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

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: 𝑢_𝑥𝑡=𝛼𝑒^𝑚𝑢+𝛽𝑒^𝑛𝑢, (1) where 𝛼, 𝛽 are two non-zero real numbers and 𝑚, 𝑛 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 𝛼=1, 𝛽=−1, 𝑚=1, 𝑛=−2 or 𝛼=−1, 𝛽=1, 𝑚=−2, 𝑛=1, especially (1) becomes classical Tzitzéica equation [1–3] as follows: 𝑢_𝑥𝑡=𝑒^𝑢−𝑒^−2𝑢, (2) 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 𝑢_𝑥𝑥−𝑢_𝑡𝑡=𝑒^𝑢−𝑒^−2𝑢; (3) see [6, 7] and the references cited therein.

When 𝛼=−1, 𝛽=−1 and 𝑚=1, 𝑛=−2, (1) becomes Dodd-Bullough-Mikhailov equation 𝑢_𝑥𝑡+𝑒^𝑢+𝑒^−2𝑢=0. (4) When 𝛼=1, 𝛽=1 and 𝑚=1, 𝑛=−2, (1) becomes Tzitzéica-Dodd-Bullough equation 𝑢_𝑥𝑡−𝑒^𝑢−𝑒^−2𝑢=0. (5) The Dodd-Bullough-Mikhailov equation and Tzitzéica-Dodd-Bullough equation appeared in many problems varying from fluid flow to quantum field theory.

Moreover, when 𝑚=1, 𝑛=−1, 𝛼=1/2, 𝛽=−1/2 or 𝑚=−1, 𝑛=1, 𝛼=−1/2, 𝛽=1/2, (1) becomes the sinh-Gordon equation 𝑢_𝑥𝑡=sinh𝑢, (6) which was shown in [7–10] and the references cited therein. When 𝑚=1, 𝑛=−1, 𝛼=𝛽=1/2 or 𝑚=−1, 𝑛=1, 𝛼=𝛽=1/2, (1) becomes the cosh-Gordon equation 𝑢_𝑥𝑡=cosh𝑢, (7) which was shown in [11, 12] and the references cited therein. When 𝑚=1, 𝑛=0, especially (1) becomes the Liouville equation 𝑢_𝑥𝑡=𝛼𝑒^𝑢. (8) 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 [17–19].

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 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we change (1) into the following form: 𝑣𝑣_𝑥𝑡−𝑣_𝑥𝑣_𝑡=𝛼𝑣^𝑚+2+𝛽𝑣^𝑛+2. (9) Let 𝑣(𝑥,𝑡)=𝜙(𝑥−𝑐𝑡)=𝜙(𝜉), substituting 𝜙(𝑥−𝑐𝑡) into (9), respectively, we obtain 𝑐𝜙^′2−𝑐𝜙𝜙^′′=𝛼𝜙^𝑚+2+𝛽𝜙^𝑛+2, (10) where ‘‘′’’ is the derivative with respect to 𝜉 and 𝑐 is wave speed.

Clearly, (10) is equivalent to the following two-dimensional systems: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=𝑑𝜉𝑐𝑦²−𝛼𝜙^𝑚+2−𝛽𝜙^2+𝑛.𝑐𝜙 (11) However, when 𝜙=0, (11) is not equivalent to (10) and the 𝑑𝑦/𝑑𝜉 cannot be defined, so we call the 𝜙=0 singular line and call (11) singular system. In order to obtain an equivalent system of (10), making a scalar transformation 𝑑𝜉=𝑐𝜙𝑑𝜏, (12) equation (11) can be changed into a regular two-dimensional system as follows: 𝑑𝜙𝑑𝜏=𝑐𝜙𝑦,𝑑𝑦𝑑𝜏=𝑐𝑦²−𝛼𝜙^𝑚+2−𝛽𝜙^𝑛+2. (13) Systems (11) and (13) are two integrable systems, and they have the same first integral as follows: 𝑦²=ℎ𝜙²−2𝛼𝜙𝑐𝑚^𝑚+2−2𝛽𝜙𝑐𝑛^𝑛+2, (14) where ℎ is integral constant.

Systems (11) and (13) are planar dynamical systems defined by the 5-parameter (𝛼,𝛽,𝑐,𝑚,𝑛). Usually their first integral (14) can be rewritten as 𝐻𝜙,𝑦≡2𝛼𝜙𝑐𝑚^𝑚+2𝛽𝜙𝑐𝑛^𝑛+𝑦²𝜙²=ℎ. (15)

When 𝑛−𝑚=2𝑘−1, (𝑘∈𝑁), system (13) has two equilibrium points 𝑂(0,0) and 𝐴((−𝛼/𝛽)^{1/(𝑛−𝑚)},0) in the 𝜙-axes. When 𝑛−𝑚=2𝑘, (𝑘∈𝑁) and 𝛼𝛽<0, system (13) has three equilibrium point 𝑂(0,0) and 𝐵_1,2(±(−𝛼/𝛽)^{1/(𝑛−𝑚)},0) in the 𝜙-axes. When 𝑛−𝑚=2𝑘, (𝑘∈𝑁) and 𝛼𝛽>0, system (13) has only one equilibrium points 𝑂(0,0). When 𝑛=𝑚=𝑘, (𝑘∈𝑁), system (13) has only one equilibrium point 𝑂(0,0).

Respectively, substituting every equilibrium point (the point 𝑂(0,0) is exception) into (15), we have ℎ_𝐴−𝛼=𝐻𝛽^{1/(𝑛−𝑚)}=(),02𝛼𝑛−𝑚−𝛼𝑐𝑚𝑛𝛽^{𝑚/(𝑛−𝑚)},ℎ_𝐵1=𝐻𝐵₁−𝛼=𝐻𝛽^{1/(𝑛−𝑚)}=(),02𝛼𝑛−𝑚−𝛼𝑐𝑚𝑛𝛽^{𝑚/(𝑛−𝑚)},ℎ_𝐵2=𝐻𝐵₂−−𝛼=𝐻𝛽^{1/(𝑛−𝑚)},0=(−1)^𝑚ℎ_𝐵1. (16)

From the transformation 𝑢(𝑥,𝑡)=ln|𝜙(𝑥−𝑐𝑡)|, we know that ln|𝜙(𝑥−𝑐𝑡)| approaches to −∞ if a solution 𝜙(𝜉) approaches to 0 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 𝜙=0. In other words, on such straight line 𝜙=0 in the phase plane (𝜙,𝑦), the function 𝜙^′′ 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 𝑛=±𝑚, ±2𝑚, 𝑚∈𝐙⁺, and 𝑚=2, 𝑛=−1

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 𝑛=2𝑚∈𝐙⁺ or 𝑛=𝑚∈𝐙⁺

(i) When ℎ>0, 𝛽𝑐<0, 𝑛=2𝑚 and 𝑚 is even, then (14) can be reduced to 𝑦=±𝜙ℎ−2𝛼𝜙𝑐𝑚^𝑚−𝛽𝜙𝑐𝑚^𝑚2. (17) Taking 𝜉₀=0 and 𝜙(0)=𝜙₁ as initial values, substituting (17) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()𝛾𝑥,𝑡=±¹Ω₁cosh𝜔₁(𝑥−𝑐𝑡)+𝛿₁^1/𝑚, (18) where 𝜙₁=(−𝛼/𝛽+(𝛼²+𝛽𝑐𝑚ℎ)/𝛽²)^1/𝑚,Ω₁=𝜙^𝑚₁(𝛼²+𝛽𝑐𝑚ℎ), 𝜔₁=−𝑚√ℎ,𝛾₁=𝑐𝑚ℎ(𝑐𝑚ℎ−𝛼𝜙^𝑚₁), 𝛿₁=𝛼(𝑐𝑚ℎ−𝛼𝜙^𝑚₁). By using program of Maple, it is easy to validate that (18) is the solution equation (9) when 𝑚 is given. Substituting (18) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain a solitary wave solutions of (1) as follows: 𝑢()=1𝑥,𝑡𝑚𝛾ln¹Ω₁cosh𝜔₁(𝑥−𝑐𝑡)+𝛿₁. (19)

(ii) When 𝑛=𝑚, ℎ>0, 𝑐/(𝛼+𝛽)>0 and 𝑚 is even, (14) can be reduced to 𝑦=±𝜙2ℎ−𝛼+𝛽𝜙𝑐𝑚^𝑚. (20) Taking 𝜉₀=0 and 𝜙(0)=(𝑚ℎ𝑐/2(𝛼+𝛽))^1/𝑚 as initial values, substituting (20) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=±𝑚ℎ𝑐sech²𝜔₂(𝑥−𝑐𝑡)2𝛼+𝛽^1/𝑚, (21) where 𝜔₂=(1/2)𝑚√ℎ. Substituting (21) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain a solution without boundary of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln𝑚ℎ𝑐sech²𝜔₂(𝑥−𝑐𝑡)2.𝛼+𝛽 (22) Equation (21) is smooth solitary wave solution of (9) but (22) is an exact solution without boundary of (1) because 𝑢→−∞ as 𝑣→0. In order to visually show dynamical behaviors of solutions 𝑣 and 𝑢 of (21) and (22), we plot graphs of their profiles which are shown in Figures 1(a) and 1(b), respectively.

(a) Smooth solitary wave

(b) Unboundary wave

Figure 1 

The phase portraits of solutions (21) and (22) for 𝑚=4, 𝛼=3, 𝛽=−1, 𝑐=2, ℎ=9, 𝑡=0.1.

(iii) When 𝑛=2𝑚, ℎ=0, 𝛽𝑐>0, 𝛼𝑐<0 and 𝑚 is even, (14) can be reduced to 𝑦=±𝜙−2𝛼𝜙𝑐𝑚^𝑚−𝛽𝜙𝑐𝑚^𝑚2. (23) Taking 𝜉₀=0 and 𝜙(0)=0 as initial values, substituting (23) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()2()𝑥,𝑡=±−𝛼𝑐𝛽𝑐+𝑚𝛼²(𝑥−𝑐𝑡)^21/𝑚. (24) Substituting (24) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain an exact solutions of rational function type of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln2𝛼𝑐𝛽𝑐+𝑚𝛼²(𝑥−𝑐𝑡)². (25)

(iv) When 𝑛=2𝑚, ℎ>0, 𝛼𝛽>0, 𝛽𝑐>0 (or 𝛼𝛽<0, 𝛽𝑐>0), and 𝑚 is odd, (14) can be reduced to 𝑦=±𝛽𝜙𝑐𝑚𝑐𝑚ℎ𝛽−2𝛼𝛽𝜙^𝑚−𝜙^𝑚2. (26) Respectively, taking 𝜉₀=0, 𝜙(0)=𝜙₁, and 𝜉₀=0, 𝜙(0)=𝜙₂ as initial values, substituting (26) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating them yield 𝑣()=𝛾𝑥,𝑡¹Ω₁cosh𝜔₁(𝑥−𝑐𝑡)+𝛿₁^1/𝑚,𝑣()=𝛾𝑥,𝑡²Ω₂cosh𝜔₂(𝑥−𝑐𝑡)+𝛿₂^1/𝑚, (27)(28) where 𝜙₁ is given above and 𝜙₂=(−𝛼/𝛽−(𝛼²+𝛽𝑐𝑚ℎ)/𝛽²)^1/𝑚, Ω₂=𝜙^𝑚₂(𝛼²+𝛽𝑐𝑚ℎ), 𝜔₂=𝑚√ℎ, 𝛾₂=𝑐𝑚ℎ(𝑐𝑚ℎ−𝛼𝜙^𝑚₂), 𝛿₂=𝛼(𝑐𝑚ℎ−𝛼𝜙^𝑚₂). Respectively, substituting (27) and (28) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain two smooth unboundary wave solutions of (1) as follows: 𝑢()=1𝑥,𝑡𝑚𝛾ln¹Ω₁cosh𝜔₁(𝑥−𝑐𝑡)+𝛿₁𝑢()=1,𝑥,𝑡𝑚𝛾ln²Ω₂cosh𝜔₂(𝑥−𝑐𝑡)+𝛿₂. (29)(30)

(v) When 𝑛=2𝑚, 0<ℎ<ℎ_𝐴, 𝛼𝛽<0, 𝛽𝑐<0 (or 𝛼𝛽>0, 𝛽𝑐<0) and 𝑚 is odd, (14) can be reduced to 𝑦=±−𝛽𝜙𝑐𝑚−𝑐𝑚ℎ𝛽+2𝛼𝛽𝜙^𝑚+𝜙^𝑚2. (31) 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 𝑛=𝑚,ℎ>0 and 𝑚 is odd, (14) can be reduced to (20), so the obtained solution is as the same as the solution (22).

(vii) When 𝑛=2𝑚, ℎ=0, 𝛽𝑐>0 and 𝑚 is odd, (14) can be reduced to 𝑦=±𝛽𝜙𝑐𝑚−2𝛼𝛽𝜙^𝑚−𝜙^𝑚2. (32) Taking 𝜉₀=0 and 𝜙(0)=(−2𝛼/𝛽)^1/𝑚 as initial values, substituting (32) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=−2𝛼𝛽1+𝑚𝛼²/𝛽𝑐(𝑥−𝑐𝑡)^21/𝑚. (33) Substituting (33) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain a smooth unboundary wave solution of rational function type of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln2𝛼𝛽1+𝑚𝛼²/𝛽𝑐(𝑥−𝑐𝑡)². (34)

3.2. Periodic Blow-Up Wave Solutions, Broken Kink Wave, and Antikink Wave Solutions in the Special Case 𝑛=2𝑚∈𝐙⁺

(1) When 𝑛=2𝑚, 𝛼𝛽<0, 𝛽𝑐>0, ℎ_𝐵1<ℎ<0 (or ℎ_𝐵2<ℎ<0) and 𝑚 is even, (14) can be reduced to 𝑦=±𝛽𝜙𝑐𝑚𝑐𝑚ℎ𝛽−2𝛼𝛽𝜙^𝑚−𝜙^𝑚2. (35) Taking 𝜉₀=0, 𝜙(0)=(−𝛼/𝛽+𝛼²/𝛽²+𝑚ℎ𝑐/𝛽)^1/𝑚 and 𝜉₀=0, 𝜙(0)=(−𝛼/𝛽−𝛼²/𝛽²+𝑚ℎ𝑐/𝛽)^1/𝑚 as the initial values, substituting (35) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating them yield 𝑣(𝑥,𝑡)=𝑐𝑚ℎ𝛼−𝛽Ω₃sinarcsin𝐷₁−𝜔₃(𝑥−𝑐𝑡)^1/𝑚,𝑣(𝑥,𝑡)=−𝑐𝑚ℎ𝛼+𝛽Ω₃sinarcsin𝐷₂+𝜔₃(𝑥−𝑐𝑡)^1/𝑚, (36)(37) where Ω₃=(𝛼²+𝑐𝑚ℎ𝛽)/𝛽², 𝜔₃=𝑚√−ℎ, 𝐷₁=(𝛼²+𝛼𝛽Ω₃+𝛽𝑐𝑚ℎ)/𝛽Ω₃(𝛼+𝛽Ω₃), 𝐷₂=(𝛼²−𝛼𝛽Ω₃+𝛽𝑐𝑚ℎ)/𝛽Ω₃(−𝛼+𝛽Ω₃). Respectively, substituting (36) and (37) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain two periodic blow-up wave solutions of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln𝑐𝑚ℎ𝛼−𝛽Ω₃sinarcsin𝐷₁−𝜔₃()𝑢()=1𝑥−𝑐𝑡,𝑥,𝑡𝑚ln𝑐𝑚ℎ𝛼+𝛽Ω₃sinarcsin𝐷₂+𝜔₃(𝑥−𝑐𝑡). (38)(39)

(2) When 𝑛=2𝑚, 𝛼𝛽<0, 𝛽𝑐>0, ℎ_𝐵2<ℎ<0 and 𝑚 is odd, (14) can be reduced to 𝑦=±𝛽𝜙𝑐𝑚𝑐𝑚ℎ𝛽−2𝛼𝛽𝜙^𝑚−𝜙^𝑚2, (40) which is the same as (35), so the obtained solution is the same as solution (39). Similarly, when 𝑛=2𝑚, 𝛼𝛽<0, 𝛽𝑐>0, ℎ_𝐵1<ℎ<0 and 𝑚 is odd, the obtained solution is also the same as solution (38).

(3) when 𝑛=2𝑚, 𝛼𝛽<0, 𝛽𝑐<0, ℎ=ℎ_𝐵1=ℎ_𝐵2=−𝛼²/𝛽𝑐𝑚 and 𝑚 is even, (14) can be reduced to 𝑦=±−1𝜙𝑐𝑚𝛽𝛽𝜙^𝑚+𝛼. (41) Substituting (41) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11) and then integrating it and setting the integral constants as zero yield 𝑣()=−𝛼𝑥,𝑡Ω2𝛽1±tanh⁴2(𝑥−𝑐𝑡)^1/𝑚, (42) where Ω₄=𝛼−𝑚/𝑐𝛽. Substituting (42) into the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)|, we obtain an unboundary wave solution of (1) as follows: 𝑢()=1𝑥,𝑡𝑚𝛼lnΩ2𝛽1±tanh⁴2(𝑥−𝑐𝑡). (43) Equation (42) is smooth kink wave solution of (9), but (43) is a broken kink wave solution without boundary of (1) because 𝑢→−∞ as 𝑣→0. In order to visually show dynamical behaviors of solutions 𝑣 and 𝑢 of (42) and (43), we plot graphs of their profiles which are shown in Figures 2(a) and 2(b), respectively.

(a) Smooth kink wave

(b) Unboundary wave

Figure 2 

The phase portraits of solutions (42) and (43) for 𝑚=4,𝛼=0.3,𝛽=−0.3,𝑐=2,𝑡=0.1.

(4) When 𝑛=2𝑚, 𝛼𝛽>0, 𝛽𝑐<0, ℎ=ℎ_𝐴 or 𝛼𝛽<0, 𝛽𝑐<0, ℎ=ℎ_𝐴 and 𝑚 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 𝑚=2, 𝑛=−1

When 𝑛=−1, (14) can be reduced to 𝑦=±ℎ𝜙²−2𝛼𝜙𝑐𝑚^𝑚+2+2𝛽𝑐𝜙. (44)

(a) Under the conditions 𝛼𝛽>0, 𝛼𝑐>0, ℎ∈(−∞,+∞) and 𝑚=2, (44) becomes 𝑦=±𝛼𝑐𝜗₁−𝜙𝜙−0𝜙−𝛾₁𝜙−𝛿₁,𝜙∈0,𝜗₁, (45) where 𝜗₁, 𝛾₁, 𝛿₁ are three roots of equation 2𝛽/𝛼+(𝑐ℎ/𝛼)𝜙−𝜙³ and 𝜗₁>0>𝛾₁>𝛿₁. These three roots 𝜗₁, 𝛾₁, 𝛿₁ can be obtained by Cardano formula as long as the parameters 𝛼, 𝛽, 𝑐, ℎ are fixed (or given) concretely. For example, taking 𝛼=4.0, 𝛽=2.0, 𝑐=4.0, ℎ=9.0, the 𝜗₁≐3.054084215, 𝛾₁≐−0.1112641576, 𝛿₁≐−2.942820058. Taking (0,𝜗₁) as the initial values of the variables (𝜉,𝜙), substituting (45) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜗𝑥,𝑡=𝜙¹−𝛿₁cn²Ω₁(𝜉),𝑘₁−𝛿₁+𝜗₁sn²Ω₁𝜉,𝑘₁,𝜉≠𝑇₁, (46) where Ω₁=(1/2)−𝛼𝛿₁(𝜗₁−𝛿₁)/𝑐, 𝑘₁=𝜗₁(𝛾₁−𝛿₁)/−𝛿₁(𝜗₁−𝛾₁), 𝑇₁=(1/Ω₁)sn⁻¹(1,𝑘₁). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (46), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝜗𝑥,𝑡=ln¹−𝛿₁cn²Ω₁(𝑥−𝑐𝑡),𝑘₁−𝛿₁+𝜗₁sn²Ω₁(𝑥−𝑐𝑡),𝑘₁,𝑥−𝑐𝑡≠𝑇₁. (47)

(b) Under the conditions 𝛼𝛽>0, 𝛼𝑐>0, ℎ∈(ℎ_𝐴,+∞) and 𝑚=2, (44) becomes 𝑦=±𝛼𝑐𝜗₁−𝜙0−𝜙𝛾₁−𝜙𝜙−𝛿₁,𝜙∈𝛿₁,𝛾₁. (48) Similarly, taking (0,𝛿₁) as the initial values of the variables (𝜉,𝜙), substituting (48) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝛾𝑥,𝑡=𝜙¹𝛿₁𝛿₁+𝛾₁−𝛿₁sn²Ω₁𝜉,𝑘₁. (49) By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (49), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝛾𝑥,𝑡=ln¹𝛿₁𝛿₁+𝛾₁−𝛿₁sn²Ω₁(𝑥−𝑐𝑡),𝑘₁. (50)

(c) Under the conditions 𝛼𝛽<0, 𝛼𝑐>0, ℎ∈(−∞,+∞) and 𝑚=2, (44) becomes 𝑦=±𝛼𝑐𝜗₂−𝜙𝜂₂−𝜙0−𝜙𝜙−𝛿₂,𝜙∈𝛿₂,0, (51) where 𝜗₂, 𝜂₂, 0, 𝛿₂ are roots of equation 𝜙(2𝛽/𝛼+(𝑐ℎ/𝛼)𝜙−𝜙³)=0 and 𝜗₂>𝜂₂>0>𝛿₂. As in the above cases, these three roots can be obtained by Cardano formula as long as the parameters 𝛼, 𝛽, 𝑐, ℎ are given concretely, so the similar cases will be not commentated anymore in the following discussions. Similarly, taking (0,𝛿₂) as the initial values of the variables (𝜉,𝜙), substituting (51) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜂𝑥,𝑡=𝜙²𝛿₂sn²Ω₂(𝑥−𝑐𝑡),𝑘₂𝜂₂−𝛿₂+𝛿₂sn²Ω₂𝜉,𝑘₂,𝜉≠𝑇₂, (52) where Ω₂=(1/2)𝛼𝜗₂(𝜂₂−𝛿₂)/𝑐, 𝑘₂=−𝛿₂(𝜗₂−𝜂₂)/𝜗₂(𝜂₂−𝛿₂), 𝑇₂=(1/Ω₂)sn⁻¹(0,𝑘₂). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (52), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝜂𝑥,𝑡=ln²𝛿₂sn²Ω₂(𝑥−𝑐𝑡),𝑘₂𝜂₂−𝛿₂+𝛿₂sn²Ω₂𝜉,𝑘₂,𝑥−𝑐𝑡≠𝑇₂. (53)

(d) Under the conditions 𝛼𝛽<0, 𝛼𝑐>0, ℎ∈(ℎ_𝐴,+∞) and 𝑚=2, (44) becomes 𝑦=±𝛼𝑐𝜗₂−𝜙𝜙−𝜂₂𝜙−0𝜙−𝛿₂. (54) Taking the (0,𝜗₂) as the initial values of the variables (𝜉,𝜙), substituting (54) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜗𝑥,𝑡=𝜙²𝜂₂−𝛿₂dn²Ω₂(𝑥−𝑐𝑡),𝑘₂𝜂₂−𝛿₂+𝜗₂−𝜂₂sn²Ω₂𝜉,𝑘₂. (55) By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (55), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝜗𝑥,𝑡=ln²𝜂₂−𝛿₂dn²Ω₂(𝑥−𝑐𝑡),𝑘₂𝜂₂−𝛿₂+𝜗₂−𝜂₂sn²Ω₂(𝑥−𝑐𝑡),𝑘₂. (56)

(e) Under the conditions 𝛼𝛽<0, 𝛼𝑐<0, ℎ∈(−∞,ℎ_𝐴) and 𝑚=2, (44) becomes 𝑦=±−𝛼𝑐𝜗₃−𝜙𝜂₃−𝜙𝜙−0𝜙−𝛿₃,𝜙∈0,𝜂₃, (57) where 𝜗₃, 𝜂₃, 0, 𝛿₃ are roots of equation 𝜙(−2𝛽/𝛼−(𝑐ℎ/𝛼)𝜙+𝜙³)=0 and 𝜗₃>𝜂₃>0>𝛿₃. Taking the (0,𝜂₃) as the initial values of the variables (𝜉,𝜙), substituting (57) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜗𝑥,𝑡=𝜙³𝜂₃cn²Ω₃𝜉,𝑘₃𝛿₃−𝜂₃sn²Ω₃𝜉,𝑘₃,𝜉≠𝑇₃, (58) where Ω₃=(1/2)−𝛼𝜗₃(𝜂₃−𝛿₃)/𝑐, 𝑘₃=𝜂₃(𝜗₃−𝛿₃)/𝜗₃(𝜂₃−𝛿₃), 𝑇₃=(1/Ω₃)sn⁻¹(1,𝑘₃). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (58), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝜗𝑥,𝑡=ln³𝜂₃cn²Ω₃(𝑥−𝑐𝑡),𝑘₃𝛿₃−𝜂₃sn²Ω₃(𝑥−𝑐𝑡),𝑘₃,𝑥−𝑐𝑡≠𝑇₃. (59)

(f) Under the conditions 𝛼𝛽<0, 𝛼𝑐<0, ℎ∈(−∞,ℎ_𝐴) and 𝑚=2, (44) becomes 𝑦=±−𝛼𝑐𝜗₄−𝜙0−𝜙𝜙−𝛾₄𝜙−𝛿₄,𝜙∈𝛾₄,0, (60) where 𝜗₄, 0, 𝛾₄, 𝛿₄ are roots of equation 𝜙(−2𝛽/𝛼−(𝑐ℎ/𝛼)𝜙+𝜙³)=0 and 𝜗₄>0>𝛾₄>𝛿₄. Taking the (0,𝛾₄) as the initial values of the variables (𝜉,𝜙), substituting (60) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝛿𝑥,𝑡=𝜙⁴𝛾₄cn²Ω₄𝜉,𝑘₄𝛿₄−𝛾₄cn²Ω₄𝜉,𝑘₄,𝜉≠𝑇₄, (61) where Ω₄=(1/2)𝛼𝛿₄(𝜗₄−𝛾₄)/𝑐, 𝑘₄=𝛾₄(𝜗₄−𝛿₄)/𝜗₄(𝜗₄−𝛾₄), 𝑇₄=(1/Ω₄)sn⁻¹(1,𝑘₄). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (61), we obtain a periodic below-up wave solutions of (1) as follows: 𝑢()𝛿𝑥,𝑡=ln⁴𝛾₄cn²Ω₄(𝑥−𝑐𝑡),𝑘₄𝛿₄−𝛾₄cn²Ω₄(𝑥−𝑐𝑡),𝑘₄,𝑥−𝑐𝑡≠𝑇₄. (62)

(g) Under the conditions 𝑚=2, 𝛼𝑐<0, ℎ=ℎ_𝐴=−𝛼²/2𝑐𝛽, (44) becomes 𝑦=±−𝛼𝑐𝜙₁−𝜙𝜙𝜙+2𝜙₁,𝜙≠0, (63) where 𝜙₁=(−𝛽/𝛼)^1/3. Taking (0,(1/2)𝜙₁) as the initial values of the variables (𝜉,𝜙), substituting (63) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜙₁31−coshΩ₅𝜉+2,𝜉≠𝑇₅, (64) where Ω₅=𝜙₁−3𝛼/𝑐, 𝑇₅=(1/Ω₅)cosh⁻¹(1). Clearly, we have 𝜙(𝑇₅)=0, 𝜙(±∞)=𝜙₁. By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (64), we obtain a solitary wave solution of blow-up form of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙₁31−coshΩ₅(𝑥−𝑐𝑡)+2,𝑥−𝑐𝑡≠𝑇₅. (65) Equation (64) is smooth solitary wave solution of (9), but (65) is a solitary wave solution of blow-up form for (1) because 𝑢→−∞ as 𝑣→0. In order to visually show dynamical behaviors of solutions 𝑣 and 𝑢 of (64) and (65), we plot graphs of their profiles which are shown in Figures 3(a) and 3(b), respectively.

(a) Smooth solitary wave

(b) Solitary wave of blow-up form

Figure 3 

The phase portraits of solutions (64) and (65) for 𝑚=2, 𝛼=0.3, 𝛽=−0.3, 𝑐=−2, 𝑡=0.1.

3.4. Different Kinds of Periodic Wave, Broken Solitary Wave Solutions in the Special Case 𝑚=1, 𝑛=−1

(1) Under the conditions ℎ∈(ℎ_𝐴,+∞), 𝛼𝛽>0, 𝛽𝑐>0 (or 𝛼𝛽<0, 𝛽𝑐>0), 𝛼𝑐>0 and 𝑚=1, (44) becomes 𝑦=±2𝛼𝑐𝜙_𝑀−𝜙𝜙−𝜙_𝑚𝜙−0,𝜙∈𝜙_𝑚,𝜙_𝑀, (66) where 𝜙_𝑀,𝑚=(ℎ𝑐±ℎ²𝑐²+16𝛼𝛽)/4𝛼 and 0 are three roots of equation ℎ𝜙²−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 𝜙_𝑀>𝜙_𝑚>0. Taking (0,𝜙_𝑚) as the initial values of the variables (𝜉,𝜙), substituting (66) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜙_𝑚−𝜙_𝑀−𝜙_𝑚sn²Ω₆𝜉,𝑘₆, (67) where Ω₆=𝛼𝜙_𝑀/2𝑐, 𝑘₆=(𝜙_𝑀−𝜙_𝑚)/𝜙_𝑀. By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (67), we obtain a double periodic wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙_𝑚−𝜙_𝑀−𝜙_𝑚sn²Ω₆(𝑥−𝑐𝑡),𝑘₆. (68) Equation (67) is smooth periodic wave solution of (9), but (68) is a double periodic wave solution of blow-up form of (1) because 𝑢→−∞ as 𝑣→0. In order to visually show dynamical behaviors of solutions 𝑣 and 𝑢 of (67) and (68), we plot graphs of their profiles which are shown in Figures 4(a) and 4(b), respectively.

(a) Smooth periodic wave

(b) Double periodic wave of blow-up type

Figure 4 

The phase portraits of solutions (67) and (68) for ℎ=8, 𝛼=2.5, 𝛽=−2.5, 𝑐=2, 𝑡=0.1.

(2) Under the condition ℎ∈(−∞,ℎ_𝐴),𝛼𝛽<0, 𝛽𝑐>0 (or 𝛼𝛽<0, 𝛽𝑐<0),𝛼𝑐>0 and 𝑚=1, (44) becomes 𝑦=±2𝛼𝑐0−𝜙𝜙−𝜙_𝑀𝜙−𝜙_𝑚,𝜙∈𝜙_𝑀,0, (69) where 𝜙_𝑀,𝜙_𝑚,0 are roots of equation ℎ𝜙²−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 𝜙_𝑚<𝜙_𝑀<0. Taking (0,𝜙_𝑀) as the initial values of the variables (𝜉,𝜙), substituting (69) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜙_𝑀sn²Ω₇𝜉,𝑘₇,𝜉≠𝑇₇, (70) where Ω₇=−𝛼𝜙_𝑀/2𝑐, 𝑘₇=𝜙_𝑀/𝜙_𝑚, 𝑇₇=(1/Ω₇)sn⁻¹(0,𝑘₇). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (70), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙_𝑀sn²Ω₇(𝑥−𝑐𝑡),𝑘₇,𝑥−𝑐𝑡≠𝑇₇. (71)

(3) Under the conditions ℎ∈(ℎ_𝐴,+∞), 𝛼𝛽<0, 𝛽𝑐>0 (or 𝛼𝛽>0, 𝛽𝑐>0) and 𝑚=1, (44) becomes 𝑦=±−2𝛼𝑐0−𝜙𝜙_𝑀−𝜙𝜙−𝜙_𝑚,𝜙∈𝜙_𝑚,𝜙_𝑀, (72) where 𝜙_𝑀, 𝜙_𝑚, 0 are roots of equation ℎ𝜙²−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 0>𝜙_𝑀>𝜙_𝑚. Taking (0,𝜙_𝑀) as the initial values of the variables (𝜉,𝜙), substituting (72) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜙𝑥,𝑡=𝜙^𝑀dn²Ω₈𝜉,𝑘₈, (73) where Ω₈=𝛼𝜙_𝑚/2𝑐, 𝑘₈=(𝜙_𝑀−𝜙_𝑚)/−𝜙_𝑚. By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (73), we obtain a periodic blow-up wave solutions of (1) as follows: 𝑢()𝜙𝑥,𝑡=ln^𝑀dn²Ω₈(𝑥−𝑐𝑡),𝑘₈. (74)

(4) Under the conditions ℎ∈(−∞,ℎ_𝐴),𝛼𝛽<0, 𝛽𝑐>0 (or 𝛼𝛽<0, 𝛽𝑐<0) and 𝑚=1, (44) becomes 𝑦=±−2𝛼𝑐𝜙_𝑀−𝜙𝜙_𝑚−𝜙𝜙−0,𝜙∈0,𝜙_𝑚, (75) where 𝜙_𝑀, 𝜙_𝑚, 0 are roots of equation ℎ𝜙²−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 𝜙_𝑀>𝜙_𝑚>0. Taking (0,𝜙_𝑚) as the initial values of the variables (𝜉,𝜙), substituting (75) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()(𝜉)=𝜙𝑥,𝑡=𝜙^𝑚cn²Ω₉𝜉,𝑘₉dn²Ω₉𝜉,𝑘₉,𝜉≠𝑇₉, (76) where Ω₉=−𝛼𝜙_𝑀/2𝑐, 𝑘₉=𝜙_𝑚/𝜙_𝑀, 𝑇₉=(1/Ω₉)sn⁻¹(0,𝑘₉). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (76), we obtain periodic blow-up wave solutions of (1) as follows: 𝑢()𝜙𝑥,𝑡=ln^𝑚cn²Ω₉(𝑥−𝑐𝑡),𝑘₉dn²Ω₉(𝑥−𝑐𝑡),𝑘₉,𝑥−𝑐𝑡≠𝑇₉. (77)

(5) Under the conditions ℎ=ℎ_𝐴=−4𝛼𝜙₁/𝑐, 𝛼𝛽<0, 𝛽𝑐>0 and 𝑚=1, (44) becomes 𝑦=±2𝛼𝑐𝜙₁+𝜙−𝜙,𝜙∈−𝜙₁,0, (78) where 𝜙₁=(−𝛽/𝛼)^1/2. Taking (0,−𝜙₁) as the initial values of the variables (𝜉,𝜙), substituting (78) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=−𝜙₁tanh²0.5Ω₁₀𝜉,𝜉≠0, (79) where Ω₁₀=2𝛼𝜙₁/𝑐. By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (79), we obtain a solitary wave solution of blow-up form of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙₁tanh²0.5Ω₁₀(𝑥−𝑐𝑡),𝑥−𝑐𝑡≠0. (80)

(6) Under the conditions ℎ=ℎ_𝐴=4𝛼𝜙₁/𝑐, 𝛼𝛽<0, 𝛽𝑐>0 and 𝑚=1, (44) becomes 𝑦=±−2𝛼𝑐𝜙−𝜙₁𝜙,𝜙∈0,𝜙₁, (81) where 𝜙₁=(−𝛽/𝛼)^1/2. Taking (0,𝜙₁) as the initial values of the variables (𝜉,𝜙), substituting (81) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜙₁tanh²0.5Ω₁₁𝜉,𝜉≠0, (82) where Ω₁₁=−2𝛼𝜙₁/𝑐. By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (82), we obtain a broken solitary wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙₁tanh²0.5Ω₁₁(𝑥−𝑐𝑡),𝑥−𝑐𝑡≠0. (83)

(7) Under the conditions ℎ∈(−∞,0], 𝛼𝛽>0, 𝛽𝑐<0 and 𝑚=1, (44) becomes 𝑦=±−2𝛼𝑐𝜙_𝑀−𝜙0−𝜙𝜙−𝜙_𝑚,𝜙∈𝜙_𝑚,0, (84) where 𝜙_𝑀, 𝜙_𝑚, 0 are roots of equation 𝜙²ℎ−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 𝜙_𝑀>0>𝜙_𝑚. Taking (0,𝜙_𝑀) as the initial values of the variables (𝜉,𝜙), substituting (84) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=−𝜙_𝑀𝑘²₁₂sn²Ω₁₂𝜉,𝑘₁₂dn²Ω₁₂𝜉,𝑘₁₂,𝜉≠𝑇₁₂, (85) where Ω₁₂=−𝛼(𝜙_𝑀−𝜙_𝑚)/2𝑐, 𝑘₁₂=−𝜙_𝑚/(𝜙_𝑀−𝜙_𝑚), 𝑇₁₂=(1/Ω₁₂)sn⁻¹(0,𝑘₁₂). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (85), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln−𝜙_𝑀𝑘²₁₂sn²Ω₁₂(𝑥−𝑐𝑡),𝑘₁₂dn²Ω₁₂(𝑥−𝑐𝑡),𝑘₁₂,𝑥−𝑐𝑡≠𝑇₁₂. (86)

(8) Under the conditions ℎ∈(−∞,0], 𝛼𝛽>0, 𝛽𝑐>0 and 𝑚=1, (44) becomes 𝑦=±2𝛼𝑐𝜙_𝑀−𝜙𝜙−0𝜙−𝜙_𝑚,𝜙∈0,𝜙_𝑀, (87) where 𝜙_𝑀, 𝜙_𝑚, 0 are roots of equation ℎ𝜙²−(2𝛼/𝑐)𝜙³+(2𝛽/𝑐)𝜙=0 and 𝜙_𝑀>0>𝜙_𝑚. Taking (0,𝜙_𝑀) as the initial values of the variables (𝜉,𝜙), substituting (87) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙_𝑀cn²Ω₁₃𝜉,𝑘₁₃,𝜉≠𝑇₁₃, (88) where Ω₁₃=𝛼(𝜙_𝑀−𝜙_𝑚)/2𝑐, 𝑘₁₃=𝜙_𝑀/(𝜙_𝑀−𝜙_𝑚), 𝑇₁₃=(1/Ω₁₃)sn⁻¹(0,𝑘₁₃). By using the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (88), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙_𝑀cn²Ω₁₃(𝑥−𝑐𝑡),𝑘₁₃,𝑥−𝑐𝑡≠𝑇₁₃. (89)

3.5. Periodic Blow-Up Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Cases 𝑚=2 or 𝑚=1 and 𝑛=−2

When 𝑚=2 is even and 𝑛=−2, (14) can be reduced to 𝑦=±ℎ𝜙²−2𝛼𝜙𝑐𝑚^𝑚+2+𝛽𝑐. (90) Because the singular straight line 𝜙=0, this implies that (1) has broken kink and antikink wave solutions: see the following discussions.

(i) Under the conditions 𝑚=2, 𝑛=−2, ℎ=−4𝛼𝛽/𝑐², 𝛼𝛽<0, 𝛼𝑐<0, (90) becomes 𝑦=±−𝛼𝑐𝜙²−𝜙²₁,𝜙≠0, (91) where 𝜙₁=−𝛽/𝛼. Taking (0,𝜙₁) as the initial values of the variables (𝜉,𝜙), substituting (91) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=±𝜙₁tanhΩ₁₄𝜉, (92) where Ω₁₄=𝜙₁−𝛼/𝑐. By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (92), we obtain a broken kink wave solutions of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙₁tanhΩ₁₄(𝑥−𝑐𝑡), (93) where (𝑥−𝑐𝑡)∈(−∞,0) and (𝑥−𝑐𝑡)∈(0,+∞).

(ii) Under the conditions ℎ∈(ℎ_𝐵1,+∞), 𝛼𝛽>0, 𝛼𝑐>0 and 𝑚=2, 𝑛=−2, (90) becomes 𝑦=±𝛼𝑐𝑎²−𝜙²𝜙²−𝑏², (94) where 𝑎²=(1/2𝛼)(𝑐ℎ+𝑐²ℎ²+4𝛼𝛽), 𝑏²=(1/2𝛼)(𝑐ℎ−𝑐²ℎ²+4𝛼𝛽). Taking (0,𝑎) as the initial values of the variables (𝜉,𝜙), substituting (94) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=±𝑎²−𝑎²−𝑏²sn²Ω₁₅𝜉,𝑘₁₅, (95) where 𝑎²=(1/2𝛼)(𝑐ℎ+𝑐²ℎ²+4𝛼𝛽), 𝑏²=(1/2𝛼)(𝑐ℎ−𝑐²ℎ²+4𝛼𝛽), Ω₁₅=𝑎𝛼/𝑐, 𝑘₁₅=(𝑎²−𝑏²)/𝑎². By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (95), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=0.5ln𝑎²−𝑎²−𝑏²sn²Ω₁₅𝜉,𝑘₁₅. (96)

(iii) Under the conditions 𝑚=1, 𝑛=−2, ℎ=ℎ_𝐵1=3𝛼𝜙₂/𝑐, 𝛼𝛽>0, 𝛼𝑐>0, (90) becomes 𝑦=±𝛼𝑐−𝜙₂+𝜙−𝜙₂−2𝜙,𝜙≠0, (97) where 𝜙₂=(−𝛽/𝛼)^1/3. Taking (0,−0.5𝜙₂) as the initial values of the variables (𝜉,𝜙), substituting (97) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=−𝜙₂3coshΩ₁₆𝜉+1−1,𝜉≠𝑇₁₆, (98) where Ω₁₆=−3𝛼𝜙₂/𝑐, 𝑇₁₆=(1/Ω₁₆)cosh⁻¹(2). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (98), when (𝑥−𝑐𝑡)∈(−∞,𝑇₁₆)⋃(𝑇₁₆,+∞), we obtain a broken solitary wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln3𝜙₂coshΩ₁₆(𝑥−𝑐𝑡)+1−𝜙₂. (99)

(iv) Under the conditions ℎ=ℎ_𝐵1=3𝛼𝜙₂/𝑐, 𝛼𝛽<0, 𝛼𝑐<0 and 𝑚=1, 𝑛=−2, (90) becomes 𝑦=±−𝛼𝑐𝜙₂−𝜙𝜙₂+2𝜙,𝜙≠0, (100) where 𝜙₂=(−𝛽/𝛼)^1/3. Taking (0,−0.5𝜙₂) as the initial values of the variables (𝜉,𝜙), substituting (100) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜙₂31−coshΩ₁₇𝜉+1,𝜉≠𝑇₁₇, (101) where Ω₁₇=3𝛼𝜙₁/𝑐, 𝑇₁₇=(1/Ω₁₇)cosh⁻¹(2). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (101), when (𝑥−𝑐𝑡)∈(−∞,𝑇₁₇)⋃(𝑇₁₇,+∞), we obtain a broken solitary wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜙₂−3𝜙₂coshΩ₁₇(𝑥−𝑐𝑡)+1. (102)

(v) Under the conditions ℎ∈(ℎ_𝐵1,+∞), 𝛼𝛽<0, 𝛽𝑐<0 and 𝑚=1, 𝑛=−2, (90) becomes 𝑦=±2𝛼𝑐𝜗₅−𝜙𝜙−𝜂₅𝜙−𝛾₅,𝜙∈𝜂₅,𝜗₅, (103) where 𝜗₅, 𝜂₅, 𝛾₅ are roots of equation 𝛽/2𝛼+(𝑐ℎ/2𝛼)𝜙²−𝜙³ and 𝛾₅<0<𝜂₅<𝜗₅. Taking (0,𝜗₅) as the initial values of the variables (𝜉,𝜙), substituting (103) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜗₅−𝜗₅−𝜂₅sn²Ω₁₈𝜉,𝑘₁₈, (104) where Ω₁₈=𝛼(𝜗₅−𝛾₅)/2𝑐, 𝑘₁₈=(𝜗₅−𝜂₅)/(𝜗₅−𝛾₅). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (104), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝜗₅−𝜗₅−𝜂₅sn²Ω₁₈(𝑥−𝑐𝑡),𝑘₁₈. (105)

(vi) Under the conditions ℎ∈(ℎ_𝐵1,+∞), 𝛼𝛽>0, 𝛽𝑐<0 and 𝑚=1, 𝑛=−2, (90) becomes 𝑦=±2𝛼𝑐𝜗₅−𝜙𝜂₅−𝜙𝜙−𝛾₅,𝜙∈𝛾₅,𝜂₅, (106) where 𝛾₅<𝜂₅<0<𝜗₅. Taking (0,𝜗₅) as the initial values of the variables (𝜉,𝜙), substituting (106) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝛾₅+𝜂₅−𝛾₅sn²Ω₁₉𝜉,𝑘₁₉, (107) where 𝛾₅<𝜂₅<0<𝜗₅, Ω₁₉=−𝛼(𝜗₅−𝛾₅)/2𝑐, 𝑘₁₉=(𝜗₅−𝜂₅)/(𝜗₅−𝛾₅). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (113), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝛾₅+𝜂₅−𝛾₅sn²Ω₁₉(𝑥−𝑐𝑡),𝑘₁₉. (108)

3.6. Periodic Wave, Broken Solitary Wave, Broken Kink, and Antikink Wave Solutions in the Special Cases 𝑛=−𝑚 or 𝑛=−2𝑚 and 𝑚∈𝐙⁺

(a) When 𝑛=−𝑚, 𝑚∈𝐙⁺ and 𝛼𝛽<0, 𝛽𝑐<0, 𝛼𝑐>0, ℎ>4−𝛼𝛽/𝑐𝑚, (14) can be reduced to 𝑦=±2𝛽/𝑐𝑚𝜙^𝑚+ℎ𝜙^𝑚2−(2𝛼/𝑐𝑚)𝜙^𝑚3𝜙^𝑚−1=±2𝛼/𝑐𝑚𝜙^𝑚_𝑔−𝜙^𝑚𝜙^𝑚−𝜙^𝑚_𝑙𝜙^𝑚−0𝜙^𝑚−1, (109) where 𝜙_𝑔,𝑙=[(1/4𝛼)(𝑐𝑚ℎ±𝑐²𝑚²ℎ²+16𝛼𝛽)]^1/𝑚. Taking (0,𝜙_𝑔) as the initial values of the variables (𝜉,𝜙), substituting (109) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yields 𝑣(𝑥,𝑡)=𝜙(𝜉)=±𝜙_𝑔−𝜙_𝑔−𝜙_𝑙sn²Ω₁₉𝜉,𝑘₁₉^1/𝑚, (110) where Ω₁₉=𝑚𝛼𝜙_𝑔/2𝑐𝑚, 𝑘₁₉=(𝜙_𝑔−𝜙_𝑙)/𝜙_𝑔. By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (110), we obtain a periodic wave solution of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln𝜙_𝑔−𝜙_𝑔−𝜙_𝑙sn²Ω₁₉(𝑥−𝑐𝑡),𝑘₁₉. (111)

(b) When 𝑛=−2𝑚,𝑚∈𝐙⁺and 𝛼𝛽<0, 𝛽𝑐<0, 𝛼𝑐>0, ℎ>ℎ_𝐵1, (14) can be reduced to 𝑦=±𝛽/𝑐𝑚+ℎ𝜙^𝑚2−(2𝛼/𝑐𝑚)𝜙^𝑚3𝜙^𝑚−1=±2𝛼/𝑐𝑚𝜗^𝑚−𝜙^𝑚𝜙^𝑚−𝜂^𝑚𝜙^𝑚−𝛾^𝑚𝜙^𝑚−1, (112) where 𝜗^𝑚, 𝜂^𝑚, 𝛾^𝑚 are roots of equation 𝛽/𝑐𝑚+ℎ(𝜙^𝑚)²−(2𝛼/𝑐𝑚)(𝜙^𝑚)³=0 and 𝜗^𝑚>𝜂^𝑚>0>𝛾^𝑚. Taking (0,𝜗) as the initial values of the variables (𝜉,𝜙), substituting (112) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝜗^𝑚−𝜗^𝑚−𝜂^𝑚sn²Ω₂₀𝜉,𝑘₂₀^1/𝑚, (113) where Ω₂₀=𝑚𝛼(𝜗^𝑚−𝛾^𝑚)/2𝑐𝑚, 𝑘₂₀=(𝜗^𝑚−𝜂^𝑚)/(𝜗^𝑚−𝛾^𝑚). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (113), we obtain periodic wave solutions of (1) as follows: 𝑢()=1𝑥,𝑡𝑚ln𝜗^𝑚−𝜗^𝑚−𝜂^𝑚sn²Ω₂₀(𝑥−𝑐𝑡),𝑘₂₀. (114)

(c) When 𝑛=−2𝑚, 𝑚∈𝐙⁺ and 𝛼𝛽>0, 𝛽𝑐<0, 𝛼𝑐>0, ℎ>ℎ_𝐵2, (14) can be reduced to 𝑦=±𝛽/𝑐𝑚+ℎ𝜙^𝑚2−(2𝛼/𝑐𝑚)𝜙^𝑚3𝜙^𝑚−1=±−2𝛼/𝑐𝑚𝜗^𝑚−𝜙^𝑚𝜂^𝑚−𝜙^𝑚𝜙^𝑚−𝛾^𝑚𝜙^𝑚−1, (115) where 𝜗^𝑚>0>𝜂^𝑚>𝛾^𝑚. Taking (0,𝜂) as the initial values of the variables (𝜉,𝜙), substituting (115) into the 𝑑𝜙/𝑑𝜉=𝑦 of (11), and then integrating it yield 𝑣()=𝜂𝑥,𝑡^𝑚−𝜗^𝑚𝑘²₂₁sn²Ω₂₁𝜉,𝑘₂₁dn²Ω₂₁𝜉,𝑘₂₁^1/𝑚, (116) where Ω₂₁=𝑚−𝛼(𝜗^𝑚−𝛾^𝑚)/2𝑐𝑚, 𝑘₂₁=(𝜂^𝑚−𝜂^𝑚)/(𝜗^𝑚−𝛾^𝑚). By using the transformations 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (116), we obtain a periodic wave solution of (1) as follows: 𝑢()=1𝑥,𝑡𝑚𝜂ln^𝑚−𝜗^𝑚𝑘²₂₁sn²Ω₂₁𝜉,𝑘₂₁dn²Ω₂₁𝜉,𝑘₂₁. (117)

4. Exact Traveling Wave Solutions of (1) and Their Properties under the Conditions 𝑚∈𝐙⁻, 𝑛∈𝐙⁻

In this sections, we consider the case 𝑚∈𝐙⁻, 𝑛∈𝐙⁻ of (1); that is, we will investigate different kinds of exact traveling wave solutions of (1) under the conditions 𝑚∈𝐙⁻, 𝑛∈𝐙⁻. Letting 𝑚=−𝑞, 𝑛=−𝑝, 𝑝∈𝐙⁺, 𝑞∈𝐙⁺, (11) becomes 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=𝑑𝜉𝑐𝑦²−𝛼𝜙^2−𝑞−𝛽𝜙^2−𝑝.𝑐𝜙 (118)

First, we consider a special case 𝑝=𝑞=1. When 𝑝=𝑞=1, under the transformation 𝑑𝜉=𝑐𝜙𝑑𝜏, system (118) can be reduced to the following regular system: 𝑑𝜙𝑑𝜏=𝑐𝜙𝑦,𝑑𝑦𝑑𝜏=𝑐𝑦²−𝛼+𝛽𝜙, (119) which is an integrable system and it has the following first integral: 𝑦²=𝜙²2ℎ+𝛼+𝛽𝑐𝜙, (120) where ℎ is integral constant. When 𝜙≠0, we define 𝐻=𝑦𝜙,𝑦²𝜙²−2𝛼+𝛽=𝑐𝜙ℎ. (121) Obviously, when 𝛼≠−𝛽, system (119) has only one equilibrium point 𝑂(0,0) in the 𝜙-axis.

Second, we consider the special case 𝑝=2, 𝑞=1, and another case 𝑝=1, 𝑞=2 is very similar to the front case, so we only discuss the front case here. When 𝑝=2, 𝑞=1, under the transformation 𝑑𝜉=𝑐𝜙𝑑𝜏, system (118) can be reduced to the following regular system: 𝑑𝜙𝑑𝜏=𝑐𝜙𝑦,𝑑𝑦𝑑𝜏=𝑐𝑦²−𝛼𝜙−𝛽, (122) which is an integrable system and it has the following first integral: 𝑦²=ℎ𝜙²+2𝛼𝑐𝛽𝜙+𝑐, (123) also we define 𝐻=𝑦𝜙,𝑦²𝜙²−2𝛼−𝛽𝑐𝜙𝑐𝜙²=ℎ. (124) Indeed, system (122) has only one equilibrium point 𝐴(𝜙₀,0) in the 𝜙-axis, where 𝜙₀=−𝛽/𝛼. When 𝑐𝛽>0, system (122) has two equilibrium points 𝑆_±(0,𝑌_±) in the singular line 𝜙=0, where 𝑌_±=±𝛽/𝑐.

Finally, we consider the general case 𝑝>2. When 𝑝>2, under the transformation 𝑑𝜉=𝑐𝜙^𝑝−1𝑑𝜏, system (118) can be reduced to the following regular system: 𝑑𝜙𝑑𝜏=𝑐𝜙^𝑝−1𝑦,𝑑𝑦𝑑𝜏=𝑐𝜙^𝑝−2𝑦²−𝛼𝜙^𝑝−𝑞−𝛽 (125) which is an integrable system and it has the following first integral: 𝑦²=𝜙²ℎ+2𝛼𝜙𝑐𝑞^2−𝑞+2𝛽𝜙𝑐𝑝^2−𝑝. (126) When 𝜙≠0, we define 𝐻=𝑦𝜙,𝑦²𝜙²−2𝛼𝜙𝑐𝑞^−𝑞−2𝛽𝜙𝑐𝑝^−𝑝=ℎ. (127) Obviously, when 𝑝=𝑞, system (125) has not any equilibrium point in the 𝜙-axis. When 𝑝−𝑞 is even number and 𝛼𝛽<0, system (125) has two equilibrium points 𝐵_1,2(±𝜙₊,0) in the 𝜙-axis, where 𝜙₊=(−𝛽/𝛼)^{1/(𝑝−𝑞)}. When 𝑝−𝑞 is odd number, system (125) has only one equilibrium point at 𝐵₁(𝜙₊,0) in the 𝜙-axis.

Respectively substituting the above equilibrium points into (121), (124), and (127), we have ℎ𝐴=𝐻𝜙₀𝛼,0=−²,ℎ𝑐𝛽₁=𝐻𝜙₊,0=−2𝛼𝜙𝑐𝑞^+−𝑞−2𝛽𝜙𝑐𝑝^+−𝑝,ℎ₂=𝐻−𝜙₊,0=(−1)^𝑞+12𝛼𝜙𝑐𝑞^+−𝑞+(−1)^𝑝+12𝛽𝜙𝑐𝑝^+−𝑝. (128)(129)(130)

4.1. Nonsmooth Peakon Solutions and Periodic Blow-Up Wave Solutions under Two Special Cases 𝑝=𝑞=1 and 𝑝=2, 𝑞=1

(1) When 𝑝=𝑞=1 and 𝛼𝑐>0, ℎ<0, (120) can be reduced to 𝑦=±ℎ𝜙²−2𝛼+𝛽𝑐𝜙. (131) Substituting (131) into the first equation of (118), taking (0,𝑐ℎ/(2(𝛼+𝛽))) as initial value conditions, and integrating it, we obtain 𝑣(𝑥,𝑡)=𝜙(𝜉)=𝛼+𝛽𝑐ℎcos−ℎ𝜉−1. (132) From the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (132), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢(𝑥,𝑡)=ln𝛼+𝛽𝑐ℎcos−ℎ(𝑥−𝑐𝑡)−1, (133) where (𝑥−𝑐𝑡)≠2𝐾𝜋/−ℎ,𝐾∈𝐙.

(2) When 𝑝=2, 𝑞=1 and 𝛼𝛽>0, 𝛽𝑐>0 (or 𝛼𝛽<0, 𝛽𝑐>0), ℎ𝐴ℎ==𝛼²/𝛽𝑐, (123) can be reduced to 𝛼𝑦=±𝛽𝛽𝑐𝜙+𝛼. (134) Similarly, substituting (134) into the first equation of (118) and then integrating it, and setting the integral constant as zero, we have 𝑣()(𝜉)𝛽𝑥,𝑡=𝜙=−𝛼1−exp−𝛼𝜉𝛽𝑐. (135) From the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (135), we obtain a broken solitary cusp wave solution of (1) as follows: 𝑢()−𝛽𝑥,𝑡=ln𝛼−|𝛼(1−exp𝑥−𝑐𝑡)|,()𝛽𝑐𝑥−𝑐𝑡≠0. (136)

(3) When 𝑝=2,𝑞=1 and 𝛼𝛽>0, 𝛽𝑐>0 (or 𝛼𝛽<0,𝛽𝑐>0),ℎ<0, (123) can be reduced to 𝑦=±−ℎ−𝛽𝑐ℎ−2𝛼𝑐ℎ𝜙−𝜙²=±−ℎ𝜙_𝑀−𝜙𝜙−𝜙_𝑚, (137) where 𝜙_𝑀=(−𝛼+𝛼²−𝑐ℎ𝛽)/𝑐ℎ, 𝜙_𝑚=(−𝛼−𝛼²−𝑐ℎ𝛽)/𝑐ℎ are roots of equation −𝛽/𝑐ℎ−(2𝛼/𝑐ℎ)𝜙−𝜙²=0. Substituting (137) into the first equation of (118), taking (0,𝜙_𝑀) as initial value conditions, and then integrating it, we have 𝑣()(𝜉)=1𝑥,𝑡=𝜙2𝜙_𝑀+𝜙_𝑚−𝜙_𝑀−𝜙_𝑚cos√−ℎ𝜉,𝜉∈−𝑇₁,𝑇₁, (138) where 𝑇₁=(1/−ℎ)arccos((𝜙_𝑀+𝜙_𝑚)/(𝜙_𝑀−𝜙_𝑚)). From the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (138), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()1𝑥,𝑡=ln2𝜙_𝑀+𝜙_𝑚−𝜙_𝑀−𝜙_𝑚cos−ℎ(𝑥−𝑐𝑡), (139) where 𝑥−𝑐𝑡≠𝑇₁+𝐾𝜋,𝐾∈𝐙.

(4) When 𝑝=2,𝑞=1 and 𝛼𝛽>0, 𝛽𝑐<0 (or 𝛼𝛽<0,𝛽𝑐<0),ℎ>−𝛼²/𝑐𝛽, (123) can be reduced to 𝑦=±ℎ𝜙²+2𝛼𝑐𝛼𝜙+𝑐. (140) Similarly, we have 𝑣()(𝜉)𝛼𝑥,𝑡=𝜙=−𝑐ℎ−𝛼²−𝛽𝑐ℎ𝑐ℎcos−ℎ𝜉. (141) From the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (141), we obtain a periodic blow-up wave solution of (1) as follows: 𝑢()𝛼𝑥,𝑡=ln𝑐ℎ+𝛼²−𝛽𝑐ℎ𝑐ℎcos−ℎ(𝑥−𝑐𝑡). (142)

4.2. Periodic Blow-Up Wave Solutions in the Special Case 𝑝=2𝑞>1

When 𝑝=2𝑞>2 and 𝛼𝛽<0, 𝛼𝑐>0 (or 𝛼𝛽>0, 𝛼𝑐<0), ℎ∈(ℎ₁,0), (118) becomes 𝑦=±𝜙²ℎ+2𝛼𝜙𝑐𝑞^−𝑞+𝛽𝜙𝑐𝑞^−2𝑞=±−ℎ−𝛽/𝑐𝑞ℎ−2𝛼/𝑐𝑞ℎ𝜙^𝑞−𝜙^𝑞2𝜙^𝑞−1. (143) Substituting (143) into the first equation of (118) and then integrating it, we have 𝑣()(𝜉)=−𝛼𝑥,𝑡=𝜙ℎ+𝑐𝑞𝛼²−𝑐𝑞ℎ𝛽𝑐²𝑞²ℎ²cos𝑞−ℎ𝜉^1/𝑞. (144) From the transformation 𝑢(𝑥,𝑡)=ln|𝑣(𝑥,𝑡)| and (144), we obtain periodic blow-up wave solutions of (1) as follows: 𝑢()=1𝑥,𝑡𝑞−𝛼lnℎ+𝑐𝑞𝛼²−𝑐𝑞ℎ𝛽𝑐²𝑞²ℎ²cos𝑞−ℎ(𝑥−𝑐𝑡). (145)

5. Conclusion

In this work, by using the integral bifurcation method, a generalized Tzitzéica-Dodd-Bullough-Mikhailov (TDBM) equation is studied. In some special cases, we obtained many exact traveling wave solutions of TDBM equation. These exact solutions include smooth solitary wave solutions, singular periodic wave solutions of blow-up form, singular solitary wave solutions of blow-up form, broken solitary wave solutions, broken kink wave solutions, and some unboundary wave solutions. Comparing with the results in many references (such as [24–27]), these types of solutions in this paper are very different than those types of solutions which are obtained by Exp-function method. Though the types of solutions derived from two methods are different, they are all useful in nonlinear science. The smooth solitary wave solutions and singular solitary wave solutions obtained in this paper especially can be explained phenomenon of soliton surface that the total curvature at each point is proportional to the fourth power of the distance from a fixed point to the tangent plane.

Acknowledgments

The authors would like to thank the reviewers very much for their useful comments and helpful suggestions. This work is supported by the Natural Science Foundation of Yunnan Province (no. 2011FZ193) and the National Natural Science Foundation of China (no. 11161020). The e-mail address “[email protected]" in the author’s former publications (published articles) has been replaced by new e-mail address “[email protected]" because Yahoo Company in China will be closed on August 19, 2013. Please use the new E-mail address “[email protected]" if the readers contact him.