Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 293607, 24 pagesdoi:10.1155/2011/293607

Research Article

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

Weiguo Rui and Xinsong Yang

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 -dimensional space to the -dimensional one, even those forms of exact solution do not exist in -dimensional NDDEs. By using this extended method, the Toda lattice and -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 [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, ()-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] and the Toda lattice equation [20] It is well known that there is another form for the (2 + 1)-dimensional Toda lattice equation and there are another two forms for the Toda lattice equation Under the transformation , (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 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: respectively, where in (2.1) and in (2.2), is a discrete variable, and and 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: where is a positive integer which can be given by the homogeneous balance principle or the character of the idiographic NDDEs, the parameters are constants which need to be determined later, and .Besides the above forms of solutions, we will add three new expansions. If (2.1) has no term , then we suppose their expansions as follows: where 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: We add two new forms of solutions for (2.2) here where is a positive integer which can be given by the homogeneous balance principle or the character of the idiographic NDDEs, the parameters are constants which need to be determined later, and .

Step 2. By using the homogeneous balance principle or according to character of the idiographic NDDEs, we determine the value of then substitute it in the expressions (2.3)–(2.12) of Step 1. Sometimes, we can directly assume that or . 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 or 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 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 . Thus, we let where , , ,, ,,, are constants to be determined later, and . After substituting (3.1) into (1.1), multiplying both sides by the common denominator and dividing both sides by the common factor , it follows: where

In (3.2), setting the coefficients of all independent terms ,, to zero, we get a series of algebraic equations as follows: 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 , we let As a result, the equation becomes , that is, , which is a low-power equation. Similarly, we obtain Notice that , thus, (3.4) can be reduced to a group of low-power algebraic equations as follows: Solving (3.7) yields and 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: where are given by (3.9).

In fact, (3.10) and (3.11) are solutions of local discretization because the is a discrete variable while the 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 , when , the solution (3.10) shows a shape of discrete kink soliton, see Figure 1(a); when , the solution (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 , when , the solution (3.11) shows a shape of discrete antikink soliton, see Figure 1(c); when , the solution (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 -axes and is continuous along the -axes.

Figure 1: The solutions in (3.10) and (3.11) show four shapes of discrete kink and antikink solitons for fixed parametric values of and different kinds of values: (a) ; (b) ; (c) ; (d) .

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 . For simplicity, here we only consider the case of . Suppose that where , , , , , , are constants to be determined later, and , the 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 , it follows: where In (3.13), setting the coefficients of all independent exp-function terms to zero, it follows Let . Then Thus, (3.15) can be reduced to the following equations: Solving (3.17), we obtain and are arbitrary nonzero constants with .

Based on (3.12), (3.18), (3.19), and (3.20), we obtain four exact solutions of (1.1) as follows: where are given by (3.19), (3.20), and .

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.

Figure 2: The solutions in (3.21) and (3.22) show two shapes of discrete antikink and kink solitons for fixed parametric values: , , , , , , .

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 . So, we suppose that (1.1)'s solution has the following form: where , , , , , , are constants to be determined later, is an arbitrary constant, and 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 , it follows: where

In (3.26), setting the coefficients of all independent exp-function terms to zero, we obtain Directly solving (3.28), we obtain and are arbitrary constants. Thus, we obtain a class of exact solutions of (1.1) as follows: where is an arbitrary function, for which let us set to be , , , , , ; respectively, we obtain a series of soliton solutions and breather solutions as follows:

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 , 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 , 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 is fixed, the profiles show discrete dynamic character, see Figures 3(a), 3(c), and 3(e); while the parameter is fixed, the profiles show continuous dynamic character, see Figures 3(b), 3(d), and 3(f).

(a) Discrete soliton

(b) Continuous double soliton

(c) Discrete breather oscillation

(d) Continuous breather oscillation

(e) Discrete breather oscillation

(f) Continuous breather oscillation

Figure 3: The solutions 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 . So, we suppose that (1.1) has exact solution as the following form: where , , , , , are constants to be determined later, is an arbitrary constant, and 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: Similarly, setting to be , respectively, we obtain a series of soliton solutions and breather solutions as follows:

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 , 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 , (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).

(a) Discrete soliton

(b) Continuous double soliton

(c) Discrete breather oscillation

(d) Continuous breather oscillation

(e) Discrete breather oscillation

(f) Continuous breather oscillation

Figure 4: The solutions 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 . So, we suppose that (1.1) has exact solution as the following form: where , ,,, , are constants to be determined later, is an arbitrary constant, and is an arbitrary function. As in Section 3.3, by using (3.44) and (1.1) and the expatiatory computation, we obtain . So we obtain a family of exact solutions with arbitrary function as follows: Similarly, assume the same functions as in Section 3.3, we obtain a series of soliton solutions and breather solutions as follows:

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 , 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 , 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).

(a) Discrete kink soliton

(b) Continuous double kink soliton

(c) Discrete soliton

(d) Continuous double soliton

(e) Discrete breather oscillation

(f) Continuous breather oscillation

Figure 5: The solutions 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: After substituting (4.1) in (1.2), multiplying both sides by the common denominator , it follows: where In (4.2), setting the coefficients of all independent exp-function terms to zero, we obtain Let , then Thus, (4.4) can be reduced to Solving (4.6) yields and are arbitrary constants. From (4.1), (4.7), we obtain discrete kink or antikink soliton solution of (1.2) as follows: 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: where , ,, , , are constants to be determined later and . After substituting (4.9) in (1.2), multiplying both sides by the common denominator , it follows: where

,

.

In (4.10), we let Solving the group of (4.11), we obtain three group of parametric conditions which satisfy (4.10).

Case 1. We have the following: where , , , are arbitrary nonzero constants with .

Case 2. We have the following: where and are arbitrary nonzero constants with .

Under Case 1, substituting (4.12) in (4.9), we obtain discrete kink or antikink soliton solution of (1.2) as follows: where are arbitrary nonzero constants with .

Under Case 2, substituting (4.13) into (4.9), we obtain discrete kink or antikink soliton solution of (1.2) as follows: where denotes and denotes , and have been given above, and are arbitrary nonzero constants with .

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 , the solution (4.15) shows a shape of discrete antikink wave, see Figure 6(a). Setting , the solution (4.15) shows a shape of discrete kink wave, see Figure 6(b).

(a) Discrete antikink wave

(b) Discrete kink wave

Figure 6: The solution in (4.15) shows shapes of discrete antikink and kink solitons.

4.3. Exact Solutions of the Form (2.12)

Suppose that (1.2) has exact solution of the form (2.12) as follows: where and , , are constants to be determined later. After substituting (4.16) in (1.2), multiplying both sides by the common denominator , it follows where

In (4.17), we let Solving the group of (4.18), we obtain two groups of parametric conditions which satisfy (4.17).

Case 1. We have the following: where are two arbitrary nonzero constants.

Case 2. We have the following: where are also two arbitrary nonzero constants.

Under Case 1, substituting (4.19) in (4.16), we obtain discrete kink or antikink soliton solution of (1.2) as follows: where are arbitrary nonzero constants.

Under Case 2, substituting (4.20) in (4.16), we obtain discrete kink or antikink soliton solution of (1.2) as follows: where 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 -axes and have continuous character along the -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.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (10871073). It is also supported by the Natural Science Foundations of Yunnan Province (no. 2008CD186 and 2008ZC153M) and the Natural Science Foundation of Department Education of Yunnan Province (08Y0336).

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