Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

Abstract

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.

1. Introduction

The Benjamin-Bona-Mahony (BBM) equation [1], 𝑢_𝑡+𝑢_𝑥+𝑢𝑢_𝑥−𝑢_𝑥𝑥𝑡=0, (1.1) has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.

The Kadomtsov-Petviashvili (KP) equation [2] is given by 𝑢_𝑡+𝑎(𝑢_𝑥+𝑢_𝑥𝑥𝑥)_𝑥+𝑢_𝑦𝑦=0, (1.2) which is a weekly two-dimensional generalization of the KdV equation in the sense that it accounts for slowly varying transverse perturbations of unidirectional KdV solitons moving along the 𝑥-direction.

Wazwaz [3] presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation (𝑢_𝑡+𝑢_𝑥+𝑎(𝑢²)_𝑥−𝑏𝑢_𝑥𝑥𝑡)_𝑥+𝑟𝑢_𝑦𝑦=0, (1.3) and the generalized KP-BBM equation (𝑢_𝑡+𝑢_𝑥+𝑎(𝑢³)_𝑥−𝑏𝑢_𝑥𝑥𝑡)_𝑥+𝑟𝑢_𝑦𝑦=0. (1.4)

Wazwaz [3, 4] obtained some solitons solution and periodic solutions of (1.3) by using the sine-cosine method and the extended tanh method. Abdou [5] used the extended mapping method with symbolic computation to obtain some periodic solutions of (1.3), solitary wave solution, and triangular wave solution. Song et al. [6] obtained exact solitary wave solutions of (1.3) by using bifurcation method of dynamical systems.

The aim of this paper is to study the traveling wave solutions and their phase portraits for (1.4) by using the bifurcation method and qualitative theory of dynamical systems [6–15]. Through some special phase orbits, we obtain a number of smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded solutions, blow-up wave solutions, and solitary wave solutions.

The remainder of this paper is organized as follows. In Section 2, by using the bifurcation theory of planar dynamical systems, two phase portraits for the corresponding traveling wave system of (1.4) are given under different parameter conditions. In Section 3, we present our main results and their proofs. A short conclusion will be given in Section 4.

2. Phase Portraits

To derive our results, we give some preliminaries in this section. For given positive constant wave speed 𝑐, substituting 𝑢=𝜑(𝜉) with 𝜉=𝑥+𝑦−𝑐𝑡 into the generalized KP-BBM equation (1.4), it follows that (−𝑐𝜑^′+𝜑^′−𝑎(𝜑³)^′+𝑏𝑐𝜑^′′′)^′+𝑟𝜑^′′=0. (2.1)

Integrating (2.1) twice and letting the first integral constant be zero, we have (𝑟−𝑐+1)𝜑−𝑎𝜑³+𝑏𝑐𝜑^′′=𝑔₁, (2.2) where 𝑔₁ is the second constant of integration.

Letting 𝜙=𝜑^′, we get the following planar system: 𝑑𝜑𝑑𝜉=𝜙,𝑑𝜙𝑑𝜉=𝛼𝜑³−𝛽𝜑+𝑔, (2.3) where 𝛼=𝑎/𝑏𝑐, 𝛽=(𝑟+1−𝑐)/𝑏𝑐 and 𝑔=𝑔₁/𝑏𝑐.

Obviously, the system (2.3) is a Hamiltonian system with Hamiltonian function 𝐻𝜑,𝜙=𝜙²−𝛼2𝜑⁴+𝛽𝜑²−2𝑔𝜑=ℎ, (2.4) where ℎ is Hamiltonian.

Now we consider the phase portraits of system (2.3). Set 𝑓₀𝜑=𝛼𝜑³𝑓𝜑−𝛽𝜑,=𝛼𝜑³−𝛽𝜑+𝑔. (2.5)

Obviously, 𝑓₀(𝜑) has three zero points, 𝜑₋, 𝜑₀ and 𝜑₊, which are given as follows: 𝜑₋=−𝛽𝛼,𝜑₀=0,𝜑₊=𝛽𝛼. (2.6)

It is easy to obtain two extreme points of 𝑓₀(𝜑) as follows: 𝜑^∗₋=−𝛽3𝛼,𝜑^∗₊=𝛽.3𝛼 (2.7)

Letting 𝑔₀=𝑓₀𝜑^∗₋=𝑓₀𝜑^∗₊=2𝛽3𝛽,3𝛼 (2.8) then it is easily seen that 𝑔₀ is the extreme values of 𝑓₀(𝜑).

Let (𝜑_𝑖,0) be one of the singular points of system (2.3), then the characteristic values of the linearized system of system (2.3) at the singular points (𝜑_𝑖,0) are 𝜆_±=±𝑓′𝜑_𝑖. (2.9)

From the qualitative theory of dynamical systems, we therefore know that(i)if 𝑓′(𝜑_𝑖)>0, (𝜑_𝑖,0) is a saddle point.(ii)if 𝑓′(𝜑_𝑖)<0, (𝜑_𝑖,0) is a center point.(iii)if 𝑓′(𝜑_𝑖)=0, (𝜑_𝑖,0) is a degenerate saddle point.

Therefore, we obtain the phase portraits of system (2.3) in Figures 1 and 2.

Figure 1 

The phase portraits of system (2.3) when 𝛼>0, 𝛽>0.

Figure 2 

The phase portraits of system (2.3) when 𝛼<0, 𝛽<0.

3. Main Results and Their Proofs

In this section, we state our main results.

Proposition 3.1. For given positive constants 𝑐 and 𝑔₀, (1.4) has the following periodic wave solutions when 𝛼>0 and 𝛽>0.
(1) If 𝑔=0, we get eight periodic blow-up wave solutions 𝑢_1±𝑥,𝑦,𝑡=±2𝛽𝛼csc𝑢𝛽𝜉,_2±𝑥,𝑦,𝑡=±2𝛽𝛼sec𝑢𝛽𝜉,_3±𝜑𝑥,𝑦,𝑡=±⁶sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆,𝑢_4±𝑥,𝑦,𝑡=±𝜑²₆−𝜑²₅sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆²1−sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆², (3.1)(3.2)(3.3)(3.4) two periodic wave solutions 𝑢_5±𝑥,𝑦,𝑡=±𝜑₅sn𝜑₆𝛼2𝜑𝜉,⁵𝜑₆, (3.5) two kink wave solutions 𝑢_6±𝑥,𝑦,𝑡=±𝛽𝛼tanh𝛽2𝜉, (3.6) and two unbounded wave solutions 𝑢_7±𝑥,𝑦,𝑡=±𝛽𝛼coth𝛽2𝜉. (3.7)
(2) If 0<𝑔<𝑔₀, we get four periodic blow-up wave solutions 𝑢_8±𝑥,𝑦,𝑡=𝜑^∗₇−2𝛾₁𝛿₁±√𝜂₁cos𝛼𝛾₁,𝑢/2𝜉₉=𝜑𝑥,𝑦,𝑡¹⁰𝜑₁₁−𝜑₁₃+𝜑₁₁𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)²𝜑₁₁−𝜑₁₃+𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)²,𝑢₁₀=𝜑𝑥,𝑦,𝑡¹³𝜑₁₂−𝜑₁₀−𝜑₁₂𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)²𝜑₁₂−𝜑₁₀−𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)², (3.8)(3.9) where 𝜑^∗₇=12−𝜑₇+4𝛽𝛼−3𝜑²₇,𝛾₁=−4𝛽+3𝜑₇𝛼𝜑₇+4𝛼𝛽−3𝛼²𝜑²₇𝛼,𝛿₁=−2𝜑₇+24𝛼𝛽−3𝛼²𝜑²₇𝛼,𝜂₁=4𝜑₇𝛼𝜑₇+4𝛼𝛽−3𝛼²𝜑²₇𝛼,𝜔₁=𝛼𝜑₁₃−𝜑₁₁𝜑₁₂−𝜑₁₀2√2,𝑘₁=𝜑₁₂−𝜑₁₁𝜑₁₃−𝜑₁₀𝜑₁₃−𝜑₁₁𝜑₁₂−𝜑₁₀, (3.10) a periodic wave solution 𝑢₁₁=𝜑𝑥,𝑦,𝑡¹¹𝜑₁₀−𝜑₁₂+𝜑₁₀𝜑₁₂−𝜑₁₁sn(𝜔₁𝜉,𝑘₁)²𝜑₁₀−𝜑₁₂+𝜑₁₂−𝜑₁₁sn(𝜔₁𝜉,𝑘₁)², (3.11) a blow-up wave solution 𝑢₁₂𝑥,𝑦,𝑡=𝜑₇+2𝛽−3𝛼𝜑²₇2𝛼𝜑₇−2𝛼𝛽−𝛼𝜑²₇cosh3𝛼𝜑²₇,−𝛽𝜉 (3.12) a solitary wave solution 𝑢₁₃𝑥,𝑦,𝑡=𝜑₇+2𝛽−3𝛼𝜑²₇2𝛼𝜑₇+2𝛼𝛽−𝛼𝜑²₇cosh3𝛼𝜑²₇,−𝛽𝜉 (3.13) and an unbounded wave solution 𝑢₁₄𝑥,𝑦,𝑡=𝜑₇−2𝛽−3𝛼𝜑²₇csch3𝛼𝜑²₇−𝛽𝜉−2𝛼𝛽+6𝛼²𝜑²₇+2𝛼𝜑₇tanh3𝛼𝜑²₇.−𝛽𝜉/2 (3.14)
(3) If 𝑔=𝑔₀, we get three blow-up wave solutions 𝑢₁₅=√𝑥,𝑦,𝑡182−𝛽3𝛽𝜉³√𝛼18𝜉−3𝛽𝜉³,𝑢₁₆=√𝑥,𝑦,𝑡182+𝛽3𝛽𝜉³√𝛼−18𝜉+3𝛽𝜉³,𝑢₁₇𝑥,𝑦,𝑡=3𝛽9+2𝛽𝜉²√𝛼−9+6𝛽𝜉², (3.15) and a periodic blow-up wave solution 𝑢₁₈𝑥,𝑦,𝑡=−𝐴₁𝜑₁₇+𝐵₁𝜑₁₈+𝐴₁𝜑₁₇+𝐵₁𝜑₁₈cn𝛼𝐴₁𝐵₁/2𝜉,𝑘₂−𝐴₁+𝐵₁+𝐴₁+𝐵₁cn𝛼𝐴₁𝐵₁/2𝜉,𝑘₂, (3.16) where 𝐴₁=𝜑₁₈−𝑐₁+𝑐₁2²−(𝑐₁−𝑐₁)²4,𝐵₁=𝜑₁₇−𝑐₁+𝑐₁2²−(𝑐₁−𝑐₁)²4,𝑘₂=(𝐴₁+𝐵₁)²−(𝜑₁₈−𝜑₁₇)²4𝐴₁𝐵₁, (3.17)𝑐₁ and 𝑐₁ are conjugate complex numbers.

Proof. (1) If 𝑔=0, we will consider three kinds of orbits.
(i) From the phase portrait, we note that there are two special orbits Γ₁ and Γ₂, which have the same Hamiltonian with that of the center point (0,0). In (𝜑,𝜙) plane, the expressions of these two orbits are given as 𝜙=±𝛼2𝜑²𝜑−𝜑₁𝜑−𝜑₂, (3.18) where 𝜑₁=−2𝛽/𝛼 and 𝜑₂=2𝛽/𝛼.
Substituting (3.18) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the two orbits Γ₁ and Γ₂, it follows that ±^∞_𝜑1𝑠𝑠−𝜑₁𝑠−𝜑₂𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑21𝑠𝑠−𝜑₁𝑠−𝜑₂𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.19)
From (3.19) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get four periodic blow-up solutions 𝑢_1±(𝑥,𝑦,𝑡) and 𝑢_2±(𝑥,𝑦,𝑡) as (3.1) and (3.2).
(ii) From the phase portrait, we note that there are three special orbits Γ₃, Γ₄, and Γ₅ passing the points (𝜑₃,0),  (𝜑₄,0),  (𝜑₅,0), and (𝜑₆,0). In (𝜑,𝜙) plane, the expressions of the orbit are given as 𝜙=±𝛼2𝜑−𝜑₃𝜑−𝜑₄𝜑−𝜑₅𝜑−𝜑₆, (3.20) where 𝜑₃=−𝜑₆,𝜑₄=−2𝛽/𝛼−𝜑²₆,𝜑₅=2𝛽/𝛼−𝜑²₆  and  𝛽/𝛼<𝜑₆<2𝛽/𝛼.
Substituting (3.20) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₃, Γ₄, and Γ₅, we have ±^∞_𝜑1𝑠−𝜑₃𝑠−𝜑₄𝑠−𝜑₅𝑠−𝜑₆𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑61𝑠−𝜑₃𝑠−𝜑₄𝑠−𝜑₅𝑠−𝜑₆𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑01𝑠−𝜑₃𝑠−𝜑₄𝑠−𝜑₅𝑠−𝜑₆𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.21)
From (3.21) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get four periodic blow-up wave solutions 𝑢_3±(𝑥,𝑦,𝑡), 𝑢_4±(𝑥,𝑦,𝑡) as (3.3), (3.4) and two periodic solutions 𝑢_5±(𝑥,𝑦,𝑡) as (3.5).
(iii) From the phase portrait, we see that there are two heteroclinic orbits Γ₆ and Γ₇ connected at saddle points (𝜑₋,0) and (𝜑₊,0). In (𝜑,𝜙) plane, the expressions of the heteroclinic orbits are given as 𝜙=±𝛼2(𝜑−𝜑₋)²(𝜑−𝜑₊)². (3.22)
Substituting (3.22) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the heteroclinic orbits Γ₆ and Γ₇, it follows that ±^𝜑₀1𝑠−𝜑₋𝜑₊−𝑠𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_∞𝜑1𝑠−𝜑₋𝜑₊−𝑠𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.23)
From (3.23) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get two kink wave solutions 𝑢_6±(𝑥,𝑦,𝑡) as (3.6) and two unbounded solutions 𝑢_7±(𝑥,𝑦,𝑡) as (3.7).
(2) If 0<𝑔<𝑔₀, we set the largest solution of 𝑓(𝜑)=0 be 𝜑₇(𝛽/3𝛼<𝜑₇<𝛽/𝛼), then we can get another two solutions of 𝑓(𝜑)=0 as follows: 𝜑^∗₇=12−𝜑₇+4𝛽𝛼−3𝜑²₇,𝜑^⋆₇=12−𝜑₇−4𝛽𝛼−3𝜑²₇. (3.24)
(i) From the phase portrait, we see that there are two special orbits Γ₈ and Γ₉, which have the same Hamiltonian with that of the center point (𝜑^∗₇,0). In (𝜑,𝜙)-plane, the expressions of the orbits are given as 𝜙=±𝛼2(𝜑−𝜑^∗₇)²𝜑−𝜑₈𝜑−𝜑₉, (3.25) where 𝜑₈=12𝜑₇−4𝛽𝛼−3𝜑²₇−2𝜑²₇+𝜑₇4𝛽𝛼−3𝜑²₇,𝜑₉=12𝜑₇−4𝛽𝛼−3𝜑²₇+2𝜑²₇+𝜑₇4𝛽𝛼−3𝜑²₇. (3.26)
Substituting (3.25) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the orbits, it follows that ±^𝜑_𝜑81(𝑠−𝜑^∗₇)²𝑠−𝜑₈𝑠−𝜑₉𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑91(𝑠−𝜑^∗₇)²s−𝜑₈𝑠−𝜑₉𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.27)
From (3.27) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get two periodic blow-up wave solutions 𝑢_8±(𝑥,𝑦,𝑡) as (3.8).
(ii) From the phase portrait, we note that there are three special orbits Γ₁₀, Γ₁₁, and Γ₁₂ passing the points (𝜑₁₀,0),  (𝜑₁₁,0), (𝜑₁₂,0), and (𝜑₁₃,0). In (𝜑,𝜙) plane, the expressions of the orbit are given as 𝜙=±𝛼2𝜑−𝜑₁₀𝜑−𝜑₁₁𝜑−𝜑₁₂𝜑−𝜑₁₃, (3.28) where 𝜑₈<𝜑₁₀<𝜑₁₄<𝜑^⋆₇<𝜑₁₅<𝜑₁₁<𝜑₁₂<𝜑₇<𝜑₁₃<𝜑₉.
Substituting (3.28) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₁₀, Γ₁₁, and Γ₁₂, we have ^𝜑10_𝜑1𝜑₁₃−𝑠𝜑₁₂−𝑠𝜑₁₁−𝑠𝜑₁₀−𝑠𝑑𝑠=𝛼2^𝜉₀𝑑𝑠,_𝜑𝜑131𝑠−𝜑₁₃𝑠−𝜑₁₂𝑠−𝜑₁₁𝑠−𝜑₁₀𝑑𝑠=𝛼2^𝜉₀𝑑𝑠,_𝜑𝜑111𝜑₁₃−𝑠𝜑₁₂−𝑠𝑠−𝜑₁₁𝑠−𝜑₁₀𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.29)
From (3.29) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get two periodic blow-up wave solutions 𝑢₉(𝑥,𝑦,𝑡), 𝑢₁₀(𝑥,𝑦,𝑡) as (3.9) and a periodic wave solution 𝑢₁₁(𝑥,𝑦,𝑡) as (3.11).
(iii) From the phase portrait, we see that there are a spacial orbit Γ₁₃, which passes the point (𝜑₁₄,0), and a homoclinic orbit Γ₁₄ passing the saddle point (𝜑₇,0). In (𝜑,𝜙) plane, the expressions of the orbits are given as 𝜙=±𝛼2(𝜑−𝜑₇)²𝜑−𝜑₁₄𝜑−𝜑₁₅, (3.30) where 𝜑₁₄=−𝛼𝜑₇−2𝛼𝛽−2𝛼²𝜑²₇𝛼,𝜑₁₅=−𝛼𝜑₇+2𝛼𝛽−2𝛼²𝜑²₇𝛼. (3.31)
Substituting (3.30) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the orbits, it follows that ±^𝜑_𝜑141(𝑠−𝜑₇)²𝑠−𝜑₁₄𝑠−𝜑₁₅𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑151(𝑠−𝜑₇)²𝑠−𝜑₁₄𝑠−𝜑₁₅𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_∞𝜑1(𝑠−𝜑₇)²𝑠−𝜑₁₅𝑠−𝜑₁₅𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.32)
From (3.32) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a blow-up solution 𝑢₁₂(𝑥,𝑦,𝑡) as (3.12), a solitary wave solution 𝑢₁₃(𝑥,𝑦,𝑡) as (3.13), and an unbounded wave solution 𝑢₁₄(𝑥,𝑦,𝑡) as (3.14).
(3) If 𝑔=𝑔₀, we will consider two kinds of orbits.
(i) From the phase portrait, we see that there are two orbits Γ₁₅ and Γ₁₆, which have the same Hamiltonian with the degenerate saddle point (𝜑^∗₊,0). In (𝜑,𝜙) plane, the expressions of these two orbits are given as 𝜙=±𝛼2(𝜑−𝜑^∗₊)³𝜑−𝜑₁₆, (3.33) where 𝜑₁₆=−3𝛽𝛼. (3.34)
Substituting (3.33) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along these two orbits Γ₁₅ and Γ₁₆, it follows that ±_𝜑^+∞1(𝑠−𝜑^∗₊)³𝑠−𝜑₁₆𝑑𝑠=𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑161(𝑠−𝜑^∗₊)³𝑠−𝜑₁₆𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.35)
From (3.35) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get three blow-up solutions 𝑢₁₅(𝑥,𝑦,𝑡), 𝑢₁₆(𝑥,𝑦,𝑡), and 𝑢₁₇(𝑥,𝑦,𝑡) as (3.15).
(ii) From the phase portrait, we see that there are two special orbits Γ₁₇ and Γ₁₈ passing the points (𝜑₁₇,0) and (𝜑₁₈,0). In (𝜑,𝜙) plane, the expressions of the orbits are given as 𝜙=±𝛼2𝜑−𝜑₁₈𝜑−𝜑₁₇𝜑−𝑐₁𝜑−𝑐₁, (3.36) where 𝜑₁₇<𝜑₁₆<𝜑^∗₊<𝜑₁₈, 𝑐₁ and 𝑐₁ are conjugate complex numbers.
Substituting (3.36) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₁₇ and Γ₁₈, we have ±^𝜑_𝜑181𝑠−𝜑₁₈𝑠−𝜑₁₇𝑠−𝑐₁𝑠−𝑐₁𝑑𝑠=𝛼2^𝜉₀𝑑𝑠. (3.37)
From (3.37) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic blow-up wave solutions 𝑢₁₈(𝑥,𝑦,𝑡) as (3.16).
Thus, we obtain the results given in Proposition 3.1.

Proposition 3.2. For given positive constants 𝑐 and 𝑔₀, (1.4) has the following periodic wave solution when 𝛼<0 and 𝛽<0.
(1) If 𝑔=0, we get four periodic wave solutions 𝑢₁₉𝑥,𝑦,𝑡=𝜑₁cn𝛽−𝛼𝜑²₁𝜉,−𝜑₁𝛼−2𝛽+2𝛼𝜑²₁,𝑢₂₀𝑥,𝑦,𝑡=𝜑₂cn𝛽−𝛼𝜑²₂𝜉,𝜑₂𝛼−2𝛽+2𝛼𝜑²₂,𝑢₂₁𝑥,𝑦,𝑡=−𝜅₁𝜑₆+𝜆₁𝜑₆sn²𝜔₂𝜉,𝑘₃𝜅₁+𝜆₁sn²𝜔₂𝜉,𝑘₃,𝑢₂₂𝑥,𝑦,𝑡=𝜑²₆−2𝜑²₆−2𝛽𝛼sn²𝜑₆−𝛼2𝜉,𝑘₄, (3.38)(3.39)(3.40)(3.41) where 𝜅₁=𝜑₆+2𝛽𝛼−𝜑²₆,𝜆₁=𝜑₆−2𝛽𝛼−𝜑²₆,𝜔₂=√−𝛼𝜑₆+−2𝛽+𝛼𝜑²₆2√2,𝑘₃=𝜆₁𝜅₁,𝑘₄=2𝜑²₆−2𝛽/𝛼𝜑₆, (3.42) and two solitary wave solutions 𝑢_±23𝑥,𝑦,𝑡=±2𝛽𝛼sech−𝛽𝜉. (3.43)
(2) If −𝑔₀<𝑔<0, we get six periodic wave solutions 𝑢₂₄=𝐴𝑥,𝑦,𝑡²𝜑₁₀+𝜑₁₁𝐵₂+𝐴₂𝜑₁₀−𝜑₁₁𝐵₂cn−𝛼𝐴₂𝐵₂/2𝜉,𝑘₅𝐴₂+𝐵₂+𝐴₂−𝐵₂cn−𝛼𝐴₂𝐵₂/2𝜉,𝑘₅𝑢₂₅,𝑥,𝑦,𝑡=𝜑^∗₉+2𝜃−𝜇+√𝑞cos()𝜉,𝑢𝛼𝜃/2₂₆=𝜑𝑥,𝑦,𝑡¹⁴𝜑₁₇−𝜑₁₅+𝜑₁₇𝜑₁₅−𝜑₁₄sn²𝜔₃𝜉,𝑘₆𝜑₁₇−𝜑₁₅+𝜑₁₅−𝜑₁₄sn²𝜔₃𝜉,𝑘₆,𝑢₂₇=𝜑𝑥,𝑦,𝑡¹⁷−𝜑₁₆+𝜑₁₄−𝜑₁₄𝜑₁₇−𝜑₁₆sn²𝜔₃𝜉,𝑘₆−𝜑₁₆+𝜑₁₄−𝜑₁₇−𝜑₁₆sn²𝜔₃𝜉,𝑘₆,𝑢₂₈=𝐴𝑥,𝑦,𝑡³𝜑₁₈+𝜑₁₉𝐵₃+𝐴₃𝜑₁₈−𝜑₁₉𝐵₃cn−𝛼𝐴₃𝐵₃/2𝜉,𝑘₇𝐴₃+𝐵₃+𝐴₃−𝐵₃cn−𝛼𝐴₃𝐵₃𝑢/2𝜉,𝑘₂₉,=𝐴𝑥,𝑦,𝑡³𝜑₁₈+𝜑₁₉𝐵₃−𝐴₃𝜑₁₈−𝜑₁₉𝐵₃cn−𝛼𝐴₃𝐵₃/2𝜉,𝑘₇𝐴₃+𝐵₃−𝐴₃−𝐵₃cn−𝛼𝐴₃𝐵₃/2𝜉,𝑘₇, (3.44)(3.45)(3.46)(3.47)(3.48)(3.49) where 𝐴₂=𝜑₁₁−𝑐₃+𝑐₃2²−(𝑐₃−𝑐₃)²4,𝐵₂=𝜑₁₀−𝑐₃+𝑐₃2²−𝑐₃−𝑐₃²4,𝐴₃=𝜑₁₉−𝑐₄+𝑐₄2²−(𝑐₄−𝑐₄)²4,𝐵₃=𝜑₁₈−𝑐₄+𝑐₄2²−(𝑐₄−𝑐₄)²4,𝑘₅=(𝜑₁₁−𝜑₁₀)²−(𝐴₂−𝐵₂)²4𝐴₂𝐵₂,𝑘₆=𝜑₁₇−𝜑₁₆𝜑₁₅−𝜑₁₄𝜑₁₇−𝜑₁₅𝜑₁₆−𝜑₁₄,𝜔₃=−𝛼𝜑₁₇−𝜑₁₅𝜑₁₆−𝜑₁₄2√2,𝑘₇=(𝜑₁₉−𝜑₁₈)²−(𝐴₃−𝐵₃)²4𝐴₃𝐵₃,𝜑^∗₉=−𝛼𝜑₉−4𝛼𝛽−3𝛼²𝜑²₉2𝛼,𝜃=−4𝛽𝛼+3𝜑²₉+3𝜑₉4𝛽𝛼−3𝜑²₉,𝜇=2𝜑₉−24𝛽𝛼−3𝜑²₉,𝑞=4𝜑²₉+4𝜑₉4𝛽𝛼−3𝜑²₉, (3.50)𝑐₃, 𝑐₃, 𝑐₄ and 𝑐₄ are complex numbers.
And two solitary wave solutions 𝑢₃₀𝑥,𝑦,𝑡=𝜑₉+2𝛽−6𝛼𝜑²₉2𝛼𝜑₉+2𝛼𝛽−2𝛼²𝜑²₉cosh−𝛽+3𝛼𝜑²₉𝜉,𝑢₃₁𝑥,𝑦,𝑡=𝜑₉+2𝛽−6𝛼𝜑²₉2𝛼𝜑₉−2𝛼𝛽−2𝛼²𝜑²₉cosh−𝛽+3𝛼𝜑²₉𝜉. (3.51)(3.52)
(3) If 𝑔=−𝑔₀, we get two periodic wave solutions as follows: 𝑢₃₂=𝐴𝑥,𝑦,𝑡⁴𝜑₂₂+𝜑₂₃𝐵₄+𝐴₄𝜑₂₂−𝜑₂₃𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈𝐴₄+𝐵₄+𝐴₄−𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈𝑢₃₃,=𝐴𝑥,𝑦,𝑡⁵𝜑₂₄+𝜑₂₅𝐵₅+𝐴₅𝜑₂₄−𝜑₂₅𝐵₅cn−𝛼𝐴₅𝐵₅/2𝜉,𝑘₉𝐴₅+𝐵₅+𝐴₅−𝐵₅cn−𝛼𝐴₅𝐵₅/2𝜉,𝑘₉, (3.53)(3.54) where 𝐴₄=𝜑₂₃−𝑐₅+𝑐₅2²−(𝑐₅−𝑐₅)²4,𝐵₄=𝜑₂₂−𝑐₅+𝑐₅2²−(𝑐₅−𝑐₅)²4,𝐴₅=𝜑₂₅−𝑐₆+𝑐₆2²−(𝑐₆−𝑐₆)²4,𝐵₆=𝜑₂₄−𝑐₆+𝑐₆2²−(𝑐₆−𝑐₆)²4,𝑘₈=(𝜑₂₃−𝜑₂₂)²−(𝐴₄−𝐵₄)²4𝐴₄𝐵₄,𝑘₉=(𝜑₂₅−𝜑₂₄)²−(𝐴₅−𝐵₅)²4𝐴₅𝐵₅, (3.55)𝑐₅, 𝑐₅, 𝑐₆ and 𝑐₆ are complex numbers.
And a solitary wave solution 𝑢₃₄𝑥,𝑦,𝑡=3𝛽𝛼9+2𝛽𝜉²−9+6𝛽𝜉². (3.56)

Proof. (1) If 𝑔=0, we set 𝜑₂>2𝛽𝛼,𝛽𝛼<𝜑₆<2𝛽𝛼. (3.57)
(i) From the phase portrait, we see that there are a closed orbit Γ₁ passing the points (𝜑₁,0) and (𝜑₂,0). In (𝜑,𝜙) plane, the expressions of the closed orbits are given as 𝜙=±−𝛼2𝜑₂−𝜑𝜑−𝜑₁𝜑−𝑐₂𝜑−𝑐₂, (3.58) where 𝜑₁=−𝜑₂, 𝑐₂=𝑖𝜑²₂−2𝛽/𝛼 and 𝑐₂=−𝑖𝜑²₂−2𝛽/𝛼.
Substituting (3.58) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the orbit Γ₁, we have ±^𝜑𝜑₁1𝜑₂−𝑠𝑠−𝜑₁𝑠−𝑐₂𝑠−𝑐₂𝑑𝑠=−𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑₂1𝜑₂−𝑠𝑠−𝜑₁𝑠−𝑐₂𝑠−𝑐₂𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.59)
From (3.59) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we obtain the periodic wave solutions 𝑢₁₉(𝑥,𝑦,𝑡) as (3.38) and 𝑢₂₀(𝑥,𝑦,𝑡) as (3.39).
(ii) From the phase portrait, we see that there are two closed orbits Γ₂ and Γ₃ passing the points (𝜑₃,0), (𝜑₄,0), (𝜑₅,0), and (𝜑₆,0). In (𝜑,𝜙) plane, the expressions of the closed orbits are given as 𝜙=±−𝛼2𝜑−𝜑₃𝜑−𝜑₄𝜑−𝜑₅𝜑₆−𝜑, (3.60) where 𝜑₃=−𝜑₆, 𝜑₄=−2𝛽/𝛼−𝜑²₆ and 𝜑₅=2𝛽/𝛼−𝜑²₆.
Substituting (3.60) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₂ and Γ₃, we have ±^𝜑𝜑₃1𝜑₆−𝑠𝜑₅−𝑠𝜑₄−𝑠𝑠−𝜑₃𝑑𝑠=−𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑₆1𝜑₆−𝑠𝑠−𝜑₅𝑠−𝜑₄𝑠−𝜑₃𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.61)
From (3.61) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we obtain the periodic wave solutions 𝑢₂₁(𝑥,𝑦,𝑡) as (3.40) and 𝑢₂₂(𝑥,𝑦,𝑡) as (3.41).
(iii) From the phase portrait, we see that there are two symmetric homoclinic orbits Γ₄ and Γ₅ connected at the saddle point (0,0). In (𝜑,𝜙) plane, the expressions of the homoclinic orbits are given as 𝜙=±𝜑−𝛼2𝜑−𝜑₇𝜑₈−𝜑, (3.62) where 𝜑₇=−2𝛽/𝛼 and 𝜑₈=2𝛽/𝛼.
Substituting (3.62) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along the orbits Γ₄ and Γ₅, we have ±^𝜑𝜑₇1𝑠𝑠−𝜑₇𝜑₈−𝑠𝑑𝑠=−𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑₈1𝑠𝑠−𝜑₇𝜑₈−𝑠𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.63)
From (3.63) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we obtain the solitary wave solutions 𝑢_±23(𝑥,𝑦,𝑡) as (3.43).
(2) If −𝑔₀<𝑔<0, we set the middle solution of 𝑓(𝜑)=0 be 𝜑₉(0<𝜑₉<𝛽/3𝛼), then we can get another two solutions of 𝑓(𝜑)=0 as follows: 𝜑^∗₉=−𝛼𝜑₉−4𝛼𝛽−3𝛼²𝜑²₉,𝜑2𝛼_⋆9=−𝛼𝜑₉+4𝛼𝛽−3𝛼²𝜑²₉.2𝛼 (3.64)
(i) From the phase portrait, we see that there are a closed orbit Γ₆ passing the points (𝜑₁₀,0) and (𝜑₁₁,0). In (𝜑,𝜙) plane, the expressions of the closed orbits are given as 𝜙=±−𝛼2𝜑₁₁−𝜑𝜑−𝜑₁₀𝜑−𝑐₃𝜑−𝑐₃, (3.65) where 𝜑₁₂<𝜑₁₀<𝜑^⋆₉<𝜑₁₁<𝜑₁₃, 𝑐₃ and 𝑐₃ are conjugate complex numbers.
Substituting (3.37) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₆, we have ±^𝜑𝜑₁₀1𝜑₁₁−𝑠𝑠−𝜑₁₀𝑠−𝑐₃𝑠−𝑐₃𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.66)
From (3.66) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic wave solution 𝑢₂₄(𝑥,𝑦,𝑡) as (3.44).
(ii) From the phase portrait, we note that there is a special orbit Γ₇, which has the same Hamiltonian with that of (𝜑^∗₉,0). In (𝜑,𝜙) plane, the expressions of the orbits are given as 𝜙=±−𝛼2(𝜑−𝜑^∗₉)²𝜑−𝜑₁₂𝜑₁₃−𝜑, (3.67) where 𝜑₁₂=𝛼𝜑₉+4𝛼𝛽−3𝛼²𝜑²₉+2𝛼𝜑₉𝛼𝜑₉−4𝛼𝛽−3𝛼²𝜑²₉,𝜑2𝛼₁₃=𝛼𝜑₉+4𝛼𝛽−3𝛼²𝜑²₉−2𝛼𝜑₉𝛼𝜑₉−4𝛼𝛽−3𝛼²𝜑²₉.2𝛼 (3.68)
Substituting (3.67) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₇, it follows that ±^𝜑𝜑₁₂1𝜑₁₃−𝑠(𝑠−𝜑^∗₉)²𝑠−𝜑₁₂𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.69)
From (3.69) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic wave solution 𝑢₂₅(𝑥,𝑦,𝑡) as (3.45).
(iii) From the phase portrait, we note that there are two closed orbits Γ₈ and Γ₉ passing the points (𝜑₁₄,0), (𝜑₁₅,0), (𝜑₁₆,0), and (𝜑₁₇,0). In (𝜑,𝜙) plane, the expressions of the orbits are given as 𝜙=±−𝛼2𝜑−𝜑₁₄𝜑−𝜑₁₅𝜑−𝜑₁₆𝜑₁₇−𝜑, (3.70) where 𝜑₂₀<𝜑₁₄<𝜑₁₂<𝜑₁₀<𝜑^⋆₉<𝜑₁₁<𝜑₁₃<𝜑₁₅<𝜑₉<𝜑₁₆<𝜑^∗₉<𝜑₁₇<𝜑₂₁.
Substituting (3.70) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₈ and Γ₉, we have ±^𝜑𝜑₁₄1𝜑₁₇−𝑠𝜑₁₆−𝑠𝜑₁₅−𝑠𝑠−𝜑₁₄𝑑𝑠=−𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑₁₇1𝜑₁₇−𝑠𝑠−𝜑₁₆𝑠−𝜑₁₅𝑠−𝜑₁₄𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.71)
From (3.71) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get two periodic wave solutions 𝑢₂₆(𝑥,𝑦,𝑡) as (3.46) and 𝑢₂₇(𝑥,𝑦,𝑡) as (3.47).
(iv) From the phase portrait, we note that there is a special orbit Γ₁₀ passing the points (𝜑₁₈,0) and (𝜑₁₉,0). In (𝜑,𝜙) plane, the expressions of the orbit are given as 𝜙=±−𝛼2𝜑₁₉−𝜑𝜑−𝜑₁₈𝜑−𝑐₄𝜑−𝑐₄, (3.72) where 𝜑₁₈<𝜑₂₀<𝜑₂₁<𝜑₁₉, 𝑐₄ and 𝑐₄ are conjugate complex numbers.
Substituting (3.72) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating it along Γ₁₀, we have ±^𝜑𝜑₁₈1𝜑₁₉−𝑠𝑠−𝜑₁₈𝑠−𝑐₄𝑠−𝑐₄𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.73)
From (3.73) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic wave solution 𝑢₂₈(𝑥,𝑦,𝑡) as (3.48).
If 𝜑(𝜉) is a traveling wave solution, then 𝜑(𝜉+𝑞) is a traveling wave solution too. Taking 𝑞=2𝐾 and noting that cn(𝑢+2𝐾)=−cn𝑢, we get a periodic wave solution 𝑢₂₉(𝑥,𝑦,𝑡) as (3.49).
(v) From the phase portrait, we note that there are two homoclinic orbits Γ₁₁ and Γ₁₂ connected at the saddle point (𝜑₉,0). In (𝜑,𝜙) plane, the expressions of the orbits are given as 𝜙=±−𝛼2(𝜑−𝜑₉)²𝜑−𝜑₂₀𝜑₂₁−𝜑, (3.74) where 𝜑₂₀=−𝛼𝜑₉+2𝛼𝛽−2𝛼²𝜑²₉𝛼,𝜑₂₁=−𝛼𝜑₉−2𝛼𝛽−2𝛼²𝜑²₉𝛼. (3.75)
Substituting (3.74) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₁₁ and Γ₁₂, it follows that ±^𝜑𝜑₂₀1(𝑠−𝜑₉)²𝑠−𝜑₂₀𝜑₂₁−𝑠𝑑𝑠=−𝛼2^𝜉₀±𝑑𝑠,_𝜑𝜑₂₁1(𝑠−𝜑₉)²𝑠−𝜑₂₀𝜑₂₁−𝑠𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.76)
From (3.76) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get two solitary wave solutions 𝑢₃₀(𝑥,𝑦,𝑡) as (3.51) and 𝑢₃₁(𝑥,𝑦,𝑡) as (3.52).
(3) If 𝑔=−𝑔₀, we will consider two kinds of orbits.
(i) From the phase portrait, we note that there is a closed orbit Γ₁₃ passing the points (𝜑₂₂,0) and (𝜑₂₃,0). In (𝜑,𝜙) plane, the expressions of the orbit are given as 𝜙=±−𝛼2𝜑₂₃−𝜑𝜑−𝜑₂₂𝜑−𝑐₅𝜑−𝑐₅, (3.77) where −3𝛽/𝛼<𝜑₂₂<−2𝛽/3𝛼<𝜑₂₃<𝛽/3𝛼, 𝑐₅ and 𝑐₅ are conjugate complex numbers.
Substituting (3.77) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating it along Γ₁₃, we have ±^𝜑𝜑₂₂1𝜑₂₃−𝑠𝑠−𝜑₂₂𝑠−𝑐₅𝑠−𝑐₅𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.78)
From (3.78) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic wave solutions 𝑢₃₂(𝑥,𝑦,𝑡) as (3.53).
(ii) From the phase portrait, we note that there is a closed orbit Γ₁₄ passing the points (𝜑₂₄,0) and (𝜑₂₅,0). In (𝜑,𝜙) plane, the expressions of the orbit are given as 𝜙=±−𝛼2𝜑₂₅−𝜑𝜑−𝜑₂₄𝜑−𝑐₆𝜑−𝑐₆, (3.79) where 𝜑₂₄<−3𝛽/𝛼<𝜑₂₂<−2𝛽/3𝛼<𝜑₂₃<𝛽/3𝛼<𝜑₂₅, 𝑐₆ and 𝑐₆ are conjugate complex numbers.
Substituting (3.79) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₁₄, we have ±^𝜑𝜑₂₄1𝜑₂₅−𝑠𝑠−𝜑₂₄𝑠−𝑐₆𝑠−𝑐₆𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.80)
From (3.80) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a periodic wave solutions 𝑢₃₃(𝑥,𝑦,𝑡) as (3.54).
(iii) From the phase portrait, we see that there is a homoclinic orbit Γ₁₅, which passes the degenerate saddle point (𝜑^∗₊,0). In (𝜑,𝜙) plane, the expressions of the homoclinic orbit are given as 𝜙=±−𝛼2(𝜑^∗₊−𝜑)³𝜑−𝜑₂₆, (3.81) where 𝜑₂₆=−3𝛽𝛼. (3.82)
Substituting (3.81) into 𝑑𝜑/𝑑𝜉=𝜙 and integrating them along Γ₁₅, it follows that ±^𝜑𝜑₂₆1𝑠−𝜑^∗₊𝑠−𝜑^∗₊𝜑₂₆−𝑠𝑑𝑠=−𝛼2^𝜉₀𝑑𝑠. (3.83)
From (3.83) and noting that 𝑢=𝜑(𝜉) and 𝜉=𝑥+𝑦−𝑐𝑡, we get a solitary wave solution 𝑢₃₄(𝑥,𝑦,𝑡) as (3.56).
Thus, the derivation of Proposition 3.2 has been finished.

Proposition 3.3. For these solutions, the following are their relations.
(1) When 𝜑₆ tends to 2𝛽/𝛼, the periodic blow-up wave solutions 𝑢_3± and 𝑢_4± tend to periodic blow-up wave solutions 𝑢_1± and 𝑢_2±, that is, lim_{𝜑6→2𝛽/𝛼}𝑢_3±𝑥,𝑦,𝑡=𝑢_1±𝑥,𝑦,𝑡,lim_{𝜑6→2𝛽/𝛼}𝑢_4±𝑥,𝑦,𝑡=𝑢_2±𝑥,𝑦,𝑡. (3.84)
(2) When 𝜑₆ tends to 𝛽/𝛼, the periodic wave solutions 𝑢_5± tend to kink wave solutions 𝑢_6±, that is, lim_{𝜑6→𝛽/𝛼}𝑢_5±𝑥,𝑦,𝑡=𝑢_6±𝑥,𝑦,𝑡. (3.85)
(3) When 𝜑₆ tends to 𝛽/𝛼, the periodic blow-up wave solutions 𝑢_3± tend to unbounded wave solutions 𝑢_7±, that is, lim_{𝜑6→𝛽/𝛼}𝑢_3±𝑥,𝑦,𝑡=𝑢_7±𝑥,𝑦,𝑡. (3.86)
(4) When 𝜑₁₃ tends to 𝜑₉, the periodic blow-up wave solution 𝑢₉ tends to periodic blow-up wave solution 𝑢₈₋, that is, lim_{𝜑13→𝜑9}𝑢₉𝑥,𝑦,𝑡=𝑢₈₋𝑥,𝑦,𝑡. (3.87)
(5) When 𝜑₁₃ tends to 𝜑₉, the periodic blow-up wave solution 𝑢₁₀ tends to periodic blow-up wave solution 𝑢₈₊, that is, lim_{𝜑13→𝜑9}𝑢₁₀𝑥,𝑦,𝑡=𝑢₈₊𝑥,𝑦,𝑡. (3.88)
(6) When 𝜑₁₃ tends to 𝜑₇, the periodic wave solution 𝑢₁₁ tends to solitary wave solution 𝑢₁₃, that is, lim_{𝜑13→𝜑7}𝑢₁₁𝑥,𝑦,𝑡=𝑢₁₃𝑥,𝑦,𝑡. (3.89)
(7) When 𝜑₁₃ tends to 𝜑₇, the periodic blow-up wave solution 𝑢₁₀ tends to unbounded wave solution 𝑢₁₄, that is, lim_{𝜑13→𝜑7}𝑢₁₀𝑥,𝑦,𝑡=𝑢₁₄𝑥,𝑦,𝑡. (3.90)
(8) When 𝜑₁₈ tends to 𝛽/3𝛼, the periodic blow-up wave solution 𝑢₁₇ tends to blow-up wave solution 𝑢₁₆, that is, lim_{𝜑18→𝛽/3𝛼}𝑢₁₇𝑥,𝑦,𝑡=𝑢₁₆𝑥,𝑦,𝑡. (3.91)
(9) When 𝜑₂ tends to 2𝛽/𝛼, the periodic wave solutions 𝑢₁₉ and 𝑢₂₀ tend to solitary wave solutions 𝑢_±23, that is, 𝜑lim_{2→2𝛽/𝛼}𝑢₁₉𝑥,𝑦,𝑡=𝑢₋₂₃𝜑𝑥,𝑦,𝑡,lim_{2→2𝛽/𝛼}𝑢₂₀𝑥,𝑦,𝑡=𝑢₊₂₃𝑥,𝑦,𝑡. (3.92)
(10) When 𝜑₆ tend to 2𝛽/𝛼, the periodic wave solutions 𝑢₂₁ and 𝑢₂₂ tends to solitary wave solutions 𝑢_±23, that is, 𝜑lim_{6→2𝛽/𝛼}𝑢₂₁𝑥,𝑦,𝑡=𝑢₋₂₃𝜑𝑥,𝑦,𝑡,lim_{6→2𝛽/𝛼}𝑢₂₂𝑥,𝑦,𝑡=𝑢₊₂₃𝑥,𝑦,𝑡. (3.93)
(11) When 𝜑₁₁ tends to 𝜑₁₃, the periodic wave solution 𝑢₂₄ tends to periodic wave solution 𝑢₂₅, that is, 𝜑lim_11→𝜑₁₃𝑢₂₄𝑥,𝑦,𝑡=𝑢₂₅𝑥,𝑦,𝑡. (3.94)
(12) When 𝜑₁₇ tends to 𝜑^∗₉, the periodic wave solution 𝑢₂₆ tends to periodic wave solution 𝑢₂₅, that is, 𝜑lim_17→𝜑^∗₉𝑢₂₆𝑥,𝑦,𝑡=𝑢₂₅𝑥,𝑦,𝑡. (3.95)
(13) When 𝜑₁₇ tends to 𝜑₂₁, the periodic wave solutions 𝑢₂₆ and 𝑢₂₇ tend to solitary wave solutions 𝑢₃₀ and 𝑢₃₁, that is, 𝜑lim_17→𝜑₂₁𝑢₂₆𝑥,𝑦,𝑡=𝑢₃₀𝜑𝑥,𝑦,𝑡,lim_17→𝜑₂₁𝑢₂₇𝑥,𝑦,𝑡=𝑢₃₁𝑥,𝑦,𝑡. (3.96)
(14) When 𝜑₁₉ tends to 𝜑₂₁, the periodic wave solutions 𝑢₂₈ and 𝑢₂₉ tend to solitary wave solutions 𝑢₃₀ and 𝑢₃₁, that is, 𝜑lim_19→𝜑₂₁𝑢₂₈𝑥,𝑦,𝑡=𝑢₃₀𝜑𝑥,𝑦,𝑡,lim_19→𝜑₂₁𝑢₂₉𝑥,𝑦,𝑡=𝑢₃₁𝑥,𝑦,𝑡. (3.97)
(15) When 𝜑₂₂ and 𝜑₂₄ tend to 3𝛽/𝛼, the periodic wave solutions 𝑢₃₂ and 𝑢₃₃ tend to solitary wave solution 𝑢₃₄, that is, 𝜑lim_{22→3𝛽/𝛼}𝑢₃₂𝜑𝑥,𝑦,𝑡=lim_{24→3𝛽/𝛼}𝑢₃₃𝑥,𝑦,𝑡=𝑢₃₄𝑥,𝑦,𝑡. (3.98)

Proof. (1) Letting 𝜑₆→2𝛽/𝛼, it follows that 𝜑₅→0 and sn(𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆)→sn(𝛽𝜉,0)=sin𝛽𝜉, and we have lim_{𝜑6→2𝛽/𝛼}𝑢_3±𝑥,𝑦,𝑡=lim_{𝜑6→2𝛽/𝛼}±𝜑₆sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆=±2𝛽/𝛼sin𝛽𝜉=𝑢_1±𝑥,𝑦,𝑡,lim_{𝜑6→2𝛽/𝛼}𝑢_4±𝑥,𝑦,𝑡=lim_{𝜑6→2𝛽/𝛼}±𝜑²₆−𝜑²₅sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆²1−sn𝜑₆𝛼/2𝜉,𝜑₅/𝜑₆²=±2𝛽/𝛼cos𝛽𝜉=𝑢_2±𝑥,𝑦,𝑡. (3.99)
Therefore, the periodic blow-up solutions 𝑢_1±(𝑥,𝑦,𝑡) are the limit of the elliptic function periodic blow-up solutions 𝑢_3±(𝑥,𝑦,𝑡) and the periodic blow-up solutions 𝑢_2±(𝑥,𝑦,𝑡) are the limit of the elliptic function periodic blow-up solutions 𝑢_4±(𝑥,𝑦,𝑡).
(2) Letting 𝜑₆→𝛽/𝛼, it follows that 𝜑₅→𝛽/𝛼 and sn(𝜑₅𝛼/2𝜉,𝜑₅/𝜑₆)→sn(𝛽/2𝜉,1)=tanh𝛽/2𝜉, and we have lim_{𝜑6→𝛽/𝛼}𝑢_5±𝑥,𝑦,𝑡=lim_{𝜑6→𝛽/𝛼}±𝜑₅sn𝜑₆𝛼2𝜑𝜉,⁵𝜑₆=±𝛽𝛼tanh𝛽2𝜉=𝑢_6±𝑥,𝑦,𝑡. (3.100)
Therefore, the kink wave solutions 𝑢_6±(𝑥,𝑦,𝑡) are the limit of the elliptic function periodic solutions 𝑢_5±(𝑥,𝑦,𝑡).
(4) Letting 𝜑₁₃→𝜑₉, it follows that 𝜑₁₂→𝜑^∗₇, 𝜑₁₁→𝜑^∗₇, 𝜑₁₀→𝜑₈ and sn(𝜔₁𝜉,𝑘₁)→sn(𝜔₁𝜉,0)=sin𝜔₁𝜉, and we have lim_{𝜑13→𝜑9}𝑢₉𝑥,𝑦,𝑡=lim_{𝜑13→𝜑9}𝜑₁₀𝜑₁₁−𝜑₁₃+𝜑₁₁𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)²𝜑₁₁−𝜑₁₃+𝜑₁₃−𝜑₁₀sn(𝜔₁𝜉,𝑘₁)²=𝜑^∗₇−2𝛾₁𝛿₁−√𝜂₁cos𝛼𝛾₁/2𝜉=𝑢₈₋𝑥,𝑦,𝑡. (3.101)
Therefore, the trigonometric function periodic blow-up wave solution 𝑢₈₋(𝑥,𝑦,𝑡) is the limit of the elliptic function periodic blow-up solution 𝑢₉(𝑥,𝑦,𝑡).
(9) Letting 𝜑₂→2𝛽/𝛼, it follows that 𝜑₁→−2𝛽/𝛼,  𝛽−𝛼𝜑²₁→−𝛽 and −𝜑₁𝛼/(−2𝛽+2𝛼𝜑²₁)→1, and we have 𝜑lim_{2→2𝛽/𝛼}𝑢₁₉𝜑𝑥,𝑦,𝑡=lim_{2→2𝛽/𝛼}𝜑₁cn𝛽−𝛼𝜑²₁𝜉,−𝜑₁𝛼−2𝛽+2𝛼𝜑²₁=−2𝛽𝛼sech−𝛽𝜉=𝑢₋₂₃𝑥,𝑦,𝑡. (3.102)
Therefore, the solitary wave solution 𝑢₋₂₃(𝑥,𝑦,𝑡) is the limit of the elliptic function periodic wave solution 𝑢₁₉(𝑥,𝑦,𝑡).
(15) Letting 𝜑₂₂→−3𝛽/𝛼, it follows that 𝜑₂₃→𝛽/3𝛼,𝑐₅→𝛽/3𝛼,𝑐₅→𝛽/3𝛼,  𝐴₄→0,𝐵₄→4𝛽/3𝛼 and cn(−𝛼𝐴₄𝐵₄/2,𝑘₈)→cn(0,𝑘₈)=1, and we have 𝜑lim_{22→−3𝛽/𝛼}𝑢₃₂𝑥,𝑦,𝑡=lim_{𝜑22→−3𝛽/𝛼}𝐴₄𝜑₂₂+𝜑₂₃𝐵₄+𝐴₄𝜑₂₂−𝜑₂₃𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈𝐴₄+𝐵₄+𝐴₄−𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈=lim_𝐴4→0𝐴₄𝜑₂₂+𝜑₂₃𝐵₄+𝐴₄𝜑₂₂−𝜑₂₃𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈𝐴₄+𝐵₄+𝐴₄−𝐵₄cn−𝛼𝐴₄𝐵₄/2𝜉,𝑘₈=lim_𝐴4→02−2𝛼𝐴₄𝐵₄𝜑₂₂+𝜑₂₂𝜒₁+𝐵₄𝐴₄𝜑₂₂−𝐵₄𝜑₂₃𝛼𝜉𝜒₂𝜒₃2−2𝛼𝐴₄𝐵₄1+𝜒₁+𝐵₄𝐴₄−𝐵₄𝛼𝜉𝜒₂𝜒₃=3𝛽𝛼9+2𝛽𝜉²−9+6𝛽𝜉²=𝑢₃₄𝑥,𝑦,𝑡, (3.103) where  𝜒₁=cn(−𝛼𝐴₄𝐵₄/2,𝑘₈),𝜒₂=dn(−𝛼𝐴₄𝐵₄/2,𝑘₈),𝜒₃=sn(−𝛼𝐴₄𝐵₄/2,𝑘₈).
Therefore, the fractional function solitary wave solution 𝑢₃₄(𝑥,𝑦,𝑡) is the limit of the elliptic function periodic wave solution 𝑢₃₂(𝑥,𝑦,𝑡).
Similarly, we can derive the others case. This has proved Proposition 3.3.

Finally, we will show that the periodic wave solution 𝑢₅₊(𝑥,𝑦,𝑡) evolutes into the kink wave solution 𝑢₆₊(𝑥,𝑦,𝑡) when 𝜑₆→𝛽/𝛼. We take some suitable choices of the parameters, such as 𝑎=1,𝑏=1,𝑟=5𝑐=2, (3.104) as an illustrative sample and draw their plots (see Figure 3).

(a)

(b)

(c)

Figure 3 

The periodic wave solution 𝑢5+(𝑥,𝑦,𝑡) evolute into the kink wave solution 𝑢6+(𝑥,𝑦,𝑡) with the conditions (3.104). (a) 𝜑6 = 2.2; (b) 𝜑6 = 2.01; (c) 𝜑6 = 2.

Remark 3.4. One may find that we only consider the case when 𝑔≥0 in Proposition 3.1 (when 𝑔≤0 in Proposition 3.2). In fact, we may get exactly the same solutions in the opposite case.

4. Conclusion

In this paper, we have obtained many traveling wave solutions for the generalized KP-BBM equation (1.4) by employing the bifurcation method and qualitative theory of dynamical systems. The traveling wave solutions have been given in Propositions 3.1 and 3.2. On the other hand, in Proposition 3.3, we prove that the periodic wave solutions, kink wave solutions, blow-up wave solutions, unbounded solutions, and solitary wave solutions can be obtained from the limits of the smooth periodic wave solutions or periodic blow-up solutions. The method can be applied to many other nonlinear evolution equations and we believe that many new results wait for further discovery by this method.

Acknowledgment

This research is supported by the National Natural Science Foundation of China (no. 11171115) and the Natural Science Foundation of Yunnan Province (no. 2010ZC154).