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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 395628, 14 pages
http://dx.doi.org/10.1155/2013/395628
Research Article

Exact Traveling Wave Solutions for a Nonlinear Evolution Equation of Generalized Tzitzéica-Dodd-Bullough-Mikhailov Type

College of Mathematics of Honghe University, Mengzi Yunnan 661100, China

Received 18 March 2013; Accepted 9 May 2013

Academic Editor: Shiping Lu

Copyright © 2013 Weiguo Rui. 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

By using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. Under different parameters, we investigated different kinds of exact traveling wave solutions of this generalized TDBM equation. Many singular traveling wave solutions with blow-up form and broken form, such as periodic blow-up wave solutions, solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions, are obtained. In order to visually show dynamical behaviors of these exact solutions, we plot graphs of profiles for some exact solutions and discuss their dynamical properties.

1. Introduction

In this paper, we consider the following nonlinear evolution equation: where , are two non-zero real numbers and , are two integers. We call (1) generalized Tzitzéica-Dodd-Bullough-Mikhailov equation because it contains Tzitzéica equation, Dodd-Bullough-Mikhailov equation, and Tzitzéica-Dodd-Bullough equation. When , , , or , , , , especially (1) becomes classical Tzitzéica equation [13] as follows: which was originally found in the field of geometry in 1907 by G. Tzitzéica and appeared in the fields of mathematics and physics alike. Equation (2) usually called the “Dodd-Bullough equation,’’ which was initiated by Bullough and Dodd [4] and Žiber and Šabat [5]. Indeed, (2) has another form see [6, 7] and the references cited therein.

When , and , , (1) becomes Dodd-Bullough-Mikhailov equation When , and , , (1) becomes Tzitzéica-Dodd-Bullough equation The Dodd-Bullough-Mikhailov equation and Tzitzéica-Dodd-Bullough equation appeared in many problems varying from fluid flow to quantum field theory.

Moreover, when , , , or , , , , (1) becomes the -Gordon equation which was shown in [710] and the references cited therein. When , , or , , , (1) becomes the -Gordon equation which was shown in [11, 12] and the references cited therein. When , , especially (1) becomes the Liouville equation All the equations mentioned above play very significant roles in many scientific applications. And some of them were studied by many authors in recent years; see the following brief statements.

In [13], by using the tanh method, Wazwaz considered some solitary wave and periodic wave solutions for the Dodd-Bullough-Mikhailov and Dodd-Bullough equations. In [14], Andreev studied the Bäcklund transformation for Bullough-Dodd-Jiber-Shabat equation. In [15], Cherdantzev and Sharipov obtained finite-gap solutions of the Bullough-Dodd-Jiber-Shabat equation. In [16], Cherdantzev and Sharipov investigated solitons on the finite-gap background in the Bullough-Dodd-Jiber-Shabat model. In addition, the Darboux transformation, self-dual Einstein spaces and consistency and general solution of Tzitzéica equation were studied in [1719].

In this paper, by using integral bifurcation method [20], we will study (1). Indeed, the integral bifurcation method has successfully been combined with the computer method [21] and some transformations [22, 23] for investigating exact traveling wave solutions of some nonlinear PDEs. Therefore, using this method, we will obtain some new results which are different from those in the above references.

The rest of this paper is organized as follows: in Section 2, we will derive two-dimensional planar system which is equivalent to (1) and give its first integral equation. In Sections 3 and 4, by using the integral bifurcation method, we will obtain some new traveling wave solutions of (1) and discuss their dynamic properties.

2. Two-Dimensional Planar Dynamical System of (1) and Its First Integral

Applying the transformations , we change (1) into the following form: Let , substituting into (9), respectively, we obtain where ‘‘′’’ is the derivative with respect to and is wave speed.

Clearly, (10) is equivalent to the following two-dimensional systems: However, when , (11) is not equivalent to (10) and the cannot be defined, so we call the singular line and call (11) singular system. In order to obtain an equivalent system of (10), making a scalar transformation equation (11) can be changed into a regular two-dimensional system as follows: Systems (11) and (13) are two integrable systems, and they have the same first integral as follows: where is integral constant.

Systems (11) and (13) are planar dynamical systems defined by the 5-parameter . Usually their first integral (14) can be rewritten as

When , , system (13) has two equilibrium points and in the -axes. When , and , system (13) has three equilibrium point and in the -axes. When , and , system (13) has only one equilibrium points . When , , system (13) has only one equilibrium point .

Respectively, substituting every equilibrium point (the point is exception) into (15), we have

From the transformation , we know that approaches to if a solution approaches to in (14). In other words, this solution determines an unboundary wave solution or blow-up wave solution of (1). It is easy to see that the second equation of system (11) is not continuous when . In other words, on such straight line in the phase plane , the function is not defined, and this implies that the smooth wave solutions of (1) sometimes become nonsmooth wave solutions.

3. Exact Solutions of (1) and Their Properties under the Conditions , , , and ,

In this section, we investigate exact solutions of (1) under different kinds of parametric conditions and their properties.

3.1. Different Kinds of Solitary Wave Solutions and Unboundary Wave Solutions in the Special Cases or

(i) When , , and is even, then (14) can be reduced to Taking and as initial values, substituting (17) into the of (11), and then integrating it yield where ,, , . By using program of Maple, it is easy to validate that (18) is the solution equation (9) when is given. Substituting (18) into the transformation , we obtain a solitary wave solutions of (1) as follows:

(ii) When , , and is even, (14) can be reduced to Taking and as initial values, substituting (20) into the of (11), and then integrating it yield where . Substituting (21) into the transformation , we obtain a solution without boundary of (1) as follows: Equation (21) is smooth solitary wave solution of (9) but (22) is an exact solution without boundary of (1) because as . In order to visually show dynamical behaviors of solutions and of (21) and (22), we plot graphs of their profiles which are shown in Figures 1(a) and 1(b), respectively.

fig1
Figure 1: The phase portraits of solutions (21) and (22) for , , , , , .

(iii) When , , , and is even, (14) can be reduced to Taking and as initial values, substituting (23) into the of (11), and then integrating it yield Substituting (24) into the transformation , we obtain an exact solutions of rational function type of (1) as follows:

(iv) When , , , (or , ), and is odd, (14) can be reduced to Respectively, taking , , and , as initial values, substituting (26) into the of (11), and then integrating them yield where is given above and , , , , . Respectively, substituting (27) and (28) into the transformation , we obtain two smooth unboundary wave solutions of (1) as follows:

(v) When , , , (or , ) and is odd, (14) can be reduced to Similarly, by using (31), we obtain two smooth solitary wave solutions of (1) which are the same as the solutions (29) and (30).

(vi) When and is odd, (14) can be reduced to (20), so the obtained solution is as the same as the solution (22).

(vii) When , , and is odd, (14) can be reduced to Taking and as initial values, substituting (32) into the of (11), and then integrating it yield Substituting (33) into the transformation , we obtain a smooth unboundary wave solution of rational function type of (1) as follows:

3.2. Periodic Blow-Up Wave Solutions, Broken Kink Wave, and Antikink Wave Solutions in the Special Case

(1) When , , , (or ) and is even, (14) can be reduced to Taking , and , as the initial values, substituting (35) into the of (11), and then integrating them yield where , , , . Respectively, substituting (36) and (37) into the transformation , we obtain two periodic blow-up wave solutions of (1) as follows:

(2) When , , , and is odd, (14) can be reduced to which is the same as (35), so the obtained solution is the same as solution (39). Similarly, when , , , and is odd, the obtained solution is also the same as solution (38).

(3) when , , , and is even, (14) can be reduced to Substituting (41) into the of (11) and then integrating it and setting the integral constants as zero yield where . Substituting (42) into the transformation , we obtain an unboundary wave solution of (1) as follows: Equation (42) is smooth kink wave solution of (9), but (43) is a broken kink wave solution without boundary of (1) because as . In order to visually show dynamical behaviors of solutions and of (42) and (43), we plot graphs of their profiles which are shown in Figures 2(a) and 2(b), respectively.

fig2
Figure 2: The phase portraits of solutions (42) and (43) for .

(4) When , , , or , , and is odd, (14) can be reduced to (41). So the obtained solution is also the same as the solution (43).

3.3. Periodic Blow-Up Wave Solutions and Solitary Wave Solutions of Blow-Up Form in the Special Case ,

When , (14) can be reduced to

(a) Under the conditions , , and , (44) becomes where , , are three roots of equation and . These three roots , , can be obtained by Cardano formula as long as the parameters , , , are fixed (or given) concretely. For example, taking , , , , the , , . Taking as the initial values of the variables , substituting (45) into the of (11), and then integrating it yield where , , . By using the transformation and (46), we obtain a periodic blow-up wave solution of (1) as follows:

(b) Under the conditions , , and , (44) becomes Similarly, taking as the initial values of the variables , substituting (48) into the of (11), and then integrating it yield By using the transformation and (49), we obtain a periodic blow-up wave solution of (1) as follows:

(c) Under the conditions , , and , (44) becomes where , , , are roots of equation and . As in the above cases, these three roots can be obtained by Cardano formula as long as the parameters , , , are given concretely, so the similar cases will be not commentated anymore in the following discussions. Similarly, taking as the initial values of the variables , substituting (51) into the of (11), and then integrating it yield where , , . By using the transformation and (52), we obtain a periodic blow-up wave solution of (1) as follows:

(d) Under the conditions , , and , (44) becomes Taking the as the initial values of the variables , substituting (54) into the of (11), and then integrating it yield By using the transformations and (55), we obtain a periodic blow-up wave solution of (1) as follows:

(e) Under the conditions , , and , (44) becomes where , , , are roots of equation and . Taking the as the initial values of the variables , substituting (57) into the of (11), and then integrating it yield where , , . By using the transformation and (58), we obtain a periodic blow-up wave solution of (1) as follows:

(f) Under the conditions , , and , (44) becomes where , , , are roots of equation and . Taking the as the initial values of the variables , substituting (60) into the of (11), and then integrating it yield where , , . By using the transformation and (61), we obtain a periodic below-up wave solutions of (1) as follows:

(g) Under the conditions , , , (44) becomes where . Taking as the initial values of the variables , substituting (63) into the of (11), and then integrating it yield where , . Clearly, we have , . By using the transformation and (64), we obtain a solitary wave solution of blow-up form of (1) as follows: Equation (64) is smooth solitary wave solution of (9), but (65) is a solitary wave solution of blow-up form for (1) because as . In order to visually show dynamical behaviors of solutions and of (64) and (65), we plot graphs of their profiles which are shown in Figures 3(a) and 3(b), respectively.

fig3
Figure 3: The phase portraits of solutions (64) and (65) for , , , , .
3.4. Different Kinds of Periodic Wave, Broken Solitary Wave Solutions in the Special Case ,

(1) Under the conditions , , (or , ), and , (44) becomes where and are three roots of equation and . Taking as the initial values of the variables , substituting (66) into the of (11), and then integrating it yield where , . By using the transformation and (67), we obtain a double periodic wave solution of (1) as follows: Equation (67) is smooth periodic wave solution of (9), but (68) is a double periodic wave solution of blow-up form of (1) because as . In order to visually show dynamical behaviors of solutions and of (67) and (68), we plot graphs of their profiles which are shown in Figures 4(a) and 4(b), respectively.

fig4
Figure 4: The phase portraits of solutions (67) and (68) for , , , , .

(2) Under the condition ,, (or , , and , (44) becomes where are roots of equation and . Taking as the initial values of the variables , substituting (69) into the of (11), and then integrating it yield where , , . By using the transformation and (70), we obtain a periodic blow-up wave solution of (1) as follows:

(3) Under the conditions , , (or , ) and , (44) becomes where , , are roots of equation and . Taking as the initial values of the variables , substituting (72) into the of (11), and then integrating it yield where , . By using the transformation and (73), we obtain a periodic blow-up wave solutions of (1) as follows:

(4) Under the conditions ,, (or , ) and , (44) becomes where , , are roots of equation and . Taking as the initial values of the variables , substituting (75) into the of (11), and then integrating it yield where , , . By using the transformation and (76), we obtain periodic blow-up wave solutions of (1) as follows:

(5) Under the conditions , , and , (44) becomes where . Taking as the initial values of the variables , substituting (78) into the of (11), and then integrating it yield where . By using the transformation and (79), we obtain a solitary wave solution of blow-up form of (1) as follows:

(6) Under the conditions , , and , (44) becomes where . Taking as the initial values of the variables , substituting (81) into the of (11), and then integrating it yield where . By using the transformation and (82), we obtain a broken solitary wave solution of (1) as follows:

(7) Under the conditions , , and , (44) becomes where , , are roots of equation and . Taking as the initial values of the variables , substituting (84) into the of (11), and then integrating it yield where , , . By using the transformation and (85), we obtain a periodic blow-up wave solution of (1) as follows:

(8) Under the conditions , , and , (44) becomes where , , are roots of equation and . Taking as the initial values of the variables , substituting (87) into the of (11), and then integrating it yield where , , . By using the transformation and (88), we obtain a periodic blow-up wave solution of (1) as follows:

3.5. Periodic Blow-Up Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Cases or and

When is even and , (14) can be reduced to Because the singular straight line , this implies that (1) has broken kink and antikink wave solutions: see the following discussions.

(i) Under the conditions , , , , , (90) becomes where . Taking as the initial values of the variables , substituting (91) into the of (11), and then integrating it yield where . By using the transformations and (92), we obtain a broken kink wave solutions of (1) as follows: where and .

(ii) Under the conditions , , and , , (90) becomes where , . Taking as the initial values of the variables , substituting (94) into the of (11), and then integrating it yield where , , , . By using the transformations and (95), we obtain a periodic blow-up wave solution of (1) as follows:

(iii) Under the conditions , , , , , (90) becomes where . Taking as the initial values of the variables , substituting (97) into the of (11), and then integrating it yield where , . By using the transformations and (98), when , we obtain a broken solitary wave solution of (1) as follows:

(iv) Under the conditions , , and , , (90) becomes where . Taking as the initial values of the variables , substituting (100) into the of (11), and then integrating it yield where , . By using the transformations and (101), when , we obtain a broken solitary wave solution of (1) as follows:

(v) Under the conditions , , and , , (90) becomes where , , are roots of equation and . Taking as the initial values of the variables , substituting (103) into the of (11), and then integrating it yield where , . By using the transformations and (104), we obtain a periodic blow-up wave solution of (1) as follows:

(vi) Under the conditions , , and , , (90) becomes where . Taking as the initial values of the variables , substituting (106) into the of (11), and then integrating it yield where , , . By using the transformations and (113), we obtain a periodic blow-up wave solution of (1) as follows:

3.6. Periodic Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Cases or and

(a) When