Existence Analysis of Traveling Wave Solutions for a Generalization of KdV Equation

Abstract

By using the bifurcation theory of dynamic system, a generalization of KdV equation was studied. According to the analysis of the phase portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons were discussed. In some parametric conditions, exact traveling wave solutions of this generalization of the KdV equation, which are different from those exact solutions in existing references, were given.

1. Introduction

In 1995, Fokas [1] derived a generalization of KdV equation 𝑢_𝑡+𝑢_𝑥+𝜈𝑢_𝑥𝑥𝑡+𝛽𝑢_𝑥𝑥𝑥+𝛼𝑢𝑢_𝑥+13𝛼𝜈𝑢𝑢_𝑥𝑥𝑥+2𝑢_𝑥𝑢_𝑥𝑥=0, (1) which from physical considerations via a methodology introduced by Fuchssteiner and Fokas [2]. Equation (1) can also be deriven by the approaches described in [3]. By using the bifurcation theory of dynamic system (see [4–7] and references cited therein), it is easy to investigate exact traveling wave solutions of (1) under different kinds of parametric conditions. For example, in [8], by using the bifurcation theory of dynamic system, some subsection-function and implicit function solutions such as compactons, solitary waves, smooth periodic waves, and nonsmooth periodic waves with peaks as well as the existence conditions have been presented by Bi. By using the same method, Li and Zhang [9] studied a generalization form of the modified KdV equation, which is more complex than (1). In [9], the existence of solitary wave, kink and antikink wave solutions, and uncountably infinitely many smooth and nonsmooth periodic wave solutions is discussed. By using the improved method named integral bifurcation method [10, 11], Rui et al. [12] obtain all kinds of soliton-like or kink-like wave solutions, periodic wave solutions with loop or without loop, smooth compacton-like periodic wave solution, and nonsmooth periodic cusp wave solution for (1). The integral bifurcation method possessed some advantages of the bifurcation theory of the planar dynamic system [6, 13] and auxiliary equation method (see [14, 15] and references cited therein), it is easily combined with computer method [11] and useful for many nonlinear partial differential equations (PDEs) including some PDEs with high-power terms, such as 𝐾(𝑚,𝑛) equation [16]. However, we cannot discuss the existence of traveling wave solutions by using this method. Therefore, the bifurcation theory of the planar dynamic system bulks large with the existence analysis of traveling wave solutions of nonlinear PDEs.

In this paper, by using the bifurcation theory of dynamic system and a translation transformation, we will restudy (1). According to the analysis of the phase portraits, the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons will be discussed. The obtained results are different from those in [8, 12].

2. Bifurcations of Phase Portraits of the System and the Existence of Traveling Wave Solutions for (1)

2.1. Bifurcations of Phase Portraits for the 2-Dimensional Planar System of (1)

In this subsection, we discuss the first integral and bifurcations of phase portraits of (1).

Making a traveling wave transformation 𝑢=𝜙(𝜉)+𝜆 with 𝜉=𝑥−𝑐𝑡, (1) can be reduced to the following ordinary differential equation: (1−𝑐)𝜙^′+𝛽−𝑐𝜈𝜙^′′′+𝛼𝜙+𝜆𝜙^′+13𝛼𝜈𝜙+𝜆𝜙^′′′+2𝜙^′𝜙^′′=0. (2) We call the solution 𝜙(𝜉)+𝜆 traveling wave solution.

Integrating (2) with respect to 𝜉, we have 1𝛽−𝑐𝜈+31𝛼𝜈𝜆+3𝛼𝜈𝜙𝜙^′′()1=𝑔+𝑐−1−𝛼𝜆𝜙−2𝛼𝜙²−16𝛼𝜈𝜙^′2, (3) where 𝑔 is an integral constant. Let 𝑑𝜙/𝑑𝜉=𝑦. Thus, (3) can be rewritten as the following two-dimensional planar system: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=()()𝑑𝜉𝑔+𝑐−1−𝛼𝜆𝜙−1/2𝛼𝜙²−(1/6)𝛼𝜈𝑦²()(),𝛽−𝑐𝜈+1/3𝛼𝜈𝜆+1/3𝛼𝜈𝜙 (4) which is a singular system. Equation (4) has the following first integral: 𝐻𝜙,𝑦≡3𝑐𝜈−3𝛽−𝛼𝜈𝜆−𝛼𝜈𝜙𝑦²−𝛼𝜙³+3(𝑐−1−𝛼𝜆)𝜙²+6𝑔𝜙=ℎ. (5) Because the 𝑑𝑦/𝑑𝜉 is not defined when 𝛽−𝑐𝜈+(1/3)𝛼𝜈𝜆+(1/3)𝛼𝜈𝜙=0, so we make a transformation as follows: 1𝑑𝜉=𝛽−𝑐𝜈+31𝛼𝜈𝜆+3𝛼𝜈𝜙𝑑𝜏, (6) where 𝜏 is a parameter. Under the transformation (6), (4) becomes the following regular system: 𝑑𝜙=1𝑑𝜏𝛽−𝑐𝜈+31𝛼𝜈𝜆+3𝛼𝜈𝜙𝑦,𝑑𝑦()1𝑑𝜏=𝑔+𝑐−1−𝛼𝜆𝜙−2𝛼𝜙²−16𝛼𝜈𝑦². (7) Obviously, (7) has the same first integral as (4) which is (5).

Obviously, system (7) has two equilibrium points 𝑀_1,2(𝜙_1,2,0) on 𝜙-axes, where 𝜙_1,2=((𝑐−1−𝛼𝜆)±(𝑐−1−𝛼𝜆)²+2𝛼𝑔)/𝛼. System (7) has two equilibrium points 𝑆_±(𝜙_𝑠,𝑌_1,2) on singular line 𝜙=𝜙_𝑠, where 𝜙_𝑠=(3𝑐𝜈−3𝛽−𝛼𝜈𝜆)/𝛼𝜈, 𝑌_1,2=±3𝛼𝜈(2𝑔+2(𝑐−1−𝛼𝜆)𝜙_𝑠−𝛼𝜙²_𝑠)/𝛼𝜈. By using (5), we obtain Hamiltonian of these equilibrium points as follows: ℎ_1,2≡𝐻𝜙_1,2,0=−𝛼𝜙³_1,2+3(𝑐−1−𝛼𝜆)𝜙²_1,2+6𝑔𝜙_1,2,ℎ₃≡ℎ_𝑠=𝐻𝜙_𝑠,𝑌_1,2=3𝑐𝜈−3𝛽−𝛼𝜈𝜆−𝛼𝜈𝜙_𝑠𝑌²_1,2−𝛼𝜙³_𝑠+3(𝑐−1−𝛼𝜆)𝜙²_𝑠+6𝑔𝜙_𝑠. (8) According to the characteristic and the relation of Hamiltonian of each equilibrium points, we obtain three bifurcations of parameters as follows: 𝑔₁𝑐()+()=−𝑐−2−2𝛼𝜆1+𝛼𝜆²,𝑔2𝛼₂=3𝑐𝜈−3𝛽−𝛼𝜈𝜆2𝜈+𝑐𝜈−3𝛽+𝛼𝜈𝜆2𝛼𝜈²,𝑔₃=3𝜈+2𝛼𝜈𝜆−3𝛽4𝑐𝜈−𝜈−2𝛼𝜈𝜆−3𝛽8𝛼𝜈². (9)

Under different parametric conditions, that is to say, according to different bifurcations of parameters, in the different area of parameters and on the different bifurcation curves, we derive all kinds of bifurcations of phase portraits for the system (7), which are shown in Figures 1 and 2.

(a) 𝑔=𝑔1

(b) 𝑔1<𝑔<𝑔3

(c) 𝑔=𝑔3

(d) 𝑔3<𝑔<𝑔2

(e) 𝑔=𝑔2

(f) 𝑔>𝑔2

Figure 1 

The phase portraits of (7) under different parametric conditions for 𝜈<0.

(a) 𝑐<0,𝑔>𝑔2

(b) 𝑐<0,𝑔=𝑔2

(c) 𝑐<0,𝑔1<𝑔<𝑔2

(d) 𝑐<0,𝑔=𝑔1

(e) 𝑐>0,𝑔=𝑔1

(f) 𝑐>0,𝑔1<𝑔<𝑔2

(g) 𝑐>0,𝑔=𝑔2

(h) 𝑐>0,𝑔>𝑔2

Figure 2 

The phase portraits of (7) under different parametric conditions for 𝜈<0.

Suppose that 𝑢(𝑥,𝑡)=𝜙(𝜉)+𝜆 is a continuous traveling wave solution of (1) for 𝜉∈(−∞,∞), and lim_𝜉→∞𝜙(𝜉)=𝑎, lim_{𝜉→−∞}𝜙(𝜉)=𝑏. It is well known that (i) 𝜙(𝑥,𝑡) is called a solitary wave solution if 𝑎=𝑏; (ii) 𝜙(𝑥,𝑡) is called a kink or antikink solution if 𝑎≠𝑏. Usually, a solitary wave solution of a nonlinear wave equation corresponds to a homoclinic orbit of its traveling wave equation; a kink (or antikink) wave solution corresponds to a heteroclinic orbit (or connecting orbit). Similarly, a periodic orbit of a traveling wave equation corresponds to a periodic travelling wave solution of the nonlinear wave equation. To find all possible bifurcations of solitary waves, and periodic waves, kink and antikink wave of a nonlinear wave equation, we need to investigate the existence of all homoclinic, heteroclinic orbits, and periodic orbits for its traveling wave equation in the parameter space. The solitary wave solution is also called soliton solution. Recently, some new phenomena of soliton are revealed by many works, see [17–26] and the references cited therein.

2.2. Existence of Traveling Wave Solutions for (1)

In this subsection, according to the bifurcations of phase portraits in Figures 1 and 2, by using analysis of the phase portraits, we will discuss the existence of smooth solitary traveling wave solutions, kink wave solutions, peakon solutions, compacton solutions, and periodic traveling wave solutions of (1).

Theorem 1. Existence theorem of traveling wave solution.
Suppose that 𝜈<0. Then consider the following. (i) Equation (1) has infinitely many periodic wave solutions corresponding to a family of close orbits 𝐻(𝜙,𝑦)=ℎ under the following parametric conditions: (1)𝑔₁<𝑔<𝑔₃, ℎ₁<ℎ<ℎ₂, see Figure 1(b); (2)𝑔=𝑔₃, ℎ₁<ℎ<ℎ₂, see Figure 1(c);(3)𝑔₃<𝑔<𝑔₂, ℎ₃<ℎ<ℎ₂, see Figure 1(d).(ii) Equation (1) has a solitary wave solution corresponding to the homoclinic orbit 𝐻(𝜙,𝑦)=ℎ₁ under the parametric conditions 𝑔₁<𝑔<𝑔₃, see Figure 1(b). (iii) Equation (1) has a solitary cusp wave (peakon) solution corresponding to two heteroclinic orbits 𝐻(𝜙,𝑦)=ℎ_𝑠 under the following parametric conditions 𝑔=𝑔₃, ℎ₁=ℎ_𝑠, see Figure 1(c). (iv) Equation (1) has a periodic cusp wave solution corresponding to a semilunar orbits 𝐻(𝜙,𝑦)=ℎ₃ under the following parametric conditions 𝑔₃<𝑔<𝑔₂, ℎ₃=ℎ₄, see Figure 1(c). (v) Equation (1) has infinitely many compacton solutions corresponding to a  family of bounded open orbits 𝐻(𝜙,𝑦)=ℎ under the following parametric conditions: (1)𝑔₃<𝑔<𝑔₂, ℎ₁<ℎ<ℎ₃, see Figure 1(d); (2)𝑔=𝑔₂, ℎ₁<ℎ<ℎ₃, see Figure 1(e); (3)𝑔>𝑔₂, ℎ₁<ℎ<ℎ₃, see Figure 1(f). (vi) Equation (1) has two-half kink wave (generalized kink wave) solutions  corresponding to two heteroclinic orbits 𝐻(𝜙,𝑦)=ℎ under the following  parametric conditions: (1)𝑔₃<𝑔<𝑔₂, ℎ=ℎ₁, see Figure 1(d); (2)𝑔>𝑔₂, ℎ=ℎ₁  or  ℎ=ℎ₂, see Figure 1(f).

Theorem 2. Existence theorem of traveling wave solution.
Suppose that 𝜈>0. Then consider the following.(i) Equation (1) has infinitely many periodic wave solutions corresponding to a family of close orbits 𝐻(𝜙,𝑦)=ℎ under the following parametric conditions:(1)𝑐<0, 𝑔>𝑔₂, ℎ₁<ℎ<ℎ₃ and ℎ₃<ℎ<ℎ₂, see Figure 2(a);(2)𝑐<0, 𝑔=𝑔₂, ℎ₃<ℎ<ℎ₂, see Figure 2(b);(3)𝑐<0, 𝑔₁<𝑔<𝑔₂, ℎ₁<ℎ<ℎ₂, see Figure 2(c);(4)𝑐>0, 𝑔₁<𝑔<𝑔₂, ℎ₁<ℎ<ℎ₂, see Figure 2(f);(5)𝑐>0, 𝑔=𝑔₂, ℎ₁<ℎ<ℎ₂, see Figure 2(g);(6)𝑐>0, 𝑔>𝑔₂, ℎ₁<ℎ<ℎ₃ and ℎ₃<ℎ<ℎ₂, see Figure 2(g).(ii) Equation (1) has a solitary wave solution corresponding to the homoclinic orbit 𝐻(𝜙,𝑦)=ℎ₁ or 𝐻(𝜙,𝑦)=ℎ₂ under the parametric conditions:(1)𝑐<0, 𝑔₁<𝑔<𝑔₂, ℎ=ℎ₁, see Figure 2(c);(2)𝑐>0, 𝑔₁<𝑔<𝑔₂, ℎ=ℎ₂, see Figure 2(f). (iii) Equation (1) has a periodic cusp wave solution corresponding to a semilunar orbits 𝐻(𝜙,𝑦)=ℎ₃ under the following parametric conditions:(1)𝑐<0, 𝑔>𝑔₂, ℎ₂=ℎ₃, see Figure 2(a);(2)𝑐>0, 𝑔>𝑔₂, ℎ₂=ℎ₃, see Figure 2 (h).(iv) Equation (1) has infinitely many compacton solutions corresponding to a family of bounded open orbits 𝐻(𝜙,𝑦)=ℎ under the following parametric conditions:(1)𝑐<0, 𝑔>𝑔₂, ℎ>ℎ₂ and −∞<ℎ<ℎ₁, see Figure 2(a);(2)𝑐<0, 𝑔=𝑔₂, ℎ>ℎ₃ and 0<ℎ<ℎ₃, see Figure 2(b);(3)𝑐<0, 𝑔₁<𝑔<𝑔₂, ℎ>ℎ₃ and −∞<ℎ<ℎ₁, see Figure 2(c);(4)𝑐<0, 𝑔=𝑔₁, ℎ>ℎ₃ and −∞<ℎ<ℎ₂, see Figure 2(d);(5)𝑐>0, 𝑔=𝑔₁, ℎ₁<ℎ<+∞ and −∞<ℎ<ℎ₁, see Figure 2(e);(6)𝑐>0, 𝑔₁<𝑔<𝑔₂, ℎ₂<ℎ<+∞ and −∞<ℎ<ℎ₁, see Figure 2(f);(7)𝑐>0, 𝑔=𝑔₂, ℎ₂<ℎ<+∞ and ℎ₁<ℎ<ℎ₂, see Figure 2(g);(8)𝑐>0, 𝑔>𝑔₂, ℎ₂<ℎ<+∞ and −∞<ℎ<ℎ₁, see Figure 2(g).(v) Equation (1) has two-half kink wave (generalized kink wave) solutions corresponding to two heteroclinic orbits 𝐻(𝜙,𝑦)=ℎ under the following parametric conditions:(1)𝑐<0, 𝑔=𝑔₁, ℎ=ℎ₁=ℎ₂, see Figure 2(d);(2)𝑐>0, 𝑔=𝑔₁, ℎ=ℎ₁=ℎ₂, see Figure 2(e).

3. The Exact Travelling Wave Solutions of (1)

In this section, we discuss exact travelling wave solutions of (1) under some parametric conditions.

Case 1. Whenℎ=ℎ_𝑠, 𝑔=𝑔₃, system (7) has two heteroclinic orbits, see Figure 1(c); by using (5), their expressions can be written as 𝑦=±1−𝜈𝜙−3𝛽−3𝜈−2𝛼𝜈𝜆.2𝛼𝜈 (10) Corresponding to these two heteroclinic orbits, (1) has a peakon solution as follows: 3𝑢=𝜆+2𝑐𝜈−𝜈−3𝛽𝑒2𝛼𝜈^{−1/−𝜈|𝜉|}. (11)

Case 2. When ℎ=ℎ₁, 𝑔₁<𝑔<𝑔₃, 𝜆=(𝑐−1)/𝛼, 𝑔=𝛼, 𝑐=−(√2𝛼𝜈−3𝛽+𝜈)/2𝜈, system (7) has a homoclinic orbit, see Figure 1(b); by using (5), the expression can be written as 𝑦=±−1𝜈−2√2+𝜙√2+𝜙. (12) Corresponding to the homoclinic orbit, (1) has a solitary wave solution as follows: √𝑢=𝜆+2tanh²(1/2)−1/𝜈𝜉+2√2sech²(1/2)−1/𝜈𝜉. (13)

Case 3. When ℎ=ℎ₁, 𝑔=𝑔₂, system (7) has two heteroclinic orbits, see Figure 1(e); by using (5), the expression can be written as 𝑦=±−1𝜈𝜙−𝜙₁𝑧₁−𝜙𝜙_𝑠𝜙−𝜙¹<𝜙<𝜙_𝑠<𝑧₁, (14) where 𝑧₁=(5𝑐𝜈+𝜈−6𝛽−𝛼𝜈𝜆)/𝛼𝜈. Corresponding to these two heteroclinic orbits, taking (0,𝜙⁰_𝑖), (𝑖=1,2) as the initial values and substituting (14) into the first expression of (4), we have ^𝜙_𝜙0𝑖1𝜑−𝜙₁𝜙_𝑠−𝜑𝑧₁−𝜑𝑑𝜑=±^𝜉₀−1𝜈𝑑𝜉̃. (15) In order to compute the left integral of (15), we let (𝜙_𝑠−𝜑)/(𝑧₁−𝜑)=𝜓. First, using the integration by substitution, we compute the indefinite integral ∫(1/(𝜑−𝜙₁))(𝜙_𝑠−𝜑)/(𝑧₁−𝜑)𝑑𝜑, see the following computation: 1𝜑−𝜙₁𝜙_𝑠−𝜑𝑧₁=2𝜙−𝜑𝑑𝜑_𝑠−𝑧₁𝑧₁−𝜙₁𝜓²𝜓²−1𝜓²−𝜙_𝑠−𝜙₁/𝑧₁−𝜙₁=1𝑑𝜓−1𝜓+11𝜓+1𝑑𝜓+𝛿−1𝜓−𝛿𝜓+𝛿𝑑𝜓=ln𝜓+1𝜓−1𝜓−𝛿𝜓+𝛿^𝛿=ln𝜙_𝑠−𝜙+𝑧₁−𝜙𝜙_𝑠−𝜙−𝑧₁−𝜙𝜙_𝑠−𝜙−𝛿𝑧₁−𝜙𝜙_𝑠−𝜙+𝛿𝑧₁−𝜙^𝛿, (16) where 𝛿=(𝜙_𝑠−𝑧₁)/(𝑧₁−𝜙₁). Second, by using (16), completing the integral equation (15), we obtain ln𝜙_𝑠−𝜙+𝑧₁−𝜙𝜙_𝑠−𝜙−𝑧₁−𝜙𝜙_𝑠−𝜙−𝛿𝑧₁−𝜙𝜙_𝑠−𝜙+𝛿𝑧₁−𝜙^𝛿=lnΩ_𝑖±−1𝜈𝜉, (17) where Ω_𝑖=𝜙_𝑠−𝜙⁰_𝑖+𝑧₁−𝜙⁰_𝑖𝜙_𝑠−𝜙⁰_𝑖−𝑧₁−𝜙⁰_𝑖𝜙_𝑠−𝜙⁰_𝑖−𝛿𝑧₁−𝜙⁰_𝑖𝜙_𝑠−𝜙⁰_𝑖+𝛿𝑧₁−𝜙⁰_𝑖^𝛿𝑖=1,2, (18) with 𝜙⁰_𝑖∈(𝜙₁,𝜙_𝑠). Finally, substituting 𝜙=𝑢−𝜆 into (18), and then simplifying them, we obtain two generalized kink wave solutions of (1) as follows: 𝜙_𝑠−𝑢+𝜆+𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆−𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆−𝛿𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆+𝛿𝑧₁−𝑢+𝜆^𝛿=Ω₁𝑒^{−1/𝜈𝜉},𝜉∈−∞,𝑙₁,𝜙_𝑠−𝑢+𝜆+𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆−𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆−𝛿𝑧₁−𝑢+𝜆𝜙_𝑠−𝑢+𝜆+𝛿𝑧₁−𝑢+𝜆^𝛿=Ω₂𝑒^{−−1/𝜈𝜉},𝜉∈𝑙₂,+∞, (19)(20) where 𝑙_𝑖=−√−𝜈lnΩ_𝑖, (𝑖=1,2).

Case 4. When 𝑔=𝑔₂, ℎ=ℎ_𝑠, system (7) has a close orbit, see Figure 2(b); by using (5), the expression can be written as 𝑦=±1𝜈𝑧₂−𝜙𝜙−𝜙_𝑠, (21) where 𝑧₂=(6𝛽−3𝜈−3𝑐𝜈−𝛼𝜈𝜆)/𝛼𝜈. Corresponding to this close orbit, (1) has a periodic wave solution as follows: 𝑢=𝜆+3𝜈+2𝛼𝜈𝜆−3𝛽+2𝛼𝜈3𝜈+6𝑐𝜈−6𝛽2𝛼𝜈cos1𝜈𝜉. (22)

Case 5. When 𝑔=𝑔₂, ℎ=0, system (7) has a open orbit, see Figure 2(g); by using (5), the expression can be written as 𝑦=±1𝜈𝜙−𝑧₃𝜙−𝑧₄0−𝜙𝜙−𝜙_𝑠𝑧₃<𝑧₄<𝜙_𝑠<𝜙≤0, (23) where 𝑧_3,4=(3𝜈(𝑐−1−𝛼𝜆)∓√Δ)/2𝛼𝜈, Δ=3(5𝑐𝜈+𝜈−6𝛽−𝛼𝜈𝜆)(3𝑐𝜈+3𝜈−6𝛽+𝛼𝜈𝜆). Corresponding to this open orbit, substituting (23) into the first equation of (4) yields ^𝜙_𝜙𝑠𝜑−𝜙_𝑠𝜑−𝑧₃𝜑−𝑧₄0−𝜑𝑑𝜑=±^𝜉₀1𝜈𝑑𝜉̃. (24) By using the formulas 256.02th and 400th in [27], we integrate ∫^𝜙_(𝜙)𝑠(𝜑−𝜙_𝑠)/[(𝜑−𝑧₃)(𝜑−𝑧₄)(0−𝜑)]𝑑𝜑 of (24), thus we obtain a compacton solution of (1) as follows: Πsn⁻¹𝑧₄𝑢−𝜆−𝜙_𝑠𝜙_𝑠𝑢−𝜆−𝑧₄,𝑘₂,𝛼²₂−sn⁻¹𝑧₄𝑢−𝜆−𝜙_𝑠𝜙_𝑠𝑢−𝜆−𝑧₄,𝑘₂=Ω₃𝜉^∗₂−𝜉, (25) where Π(𝜑,𝛼²,𝑘) is the incomplete elliptic integral of the third kind in Legendre’s canonical form, and 𝛼²₂=𝜙_𝑠/𝑧₄, 𝑘²₂=𝜙_𝑠(𝑧₄−𝑧₃)/[𝑧₄(𝜙_𝑠−𝑧₃)], Ω₃=−𝑧₄(𝜙_𝑠−𝑧₃)/(2√𝜈(𝜙_𝑠−𝑧₄)), 𝜉^∗₂=(Π(𝛼²₂,𝑘₂)−𝐾(𝑘₂))/Ω₃.

4. Conclusion

In this work, by using the bifurcation theory of dynamic system, the generalization of KdV equation (1) was studied. According to the analysis of the phase portraits, the existing theorems on solitary wave, cusp wave, periodic wave, periodic cusp wave and compactons are given. In some parametric conditions, exact peakon solution (11), solitary wave solution (13), generalized kink wave solutions (15) and (16), smooth periodic wave solution (19), and compacton solution (21) were obtained. These exact solutions are different from those exact solutions in [8, 12].

Acknowledgments

The authors thank reviewers very much for their useful comments and helpful suggestions. This research is supported by the National Natural Science Foundation of China (11161020) and the Natural Science Foundation of Yunnan Province (2011FZ193).