎2𝛽+𝑎0𝑏2𝛽+𝛽𝑏2𝑎0𝑎2𝑏0𝑎0𝑏2𝛽+2𝑏0𝑏2𝑎2𝑏0𝛽𝑏0𝑏2𝑤,(3.19)3,4=±𝑏0𝑏2𝑏0𝑏2𝑏0𝑎2𝛽+𝑎0𝑏2𝛽𝛽𝑏2𝑎0𝑎2𝑏0𝑎0𝑏2𝛽+2𝑏0𝑏2𝑎2𝑏0𝛽𝑏0𝑏2,(3.20) and 𝑎0,𝑎2,𝑏0,𝑏2,𝛽,𝛾0 are arbitrary nonzero constants with 𝑎0/𝑏0𝑎2/𝑏2.

Based on (3.12), (3.18), (3.19), and (3.20), we obtain four exact solutions of (1.1) as follows:𝑢𝑛=𝑎0+𝑎2𝑤12𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑏0+𝑏2𝑤12𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑢,(3.21)𝑛=𝑎0+𝑎2𝑤22𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑏0+𝑏2𝑤22𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑢,(3.22)𝑛=𝑎0+𝑎2𝑤32𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑏0+𝑏2𝑤32𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑢,(3.23)𝑛=𝑎0+𝑎2𝑤42𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0𝑏0+𝑏2𝑤42𝑛𝑏exp0𝑎2𝑎0𝑏2/𝑏0𝑏2𝑥2𝛽𝑡+2𝛾0,(3.24) where 𝑤1,𝑤2,𝑤3,𝑤4 are given by (3.19), (3.20), and 𝑤1,2,3,41,𝑏0𝑎2𝑎0𝑏2.

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.

fig2
Figure 2: The solutions 𝑢𝑛 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:𝑢𝑛=𝐴𝑛𝑓(𝑥)+𝑏0+𝑏1𝑤𝑛𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑤𝑛𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0,(3.25) 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+𝐶𝑒𝛽𝑡𝛾0)3(𝑤𝑛1𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0)(𝑤𝑛+1𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0), it follows:𝐴3𝑒3(𝛽𝑡+𝛾0)+𝐴1𝑒(𝛽𝑡+𝛾0)+𝐴1𝑒𝛽𝑡+𝛾0+𝐴3𝑒3(𝛽𝑡+𝛾0)=0,(3.26) where 𝐴3=𝑎1𝐶3(𝐵+𝐶)𝑤𝑛1(𝑤1)2,𝐴1=𝑎1𝐶(𝐵+𝐶)2𝑎1𝛽𝐵+𝐶+2𝑎1𝑤𝛽𝐶2𝑛1(𝑤1)2,𝐴1=𝑎1(𝐵+𝐶)2𝑎1𝛽𝐵+𝐶+2𝑎1𝑤𝛽𝐶3𝑛1(𝑤1)2,𝐴3=𝑎1(𝐵+𝐶)𝑤4𝑛1(𝑤1)2.(3.27)

In (3.26), setting the coefficients of all independent exp-function terms to zero, we obtain𝐴3=0,𝐴1=0,𝐴1=0,𝐴3=0.(3.28) 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:𝑢𝑛=𝐴𝑛𝑓(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0,(3.29) 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:𝑢𝑛=𝐴𝑛tanh(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0𝑢,(3.30)𝑛=𝐴𝑛sech(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0𝑢,(3.31)𝑛=𝐴𝑛sin(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0𝑢,(3.32)𝑛=𝐴𝑛cos(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0𝑢,(3.33)𝑛[]=𝐴𝑛tanh(𝑥)+sin(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0𝑢,(3.34)𝑛[]=𝐴𝑛sech(𝑥)+cos(𝑥)+𝑏0+𝑏1𝑒𝛽𝑡+𝛾0𝐵𝑒𝛽𝑡𝛾0𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾0.(3.35)

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).

fig3
Figure 3: The solutions 𝑢𝑛 in (3.31), (3.34), and (3.35) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
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:𝑢𝑛=𝐴𝑛𝑓(𝑥)+𝑏0+𝑏2𝑤𝑛𝑤2𝑛𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02,(3.36) 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:𝑢𝑛=𝐴𝑛𝑓(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02.(3.37) 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:𝑢𝑛=𝐴𝑛tanh2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02,𝑢(3.38)𝑛=𝐴𝑛sech2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02𝑢,(3.39)𝑛=𝐴𝑛sin2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02𝑢,(3.40)𝑛=𝐴𝑛cos2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02𝑢,(3.41)𝑛=𝐴𝑛tanh2(𝑥)+sin2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02𝑢,(3.42)𝑛=𝐴𝑛sech2(𝑥)+cos2(𝑥)+𝑏0+𝑏21𝑒𝛽𝑡+𝛾0+𝐶𝑒𝛽𝑡𝛾02.(3.43)

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).

fig4
Figure 4: The solutions 𝑢𝑛 in (3.39), (3.42), and (3.43) show six shapes of discrete and continuous solitons and discrete and continuous breather oscillations.
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:𝑢𝑛𝑐=𝐴𝑛𝑓(𝑥)+0+𝑐1+𝐵2𝑤2𝑛𝑒2(𝛽𝑡+𝛾0)+𝐵𝑤2𝑛,(3.44) 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:𝑢𝑛𝑐=𝐴𝑛𝑓(𝑥)+0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0).+𝐵(3.45) Similarly, assume 𝑓(𝑥) the same functions as in Section 3.3, we obtain a series of soliton solutions and breather solutions as follows:𝑢𝑛𝑐=𝐴𝑛tanh(𝑥)+0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0),𝑢+𝐵(3.46)𝑛𝑐=𝐴𝑛sech(𝑥)+0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0)𝑢+𝐵,(3.47)𝑛𝑐=𝐴𝑛sin(𝑥)+0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0)𝑢+𝐵,(3.48)𝑛𝑐=𝐴𝑛cos(𝑥)+0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0)𝑢+𝐵,(3.49)𝑛[]+𝑐=𝐴𝑛tanh(𝑥)+sin(𝑥)0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0)𝑢+𝐵,(3.50)𝑛[]+𝑐=𝐴𝑛sech(𝑥)+cos(𝑥)0+𝑐1+𝐵2𝑒2(𝛽𝑡+𝛾0).+𝐵(3.51)

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).

fig5
Figure 5: The solutions 𝑢𝑛 in (3.46), (3.47) and (3.48) show six shapes of discrete and continuous solitons, and discrete and continuous breather oscillations.

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

In this section, using the extended method offered in Section 2, 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:𝑢𝑛=𝑎0+𝑎1𝑤𝑛𝑒Ω𝑡𝐵𝑒Ω𝑡𝑤𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡.(4.1) After substituting (4.1) in (1.2), multiplying both sides by the common denominator (𝑤𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡)3(𝑤𝑛1𝑒Ω𝑡+𝐶𝑒Ω𝑡)(𝑤𝑛+1𝑒Ω𝑡+𝐶𝑒Ω𝑡), it follows:𝐴3𝑒3Ω𝑡+𝐴1𝑒Ω𝑡+𝐴1𝑒Ω𝑡+𝐴3𝑒3Ω𝑡=0,(4.2) where 𝐴3=𝑎1𝐶3(𝐵+𝐶)𝑤𝑛1𝑤2,𝐴(4Ω+2)𝑤+11=𝑎1𝐶(𝐵+𝐶)𝑤2𝑛12𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2𝑤2+4𝐶Ω24𝑎1𝐵Ω4𝑎1𝑤𝐶Ω2𝐶+2𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2,𝐴1=𝑎1(𝐵+𝐶)𝑤3𝑛12𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2𝑤2+4𝐶Ω24𝑎1𝐵Ω4𝑎1𝑤𝐶Ω2𝐶+2𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2,𝐴3=𝑎1(𝐵+𝐶)𝑤4𝑛1𝑤2.(4Ω+2)𝑤+1(4.3) In (4.2), setting the coefficients of all independent exp-function terms to zero, we obtain𝐴3=0,𝐴1=0,𝐴1=0,𝐴3=0.(4.4) Let 𝑀3=𝐴3/[𝑎1𝐶3(𝐵+𝐶)𝑤𝑛1],𝑀1=𝐴1/[𝑎1𝐶(𝐵+𝐶)𝑤2𝑛1],𝑀1=𝐴1/[𝑎1(𝐵+𝐶)𝑤3𝑛1],𝑀3=𝐴3/[𝑎1(𝐵+𝐶)𝑤4𝑛1], then 𝑀3=𝑀3=𝑤2𝑀(4Ω+2)𝑤+1,1=𝑀1=2𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2𝑤2+4𝐶Ω24𝑎1𝐵Ω4𝑎1𝐶Ω2𝐶𝑤+2𝑎1𝐵Ω+2𝑎1𝐶Ω+𝐶4𝐶Ω2.(4.5) Thus, (4.4) can be reduced to𝑀1=0,𝑀3=0.(4.6) Solving (4.6) yields𝐶𝐵=2Ω𝑎1𝑎1,𝑤1,2=2Ω2+1±2ΩΩ2+1(4.7) 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:𝑢𝑛=𝑎0+𝑎12Ω2+1+2ΩΩ2+1𝑛𝑒Ω𝑡𝐶2Ω𝑎1/𝑎1𝑒Ω𝑡𝑤𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡,𝑢𝑛=𝑎0+𝑎12Ω2+12ΩΩ2+1𝑛𝑒Ω𝑡𝐶2Ω𝑎1/𝑎1𝑒Ω𝑡𝑤𝑛𝑒Ω𝑡+𝐶𝑒Ω𝑡.(4.8) 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:𝑢𝑛=𝑎0+𝑎2𝑤2𝑛𝑒2Ω𝑡+𝑎4𝑤4𝑛𝑒4Ω𝑡𝑏0+𝑏2𝑤2𝑛𝑒2Ω𝑡+𝑏4𝑤4𝑛𝑒4Ω𝑡,(4.9) where 𝑎0, 𝑎2,𝑎4, 𝑏0, 𝑏2, 𝑏4 are constants to be determined later and 𝑎0/𝑏0𝑎2/𝑏2𝑎4/𝑏4. After substituting (4.9) in (1.2), multiplying both sides by the common denominator (𝑏0+𝑏2𝑤2𝑛𝑒2Ω𝑡+𝑏4𝑤4𝑛𝑒4Ω𝑡)3(𝑏0+𝑏2𝑤2(𝑛1)𝑒2Ω𝑡+𝑏4𝑤4(𝑛1)𝑒4Ω𝑡)(𝑏0+𝑏2𝑤2(𝑛+1)𝑒2Ω𝑡+𝑏4𝑤4(𝑛+1)𝑒4Ω𝑡), it follows:𝐴18𝑒18Ω𝑡+𝐴16𝑒16Ω𝑡+𝐴14𝑒14Ω𝑡+𝐴12𝑒12Ω𝑡+𝐴10𝑒10Ω𝑡+𝐴8𝑒8Ω𝑡+𝐴6𝑒6Ω𝑡+𝐴4𝑒4Ω𝑡+𝐴2𝑒2Ω𝑡=0,(4.10) where

𝐴18=𝑏34(𝑎2𝑏4𝑎4𝑏2)𝑤18𝑛2[𝑤4(4Ω2+2)𝑤2+1],

𝐴16=𝑏4𝑤16𝑛4[(𝑎4𝑏0𝑏24𝑎0𝑏34)𝑤8+(𝑎2𝑏4𝑎4𝑏2)(2Ω𝑏4𝑎2𝑏2𝑏4+4𝑏4Ω2𝑏22Ω4𝑏2)𝑤6+(2𝑎4𝑏0𝑏24+16Ω2𝑎0𝑏344Ω𝑎22𝑏242𝑎4𝑏22𝑏44Ω𝑎24𝑏22+8Ω𝑎2𝑏4𝑎4𝑏2+2𝑎2𝑏24𝑏2+2𝑎0𝑏3416Ω2𝑎4𝑏0𝑏244Ω2𝑎2𝑏24𝑏2+4Ω2𝑎4𝑏22𝑏4)𝑤4+(𝑎2𝑏4𝑎4𝑏2)(2Ωb4𝑎2𝑏2𝑏4+4𝑏4Ω2𝑏22Ω4𝑏2)𝑤2+𝑎4𝑏0𝑏24𝑎0𝑏34],

𝐴14=𝑤14𝑛4[(3𝑎0𝑏2𝑏34+2𝑎2𝑏0𝑏34+4Ω2𝑎2𝑏0𝑏34+2Ω𝑎2𝑏34𝑎02Ω𝑎4𝑏2𝑎0𝑏24+𝑎4𝑏0𝑏2𝑏24+2Ω𝑎24𝑏0𝑏2𝑏42Ω𝑎2𝑏24𝑎4𝑏04Ω2𝑎4𝑏0𝑏2𝑏24)𝑤8+(4Ω𝑎2𝑏34𝑎0+𝑎0𝑏2𝑏34+4Ω2𝑎4𝑏32𝑏4𝑎4𝑏32𝑏42Ω𝑎24𝑏322Ω𝑎22𝑏24𝑏2+𝑎2𝑏22𝑏244𝑎2𝑏0𝑏34+4Ω𝑎2𝑏4𝑎4𝑏2216Ω2𝑎4𝑏0𝑏2𝑏24+16Ω2𝑎0𝑏2𝑏34+4Ω𝑎24𝑏0𝑏2𝑏44Ω2𝑎2𝑏24𝑏22+3𝑎4𝑏0𝑏2𝑏244Ω𝑎4𝑏2𝑎0𝑏244Ω𝑎2𝑏24𝑎4𝑏0)𝑤6+(4𝑎0𝑏2𝑏34+4𝑎2𝑏0𝑏3412Ω𝑎2𝑏34𝑎012Ω𝑎24𝑏0𝑏2𝑏4+12Ω𝑎4𝑏2𝑎0𝑏242𝑎2𝑏22𝑏24+4Ω𝑎24𝑏328𝑎4𝑏0𝑏2𝑏24+2𝑎4𝑏32𝑏424Ω2𝑎2𝑏0𝑏34+12Ω𝑎2𝑏24𝑎4𝑏0+4Ω2𝑎2𝑏24𝑏22+12Ω2𝑎4𝑏0𝑏2𝑏24+12Ω2𝑎0𝑏2𝑏34+4Ω𝑎22𝑏24𝑏24Ω2𝑎4𝑏32𝑏48Ω𝑎2𝑏4𝑎4𝑏22)𝑤4+(4Ω𝑎2𝑏34𝑎0+𝑎0𝑏2𝑏34+4Ω2𝑎4𝑏32𝑏4𝑎4𝑏32𝑏42Ω𝑎24𝑏322Ω𝑎22𝑏24𝑏2+𝑎2𝑏22𝑏244𝑎2𝑏0𝑏34+4Ω𝑎2𝑏4𝑎4𝑏2216Ω2𝑎4𝑏0𝑏2𝑏24+16Ω2𝑎0𝑏2𝑏34+4Ω𝑎24𝑏0𝑏2𝑏44Ω2𝑎2𝑏24𝑏22+3𝑎4𝑏0𝑏2𝑏244Ω𝑎4𝑏2𝑎0𝑏244Ω𝑎2𝑏24𝑎4𝑏0)𝑤23𝑎0𝑏2𝑏34+2𝑎2𝑏0𝑏34+4Ω2𝑎2𝑏0𝑏34+2Ω𝑎2𝑏34𝑎02Ω𝑎4𝑏2𝑎0𝑏24+𝑎4𝑏0𝑏2𝑏24+2Ω𝑎24𝑏0𝑏2𝑏42Ω𝑎2𝑏24𝑎4𝑏04Ω2𝑎4𝑏0𝑏2𝑏24],

𝐴12=𝑤12𝑛4[(3𝑎0𝑏22𝑏24+𝑎4𝑏24𝑏20𝑎0𝑏34𝑏0+4Ω𝑎20𝑏34+4𝑎2𝑏0𝑏2𝑏24𝑎4𝑏0𝑏4𝑏2216Ω2𝑎4𝑏20𝑏24+16Ω2𝑎0𝑏34𝑏04Ω𝑎22𝑏0𝑏242Ω𝑎24𝑏0𝑏22+4Ω𝑎24𝑏20𝑏4+2Ω𝑎0𝑏2𝑎2𝑏24+6Ω𝑎2𝑏0𝑎4𝑏2𝑏48Ω𝑎4𝑏0𝑎0𝑏242Ω𝑎0𝑏22𝑎4𝑏4+4Ω2𝑎4𝑏0𝑏4𝑏224Ω2𝑎2𝑏0𝑏2𝑏24)𝑤8+(4Ω𝑎0𝑏2𝑎2𝑏24+12Ω2𝑎0𝑏22𝑏24+8Ω2𝑎4𝑏0𝑏4𝑏22+𝑎0𝑏22𝑏24+2𝑎4𝑏0𝑏4𝑏22+4Ω𝑎0𝑏22𝑎4𝑏43𝑎2𝑏0𝑏2𝑏2420Ω2𝑎2𝑏0𝑏2𝑏24+2Ω𝑎22𝑏0𝑏242Ω𝑎24𝑏0𝑏22+𝑎2𝑏32𝑏4𝑎4𝑏42)𝑤6+(4𝑎0𝑏22𝑏24+2𝑎0𝑏34𝑏04Ω2𝑎2𝑏32𝑏4+2𝑎4𝑏42+20Ω2𝑎0𝑏22𝑏24+4Ω2𝑎4𝑏422𝑎4𝑏0𝑏4𝑏2216Ω2𝑎0𝑏34𝑏0+4Ω𝑎22𝑏0𝑏244Ω2𝑎2𝑏0𝑏2𝑏242𝑎2𝑏32𝑏42𝑎2𝑏0𝑏2𝑏24+8Ω𝑎24𝑏0𝑏22+16Ω2𝑎4𝑏20𝑏242𝑎4𝑏24𝑏204Ω𝑎0𝑏22𝑎4𝑏48Ω𝑎24𝑏20𝑏416Ω2𝑎4𝑏0𝑏4𝑏2212Ω𝑎2𝑏0𝑎4𝑏2𝑏4+16Ω𝑎4𝑏0𝑎0𝑏24+4Ω𝑎0𝑏2𝑎2𝑏248Ω𝑎20𝑏34)𝑤4(4Ω𝑎0𝑏2𝑎2𝑏2412Ω2𝑎0𝑏22𝑏248Ω2𝑎4𝑏0𝑏4𝑏22𝑎0𝑏22𝑏242𝑎4𝑏0𝑏4𝑏224Ω𝑎0𝑏22𝑎4𝑏4+3𝑎2𝑏0𝑏2𝑏24+20Ω2𝑎2𝑏0𝑏2𝑏242Ω𝑎22𝑏0𝑏24+2Ω𝑎24𝑏0𝑏22𝑎2𝑏32𝑏4+𝑎4𝑏42)𝑤23𝑎0𝑏22𝑏24+𝑎4𝑏24𝑏20𝑎0𝑏34𝑏0+4Ω𝑎20𝑏34+4𝑎2𝑏0𝑏2𝑏24𝑎4𝑏0𝑏4𝑏2216Ω2𝑎4𝑏20𝑏24+16Ω2𝑎0𝑏34𝑏04Ω𝑎22𝑏0𝑏242Ω𝑎24𝑏0𝑏22+4Ω𝑎24𝑏20𝑏4+2Ω𝑎0𝑏2𝑎2𝑏24+6Ω𝑎2𝑏0𝑎4𝑏2𝑏48Ω𝑎4𝑏0𝑎0𝑏242Ω𝑎0𝑏22𝑎4𝑏4+4Ω2𝑎4𝑏0𝑏4𝑏224Ω2𝑎2𝑏0𝑏2𝑏24],

𝐴10=𝑤10𝑛4[(𝑎4𝑏0𝑏322𝑎4𝑏2𝑏4𝑏20𝑎0𝑏32𝑏4+4𝑎2𝑏24𝑏20+12Ω2𝑎0𝑏2𝑏0𝑏242𝑎0𝑏2𝑏0𝑏2424Ω2𝑎2𝑏24𝑏206Ω𝑎24𝑏2𝑏20+2𝑎2𝑏0𝑏4𝑏22+12Ω2𝑎4𝑏2𝑏4𝑏20+6Ω𝑎20𝑏2𝑏24+12Ω𝑎2𝑏20𝑎4𝑏412Ω𝑎2𝑏0𝑎0𝑏24)𝑤8(4Ω𝑎20𝑏2𝑏242𝑎2𝑏0𝑏4𝑏22+6𝑎2𝑏24𝑏204Ω2𝑎0𝑏32𝑏4+𝑎0𝑏32𝑏4+8Ω2𝑎2𝑏0𝑏4𝑏22+𝑎4𝑏0𝑏323𝑎4𝑏2𝑏4𝑏204Ω2𝑎4𝑏0𝑏323𝑎0𝑏2𝑏0𝑏248Ω𝑎2𝑏0𝑎0𝑏244Ω𝑎24𝑏2𝑏20+8Ω𝑎2𝑏20𝑎4𝑏4)𝑤6+(4𝑎2𝑏24𝑏208𝑎2𝑏0𝑏4𝑏22+4𝑎0𝑏32𝑏4+4𝑎4𝑏0𝑏324Ω2𝑎4𝑏2𝑏4𝑏204Ω2𝑎0𝑏2𝑏0𝑏24+8Ω2𝑎2𝑏24𝑏202𝑎4𝑏2𝑏4𝑏208Ω𝑎2𝑏20𝑎4𝑏4+12Ω2𝑎4𝑏0𝑏322𝑎0𝑏2𝑏0𝑏24+8Ω𝑎2𝑏0𝑎0𝑏24+4Ω𝑎24𝑏2𝑏204Ω𝑎20𝑏2𝑏2424Ω2𝑎2𝑏0𝑏4𝑏22+12Ω2𝑎0𝑏32𝑏4)𝑤4(4Ω𝑎20𝑏2𝑏242𝑎2𝑏0𝑏4𝑏22+6𝑎2𝑏24𝑏204Ω2𝑎0𝑏32𝑏4+𝑎0𝑏32𝑏4+8Ω2𝑎2𝑏0𝑏4𝑏22+𝑎4𝑏0𝑏323𝑎4𝑏2𝑏4𝑏204Ω2𝑎4𝑏0𝑏323𝑎0𝑏2𝑏0𝑏248Ω𝑎2𝑏0𝑎0𝑏244Ω𝑎24𝑏2𝑏20+8Ω𝑎2𝑏20𝑎4𝑏4)𝑤2𝑎4𝑏0𝑏322𝑎4𝑏2𝑏4𝑏20𝑎0𝑏32𝑏4+4𝑎2𝑏24𝑏20+12Ω2𝑎0𝑏2𝑏0𝑏242𝑎0𝑏2𝑏0𝑏2424Ω2𝑎2𝑏24𝑏206Ω𝑎24𝑏2𝑏20+2𝑎2𝑏0𝑏4𝑏22+12Ω2𝑎4𝑏2𝑏4𝑏20+6Ω𝑎20𝑏2𝑏24+12Ω𝑎2𝑏20𝑎4𝑏412Ω𝑎2𝑏0𝑎0𝑏24],

𝐴8=𝑤8𝑛4[(4Ω2𝑎2𝑏2𝑏4𝑏20+4Ω𝑎24𝑏30+3𝑎4𝑏22𝑏20𝑏20𝑎0𝑏24+𝑏30𝑎4𝑏42Ω𝑎4𝑏0𝑎0𝑏22+2Ω𝑎4𝑏20𝑎2𝑏24𝑎2𝑏2𝑏4𝑏204Ω𝑎22𝑏4𝑏20+𝑎0𝑏4𝑏0𝑏222Ω𝑎20𝑏4𝑏22+4Ω𝑎20𝑏24𝑏016Ω2𝑎4𝑏30𝑏44Ω2𝑎0𝑏4𝑏0𝑏22+16Ω2𝑎0𝑏24𝑏208Ω𝑎4𝑏20𝑎0𝑏4+6Ω𝑎2𝑏4𝑎0𝑏2𝑏0)𝑤8+(𝑎2𝑏0𝑏32𝑎0𝑏42+2Ω𝑎20𝑏4𝑏22+2𝑎0𝑏4𝑏0𝑏222Ω𝑎22𝑏4𝑏20+𝑎4𝑏22𝑏20+8Ω2𝑎0𝑏4𝑏0𝑏2220Ω2𝑎2𝑏2𝑏4𝑏20+12Ω2𝑎4𝑏22𝑏203𝑎2𝑏2𝑏4𝑏20+4Ω𝑎4𝑏20𝑎2𝑏24Ω𝑎4𝑏0𝑎0𝑏22)𝑤6(8Ω𝑎20𝑏4𝑏22+2𝑏20𝑎0𝑏24+4Ω𝑎22𝑏4𝑏204𝑎4𝑏22𝑏2016Ω2𝑎0𝑏24𝑏202𝑏30𝑎4𝑏4+4Ω2𝑎2𝑏2𝑏4𝑏20+16Ω2𝑎0𝑏4𝑏0𝑏22+2𝑎2𝑏2𝑏4𝑏208Ω𝑎24𝑏30+4Ω𝑎4𝑏20𝑎2𝑏2+2𝑎0𝑏4𝑏0𝑏22+16Ω𝑎4𝑏20𝑎0𝑏48Ω𝑎20𝑏24𝑏04Ω𝑎4𝑏0𝑎0𝑏22+16Ω2𝑎4𝑏30𝑏4+2𝑎2𝑏0𝑏3212Ω𝑎2𝑏4𝑎0𝑏2𝑏04Ω2𝑎0𝑏422𝑎0𝑏42+4Ω2𝑎2𝑏0𝑏3220Ω2𝑎4𝑏22𝑏20)𝑤4+(𝑎2𝑏0𝑏32𝑎0𝑏42+2Ω𝑎20𝑏4𝑏22+2𝑎0𝑏4𝑏0𝑏222Ω𝑎22𝑏4𝑏20+𝑎4𝑏22𝑏20+8Ω2𝑎0𝑏4𝑏0𝑏2220Ω2𝑎2𝑏2𝑏4𝑏20+12Ω2𝑎4𝑏22𝑏203𝑎2𝑏2𝑏4𝑏20+4Ω𝑎4𝑏20𝑎2𝑏24Ω𝑎4𝑏0𝑎0𝑏22)𝑤24Ω2𝑎2𝑏2𝑏4𝑏204Ω𝑎24𝑏30