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

Volume 2012, Article ID 363879,25pages doi:10.1155/2012/363879

Research Article

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

Ming Song

and Zhengrong Liu

1Department of Mathematics, South China University of Technology, Guangzhou 510640, China

2Department of Mathematics, Faculty of Sciences, Yuxi Normal University, Yuxi 653100, China

Correspondence should be addressed to Ming Song,songming12 15@163.com Received 17 April 2012; Accepted 15 May 2012

Academic Editor: Junjie Wei

Copyrightq2012 M. Song and Z. Liu. 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.

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-MahonyBBMequation1,

u_tu_xuu_x−u_xxt0, 1.1

has been proposed as a model for propagation of long waves where nonlinear dispersion is incorporated.

The Kadomtsov-PetviashviliKPequation2is given by

utauxuxxx_xuyy0, 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 thex-direction.

Wazwaz 3 presented the Kadomtsov-Petviashvili-Benjamin-Bona-Mahony KP- BBMequation

utuxau²_x−buxxt_xruyy0, 1.3

and the generalized KP-BBM equation

utuxau³_x−buxxt_xruyy0. 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. Abdou5used 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.6obtained exact solitary wave solutions of1.3by using bifurcation method of dynamical systems.

The aim of this paper is to study the traveling wave solutions and their phase portraits for1.4by using the bifurcation method and qualitative theory of dynamical systems6–15.

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 of1.4are given under different parameter conditions. InSection 3, we present our main results and their proofs. A short conclusion will be given inSection 4.

2. Phase Portraits

To derive our results, we give some preliminaries in this section. For given positive constant wave speedc, substitutinguϕξwithξxy−ctinto the generalized KP-BBM equation 1.4, it follows that

−cϕϕ−aϕ³bcϕrϕ0. 2.1

Integrating2.1twice and letting the first integral constant be zero, we have

r−c1ϕ−aϕ³bcϕg1, 2.2

whereg₁is the second constant of integration.

Lettingφϕ, we get the following planar system:

dϕ dξ φ, dφ

dξ αϕ³−βϕg,

2.3

whereαa/bc,β r1−c/bcandg g₁/bc.

Obviously, the system2.3is a Hamiltonian system with Hamiltonian function

H ϕ, φ

φ²−α

2ϕ⁴βϕ²−2gϕh, 2.4

wherehis Hamiltonian.

Now we consider the phase portraits of system2.3. Set

f₀ ϕ

αϕ³−βϕ, f

ϕ

αϕ³−βϕg. 2.5

Obviously,f₀ϕhas three zero points,ϕ₋,ϕ₀andϕ, which are given as follows:

ϕ₋−

β

α, ϕ₀0, ϕ

β

α. 2.6

It is easy to obtain two extreme points off₀ϕas follows:

ϕ^∗₋ −

β

3α, ϕ^∗

β

3α. 2.7

Letting

g0f0

ϕ^∗₋f0

ϕ^∗ 2β 3

β

3α, 2.8

then it is easily seen thatg0is the extreme values off0ϕ.

Letϕi,0be one of the singular points of system2.3, then the characteristic values of the linearized system of system2.3at the singular pointsϕi,0are

λ_± ± f

ϕi

. 2.9

From the qualitative theory of dynamical systems, we therefore know that iiffϕi>0,ϕi,0is a saddle point.

iiiffϕi<0,ϕi,0is a center point.

iiiiffϕi 0,ϕi,0is a degenerate saddle point.

Therefore, we obtain the phase portraits of system2.3in Figures1and2.

φ

ϕ

g <−g0 g=−g0 −g0< g <0 g=0

Γ1

φ

ϕ

0< g < g0 g=g0

Γ3

Γ4

Γ6

Γ5

Γ2

Γ7

φ

ϕ φ

ϕ ϕ

φ φ

ϕ ϕ

Γ8

Γ10Γ13

Γ14

Γ11

Γ12

Γ9

Γ17Γ15 Γ16Γ18

g > g0

ϕ1

ϕ3 ϕ4 ϕ5 ϕ6

ϕ7

ϕ8

ϕ10 ϕ13

ϕ14

ϕ11 ϕ12

ϕ9

ϕ17

ϕ15

ϕ16 ϕ18

ϕ2

ϕ− ϕ+

ϕ^∗₇ ϕ^∗+

ϕ^∗₇

Figure 1: The phase portraits of system2.3whenα >0,β >0.

g <−g0 g=−g0 −g0< g <0

g=0 0< g < g0 g=g0 g > g0

φ

ϕ

φ φ

ϕ ϕ ϕ

φ φ φ φ

ϕ ϕ ϕ

∼ ϕ24

∼ ϕ22

∼

ϕ26 ϕ∼23

∼ ϕ25

∼ ϕ18

∼ ϕ20

∼ ϕ14

∼ ϕ12 ϕ∼10

∼ ϕ11

∼ ϕ16

∼

ϕ15 ϕ∼17

∼ ϕ21

∼

ϕ9 ϕ∼19

∼

ϕ1 ϕ∼2

∼ ϕ3 ϕ∼4

∼ ϕ8

∼ ϕ5 ϕ∼6

∼ ϕ7

∼ ϕ13

Γ∼13

Γ∼14 ∼Γ15

Γ∼6

∼Γ7

Γ∼8

∼Γ9

Γ∼10

Γ∼11 ∼Γ12

∼Γ1

Γ∼2 ∼Γ3

∼Γ4 ∼Γ5

ϕ^∗₊

Figure 2: The phase portraits of system2.3whenα <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 whenα >0 andβ >0.

1Ifg0, we get eight periodic blow-up wave solutions

u_1±

x, y, t ±

2β

α csc

βξ, 3.1

u_2±

x, y, t ±

2β

α sec

βξ, 3.2

u_3±

x, y, t

± ϕ6

sn ϕ₆

α/2ξ, ϕ₅/ϕ₆

, 3.3

u_4±

x, y, t ±

ϕ²₆−ϕ²₅ sn

ϕ6

α/2ξ, ϕ5/ϕ6

2

1− sn

ϕ₆

α/2ξ, ϕ₅/ϕ_{6 2} , 3.4

two periodic wave solutions

u_5±

x, y, t

±ϕ5sn

ϕ₆ α

2ξ,ϕ₅ ϕ₆

, 3.5

two kink wave solutions

u_6±

x, y, t ±

β αtanh

β

2ξ, 3.6

and two unbounded wave solutions

u_7±

x, y, t ±

β αcoth

β

2ξ. 3.7

2If 0< g < g₀, we get four periodic blow-up wave solutions

u_8±

x, y, t

ϕ^∗₇− 2γ₁ δ1± √η1cos

αγ1/2ξ, 3.8

u₉ x, y, t

ϕ₁₀

ϕ₁₁−ϕ₁₃ ϕ₁₁

ϕ₁₃−ϕ₁₀

snω1ξ, k₁² ϕ11−ϕ13

ϕ13−ϕ10

snω1ξ, k1² ,

u₁₀ x, y, t

ϕ₁₃

ϕ₁₂−ϕ₁₀

−ϕ₁₂

ϕ₁₃−ϕ₁₀

snω1ξ, k₁² ϕ12−ϕ10−

ϕ13−ϕ10

snω1ξ, k1² ,

3.9

where

ϕ^∗₇ 1 2

⎛

⎝−ϕ7

4β α −3ϕ²₇

⎞

⎠, γ₁

−4β3ϕ₇

αϕ₇

4αβ−3α²ϕ²₇

α ,

δ₁−2ϕ72

4αβ−3α²ϕ²₇

α , η₁

4ϕ₇

αϕ₇

4αβ−3α²ϕ²₇

α ,

ω₁

α

ϕ₁₃−ϕ₁₁

ϕ₁₂−ϕ₁₀ 2√

2 , k₁

ϕ12−ϕ11

ϕ13−ϕ10

ϕ₁₃−ϕ₁₁

ϕ₁₂−ϕ₁₀,

3.10

a periodic wave solution

u₁₁ x, y, t

ϕ11

ϕ10−ϕ12

ϕ10

ϕ12−ϕ11

snω1ξ, k1² ϕ₁₀−ϕ₁₂

ϕ₁₂−ϕ₁₁

snω1ξ, k₁² , 3.11 a blow-up wave solution

u12

x, y, t

ϕ7 2

β−3αϕ²₇ 2αϕ₇−

2α

β−αϕ²₇ cosh

3αϕ²₇−βξ

, 3.12

a solitary wave solution

u13

x, y, t

ϕ7 2

β−3αϕ²₇ 2αϕ₇

2α

β−αϕ²₇ cosh

3αϕ²₇−βξ

, 3.13

and an unbounded wave solution

u₁₄ x, y, t

ϕ₇− 2

β−3αϕ²₇ csch

3αϕ²₇−βξ −2αβ6α²ϕ²₇2αϕ₇tanh

3αϕ²₇−βξ/2

. 3.14

3Ifgg₀, we get three blow-up wave solutions

u₁₅ x, y, t

18√ 2−β

3βξ³

√α

18ξ−3βξ³, u16

x, y, t 18√

2β 3βξ³

√α

−18ξ3βξ³,

u₁₇ x, y, t

3β

92βξ²

√α

−96βξ²,

3.15

and a periodic blow-up wave solution

u18

x, y, t

−A1ϕ₁₇B₁ϕ₁₈

A₁ϕ₁₇B₁ϕ₁₈

cn

αA₁B₁/2ξ, k₂

−A1B1 A1B1cn

αA1B1/2ξ, k2

, 3.16

where

A₁

ϕ₁₈− c1c1

2 2

−c1−c1²

4 , B₁

ϕ₁₇−c1c1

2 2

−c1−c1²

4 ,

k2

A1B1²−ϕ18−ϕ17² 4A1B1 ,

3.17

c₁andc₁are conjugate complex numbers.

Proof. 1Ifg0, we will consider three kinds of orbits.

iFrom the phase portrait, we note that there are two special orbitsΓ1andΓ2, which have the same Hamiltonian with that of the center point0,0. Inϕ, φplane, the expressions of these two orbits are given as

φ± α

2ϕ² ϕ−ϕ1

ϕ−ϕ2

, 3.18

whereϕ1−

2β/αandϕ2 2β/α.

Substituting3.18intodϕ/dξ φand integrating them along the two orbitsΓ1and Γ2, it follows that

± _∞

ϕ

1

s

s−ϕ1

s−ϕ2

ds α

2 _ξ

0

ds,

± _ϕ

ϕ2

1

s

s−ϕ1

s−ϕ2

ds α

2 _ξ

0

ds.

3.19

From3.19and noting thatuϕξandξxy−ct, we get four periodic blow-up solutionsu1±x, y, tandu2±x, y, tas3.1and3.2.

iiFrom the phase portrait, we note that there are three special orbitsΓ3,Γ4, andΓ5

passing the pointsϕ3,0,ϕ4,0,ϕ5,0, andϕ6,0. Inϕ, φplane, the expressions of the orbit are given as

φ± α

2

ϕ−ϕ₃

ϕ−ϕ₄

ϕ−ϕ₅ ϕ−ϕ₆

, 3.20

whereϕ₃−ϕ6, ϕ₄−

2β/α−ϕ²₆, ϕ₅

2β/α−ϕ²₆ and

β/α < ϕ₆<

2β/α.

Substituting3.20intodϕ/dξφand integrating them alongΓ3,Γ4, andΓ5, we have

± _∞

ϕ

1

s−ϕ₃

s−ϕ₄

s−ϕ₅

s−ϕ₆ds α

2 _ξ

0

ds,

± _ϕ

ϕ6

1

s−ϕ₃

s−ϕ₄

s−ϕ₅

s−ϕ₆ds α

2 _ξ

0

ds,

± _ϕ

0

1 s−ϕ3

s−ϕ4

s−ϕ5

s−ϕ6

ds α

2 _ξ

0

ds.

3.21

From3.21and noting thatuϕξandξxy−ct, we get four periodic blow-up wave solutionsu3±x, y, t,u4±x, y, tas3.3,3.4and two periodic solutionsu5±x, y, tas 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± α

2ϕ−ϕ₋²ϕ−ϕ². 3.22

Substituting3.22intodϕ/dξφand integrating them along the heteroclinic orbits Γ6andΓ7, it follows that

± _ϕ

0

1

s−ϕ₋

ϕ−sds α

2 _ξ

0

ds,

± _∞

ϕ

1

s−ϕ₋

ϕ−sds α

2 _ξ

0

ds.

3.23

From3.23and noting thatuϕξandξxy−ct, we get two kink wave solutions u6±x, y, tas3.6and two unbounded solutionsu7±x, y, tas3.7.

2If 0< g < g₀, we set the largest solution offϕ 0 beϕ₇

β/3α < ϕ₇ <

β/α, then we can get another two solutions offϕ 0 as follows:

ϕ^∗₇ 1 2

⎛

⎝−ϕ7

4β α −3ϕ²₇

⎞

⎠,

ϕ₇ 1 2

⎛

⎝−ϕ7−

4β α −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 pointϕ^∗₇,0. Inϕ, φ-plane, the expressions of the orbits are given as

φ± α

2ϕ−ϕ^∗₇²

ϕ−ϕ₈ ϕ−ϕ₉

, 3.25

where

ϕ8 1 2

⎛

⎜⎝ϕ7−

4β

α −3ϕ²₇−2

ϕ²₇ϕ7

4β

α −3ϕ²₇

⎞

⎟⎠,

ϕ9 1 2

⎛

⎜⎝ϕ7−

4β

α −3ϕ²₇2

ϕ²₇ϕ7

4β

α −3ϕ²₇

⎞

⎟⎠.

3.26

Substituting3.25intodϕ/dξ φ and integrating them along the orbits, it follows that

± _ϕ

ϕ8

1

s−ϕ^∗₇² s−ϕ8

s−ϕ9

ds α

2 _ξ

0

ds,

± _ϕ

ϕ9

1

s−ϕ^∗₇² s−ϕ8

s−ϕ9

ds α

2 _ξ

0

ds.

3.27

From3.27and noting thatuϕξandξ xy−ct, we get two periodic blow-up wave solutionsu_8±x, y, tas3.8.

iiFrom the phase portrait, we note that there are three special orbitsΓ10,Γ11, andΓ12

passing the pointsϕ10,0,ϕ11,0,ϕ12,0, andϕ13,0. Inϕ, φplane, the expressions of the orbit are given as

φ± α

2

ϕ−ϕ₁₀

ϕ−ϕ₁₁

ϕ−ϕ₁₂

ϕ−ϕ₁₃

, 3.28

whereϕ₈< ϕ₁₀< ϕ₁₄< ϕ₇ < ϕ₁₅< ϕ₁₁< ϕ₁₂< ϕ₇< ϕ₁₃ < ϕ₉.

Substituting3.28intodϕ/dξ φ and integrating them along Γ10,Γ11, andΓ12, we have

_ϕ10

ϕ

1

ϕ13−s

ϕ12−s

ϕ11−s

ϕ10−sds α

2 _ξ

0

ds, _ϕ

ϕ13

1 s−ϕ13

s−ϕ12

s−ϕ11

s−ϕ10

ds α

2 _ξ

0

ds, _ϕ

ϕ11

1

ϕ₁₃−s

ϕ₁₂−s

s−ϕ₁₁

s−ϕ₁₀ds α

2 _ξ

0

ds.

3.29

From3.29and noting thatu ϕξandξ xy−ct, we get two periodic blow- up wave solutionsu9x, y, t,u10x, y, tas3.9and a periodic wave solutionu11x, y, tas 3.11.

iiiFrom the phase portrait, we see that there are a spacial orbitΓ13, which passes the pointϕ14,0, and a homoclinic orbitΓ14 passing the saddle pointϕ7,0. Inϕ, φplane, the expressions of the orbits are given as

φ± α

2ϕ−ϕ₇²

ϕ−ϕ₁₄

ϕ−ϕ₁₅

, 3.30

where

ϕ₁₄ −αϕ7−

2αβ−2α²ϕ²₇

α ,

ϕ₁₅ −αϕ7

2αβ−2α²ϕ²₇

α .

3.31

Substituting3.30intodϕ/dξ φ and integrating them along the orbits, it follows that

± _ϕ

ϕ14

1

s−ϕ₇²

s−ϕ₁₄

s−ϕ₁₅ds α

2 _ξ

0

ds,

± _ϕ

ϕ15

1

s−ϕ₇²

s−ϕ₁₄

s−ϕ₁₅ds α

2 _ξ

0

ds,

± _∞

ϕ

1

s−ϕ₇²

s−ϕ₁₅

s−ϕ₁₅ds α

2 _ξ

0

ds.

3.32

From3.32and noting thatu ϕξandξ xy−ct, we get a blow-up solution u₁₂x, y, tas3.12, a solitary wave solutionu₁₃x, y, tas3.13, and an unbounded wave solutionu₁₄x, y, tas3.14.

3Ifgg0, we will consider two kinds of orbits.

iFrom the phase portrait, we see that there are two orbitsΓ15andΓ16, which have the same Hamiltonian with the degenerate saddle pointϕ^∗,0. Inϕ, φplane, the expressions of these two orbits are given as

φ± α

2ϕ−ϕ^∗3

ϕ−ϕ₁₆

, 3.33

where

ϕ₁₆ −

3β

α. 3.34

Substituting3.33intodϕ/dξφand integrating them along these two orbitsΓ15and Γ16, it follows that

± _∞

ϕ

1

s−ϕ^∗3

s−ϕ₁₆ds α

2 _ξ

0

ds,

± _ϕ

ϕ16

1

s−ϕ^∗3

s−ϕ₁₆ds α

2 _ξ

0

ds.

3.35

From3.35and noting thatuϕξandξxy−ct, we get three blow-up solutions u15x, y, t,u16x, y, t, andu17x, y, tas3.15.

iiFrom the phase portrait, we see that there are two special orbitsΓ17andΓ18passing the pointsϕ17,0andϕ18,0. Inϕ, φplane, the expressions of the orbits are given as

φ± α

2

ϕ−ϕ₁₈

ϕ−ϕ₁₇

ϕ−c₁ ϕ−c₁

, 3.36

whereϕ₁₇< ϕ₁₆< ϕ^∗< ϕ₁₈,c₁andc₁are conjugate complex numbers.

Substituting3.36intodϕ/dξφand integrating them alongΓ17andΓ18, we have

± _ϕ

ϕ18

1 s−ϕ18

s−ϕ17

s−c1s−c1ds α

2 _ξ

0

ds. 3.37

From3.37and noting thatu ϕξandξ xy−ct, we get a periodic blow-up wave solutionsu18x, y, tas3.16.

Thus, we obtain the results given inProposition 3.1.

Proposition 3.2. For given positive constantscandg0,1.4has the following periodic wave solution whenα <0 andβ <0.

1Ifg0, we get four periodic wave solutions

u₁₉ x, y, t

ϕ₁cn

β−αϕ²₁ξ,−ϕ₁

α

−2β2αϕ²₁

, 3.38

u₂₀ x, y, t

ϕ₂cn

β−αϕ²₂ξ,ϕ₂

α

−2β2αϕ²₂

, 3.39

u₂₁ x, y, t

−κ1ϕ6λ1ϕ6sn²ω2ξ, k3

κ₁λ₁sn²ω2ξ, k₃ , 3.40

u₂₂ x, y, t

ϕ²₆−

2ϕ²₆− 2β α

sn²

ϕ₆

−α 2ξ, k₄

, 3.41

where

κ1ϕ6

2β

α −ϕ²₆, λ1ϕ6−

2β α −ϕ²₆,

ω2

√−αϕ₆

−2βαϕ²₆ 2√

2 , k3 λ1

κ₁, k4

2ϕ²₆−2β/α

ϕ₆ ,

3.42

and two solitary wave solutions

u23± x, y, t

±

2β α sech

−βξ. 3.43

2If−g0< g <0, we get six periodic wave solutions

u24

x, y, t

A₂ϕ₁₀ϕ₁₁B₂

A₂ϕ₁₀−ϕ₁₁B₂

cn

−αA2B₂/2ξ, k₅ A2B2 A2−B2cn

−αA2B2/2ξ, k5

, 3.44

u₂₅ x, y, t

ϕ^∗₉ 2θ

−μ√qcos

αθ/2ξ, 3.45 u₂₆

x, y, t ϕ₁₄

ϕ₁₇−ϕ₁₅ ϕ₁₇

ϕ₁₅−ϕ₁₄

sn²ω3ξ, k₆

ϕ17−ϕ15 ϕ15−ϕ14

sn²ω3ξ, k6 , 3.46

u₂₇ x, y, t

ϕ17

−ϕ16ϕ14

−ϕ14

ϕ17−ϕ16

sn²ω3ξ, k6

−ϕ16ϕ14− ϕ17−ϕ16

sn²ω3ξ, k6 , 3.47

u₂₈ x, y, t

A₃ϕ₁₈ϕ₁₉B₃

A₃ϕ₁₈−ϕ₁₉B₃

cn

−αA3B₃/2ξ, k₇ A3B3 A3−B3cn

−αA3B3/2ξ, k

, 3.48

u29

x, y, t

A₃ϕ₁₈ϕ₁₉B₃−

A₃ϕ₁₈−ϕ₁₉B₃

cn

−αA3B₃/2ξ, k₇ A3B3−A3−B3cn

−αA3B3/2ξ, k7

, 3.49

where

A₂

ϕ₁₁− c₃c₃ 2

2

−c3−c₃²

4 , B₂

ϕ₁₀−c₃c₃ 2

2

−c3−c₃²

4 ,

A₃

ϕ₁₉− c4c4

2 2

−c4−c4²

4 , B₃

ϕ₁₈−c4c4

2 2

−c4−c4²

4 ,

k5

ϕ11−ϕ10²−A2−B2²

4A2B2 , k6

ϕ₁₇−ϕ₁₆

ϕ₁₅−ϕ₁₄ ϕ₁₇−ϕ₁₅

ϕ₁₆−ϕ₁₄,

ω₃ −α

ϕ₁₇−ϕ₁₅

ϕ₁₆−ϕ₁₄ 2√

2 , k₇

ϕ₁₉−ϕ₁₈²−A3−B₃² 4A₃B₃ ,

ϕ^∗₉ −αϕ9−

4αβ−3α²ϕ²₉

2α , θ−4β

α 3ϕ²₉3ϕ₉

4β α −3ϕ²₉,

μ2ϕ₉−2

4β

α −3ϕ²₉, q4ϕ²₉4ϕ₉

4β α −3ϕ²₉,

3.50

c₃,c₃,c₄andc₄are complex numbers.

And two solitary wave solutions

u30

x, y, t

ϕ9 2β−6αϕ²₉ 2αϕ9

2αβ−2α²ϕ²₉cosh

−β3αϕ²₉ξ

, 3.51

u31

x, y, t

ϕ9 2β−6αϕ²₉ 2αϕ₉−

2αβ−2α²ϕ²₉cosh

−β3αϕ²₉ξ

. 3.52

3Ifg−g0, we get two periodic wave solutions as follows:

u32

x, y, t

A4ϕ22ϕ23B4

A4ϕ22−ϕ23B4

cn

−αA4B4/2ξ, k8

A₄B₄ A₄−B₄cn

−αA4B₄/2ξ, k₈ , 3.53

u₃₃ x, y, t

A₅ϕ₂₄ϕ₂₅B₅

A₅ϕ₂₄−ϕ₂₅B₅

cn

−αA5B₅/2ξ, k₉ A5B5 A5−B5cn

−αA5B5/2ξ, k9

, 3.54

where

A4

ϕ23− c5c5

2 2

−c5−c5²

4 , B4

ϕ22−c5c5

2 2

−c5−c5²

4 ,

A5

ϕ25−c6c6

2 2

−c6−c6²

4 , B6

ϕ24−c6c6

2 2

−c6−c6²

4 ,

k₈

ϕ₂₃−ϕ₂₂²−A4−B₄² 4A4B4

, k₉

ϕ₂₅−ϕ₂₄²−A5−B₅² 4A5B5

,

3.55

c5,c5,c6andc6are complex numbers.

And a solitary wave solution

u₃₄ x, y, t

3β α

92βξ²

−96βξ². 3.56

Proof. 1Ifg 0, we set

ϕ₂>

2β

α,

β α <ϕ₆ <

2β

α. 3.57

iFrom the phase portrait, we see that there are a closed orbitΓ1passing the points ϕ₁,0andϕ₂,0. Inϕ, φplane, the expressions of the closed orbits are given as

φ±

−α 2

ϕ2−ϕ ϕ−ϕ1

ϕ−c2

ϕ−c2

, 3.58

whereϕ1−ϕ2,c2i

ϕ²₂−2β/αandc2 −i

ϕ²₂−2β/α.

Substituting3.58intodϕ/dξφand integrating them along the orbitΓ1, we have

± _ϕ

ϕ1

1 ϕ2−s

s−ϕ1

s−c2s−c2ds

−α 2

_ξ

0

ds,

± _ϕ

ϕ2

1 ϕ2−s

s−ϕ1

s−c2s−c2ds

−α 2

_ξ

0

ds.

3.59

From3.59and noting thatuϕξandξ xy−ct, we obtain the periodic wave solutionsu₁₉x, y, tas3.38andu₂₀x, y, tas3.39.

iiFrom the phase portrait, we see that there are two closed orbitsΓ2andΓ3passing the pointsϕ₃,0, ϕ₄,0,ϕ₅,0, andϕ₆,0. In ϕ, φ plane, the expressions of the closed orbits are given as

φ±

−α 2

ϕ−ϕ₃

ϕ−ϕ₄

ϕ−ϕ₅ ϕ₆−ϕ

, 3.60

whereϕ₃−ϕ₆,ϕ₄−

2β/α−ϕ²₆andϕ₅

2β/α−ϕ²₆.

Substituting3.60intodϕ/dξφand integrating them alongΓ2andΓ3, we have

± _ϕ

ϕ3

1 ϕ₆−s

ϕ₅−s

ϕ₄−s

s−ϕ₃ds

−α 2

_ξ

0

ds,

± _ϕ

ϕ6

1 ϕ₆−s

s−ϕ₅

s−ϕ₄

s−ϕ₃ds

−α 2

_ξ

0

ds.

3.61

From3.61and noting thatuϕξandξ xy−ct, we obtain the periodic wave solutionsu21x, y, tas3.40andu22x, y, tas3.41.

iiiFrom the phase portrait, we see that there are two symmetric homoclinic orbitsΓ4

andΓ5connected at the saddle point0,0. Inϕ, φplane, the expressions of the homoclinic orbits are given as

φ±ϕ

−α 2

ϕ−ϕ₇ ϕ₈−ϕ

, 3.62

whereϕ7−

2β/αandϕ8 2β/α.

Substituting3.62intodϕ/dξφand integrating them along the orbitsΓ4andΓ5, we have

± _ϕ

ϕ7

1

s

s−ϕ₇

ϕ₈−sds

−α 2

_ξ

0

ds,

± _ϕ

ϕ8

1

s

s−ϕ₇

ϕ₈−sds

−α 2

_ξ

0

ds.

3.63

From3.63and noting thatu ϕξandξ xy−ct, we obtain the solitary wave solutionsu_23±x, y, tas3.43.

2If−g0 < g <0, we set the middle solution offϕ 0 beϕ90<ϕ9<

β/3α, then we can get another two solutions offϕ 0 as follows:

ϕ^∗₉ −αϕ₉−

4αβ−3α²ϕ²₉

2α ,

ϕ₉ −αϕ9

4αβ−3α²ϕ²₉

2α .

3.64

iFrom the phase portrait, we see that there are a closed orbitΓ6passing the points ϕ10,0andϕ11,0. Inϕ, φplane, the expressions of the closed orbits are given as

φ±

−α 2

ϕ₁₁−ϕ

ϕ−ϕ₁₀

ϕ−c₃ ϕ−c₃

, 3.65

whereϕ12<ϕ10<ϕ₉<ϕ11 <ϕ13,c3andc3are conjugate complex numbers.

Substituting3.37intodϕ/dξφand integrating them alongΓ6, we have

± _ϕ

ϕ10

1 ϕ11−s

s−ϕ10

s−c3s−c3ds

−α 2

_ξ

0

ds. 3.66

From3.66and noting thatuϕξandξxy−ct, we get a periodic wave solution u₂₄x, y, tas3.44.

iiFrom the phase portrait, we note that there is a special orbitΓ7, which has the same Hamiltonian with that ofϕ^∗₉,0. Inϕ, φplane, the expressions of the orbits are given as

φ±

−α

2ϕ−ϕ^∗₉²

ϕ−ϕ₁₂ ϕ₁₃−ϕ

, 3.67

where

ϕ12

αϕ₉

4αβ−3α²ϕ²₉2

αϕ₉

αϕ₉−

4αβ−3α²ϕ²₉

2α ,

ϕ13

αϕ₉

4αβ−3α²ϕ²₉−2

αϕ₉

αϕ₉−

4αβ−3α²ϕ²₉

2α .

3.68

Substituting3.67intodϕ/dξφand integrating them alongΓ7, it follows that

± _ϕ

ϕ12

1 ϕ₁₃−s

s−ϕ^∗₉²

s−ϕ₁₂ds

−α 2

_ξ

0

ds. 3.69

From3.69and noting thatuϕξandξxy−ct, we get a periodic wave solution u₂₅x, y, tas3.45.

iiiFrom the phase portrait, we note that there are two closed orbitsΓ8andΓ9passing the pointsϕ₁₄,0,ϕ₁₅,0,ϕ₁₆,0, andϕ₁₇,0. Inϕ, φplane, the expressions of the orbits are given as

φ±

−α 2

ϕ−ϕ₁₄

ϕ−ϕ₁₅

ϕ−ϕ₁₆ ϕ₁₇−ϕ

, 3.70

whereϕ₂₀<ϕ₁₄<ϕ₁₂<ϕ₁₀<ϕ₉ <ϕ₁₁<ϕ₁₃<ϕ₁₅<ϕ₉<ϕ₁₆ <ϕ^∗₉<ϕ₁₇<ϕ₂₁.

Substituting3.70intodϕ/dξφand integrating them alongΓ8andΓ9, we have

± _ϕ

ϕ14

1 ϕ₁₇−s

ϕ₁₆−s

ϕ₁₅−s

s−ϕ₁₄ds

−α 2

_ξ

0

ds,

± _ϕ

ϕ17

1 ϕ₁₇−s

s−ϕ₁₆

s−ϕ₁₅

s−ϕ₁₄ds

−α 2

_ξ

0

ds.

3.71

From3.71and noting thatu ϕξandξ xy−ct, we get two periodic wave solutionsu₂₆x, y, tas3.46andu₂₇x, y, tas3.47.

ivFrom the phase portrait, we note that there is a special orbitΓ10passing the points ϕ₁₈,0andϕ₁₉,0. Inϕ, φplane, the expressions of the orbit are given as

φ±

−α 2

ϕ19−ϕ ϕ−ϕ18

ϕ−c4

ϕ−c4

, 3.72

whereϕ₁₈<ϕ₂₀<ϕ₂₁<ϕ₁₉,c₄andc₄are conjugate complex numbers.

Substituting3.72intodϕ/dξφand integrating it alongΓ10, we have

± _ϕ

ϕ18

ϕ19−s 1 s−ϕ18

s−c4s−c4ds

−α 2

_ξ

0

ds. 3.73

From3.73and noting thatuϕξandξxy−ct, we get a periodic wave solution u28x, y, tas3.48.

Ifϕξ is a traveling wave solution, thenϕξ q is a traveling wave solution too.

Takingq2Kand noting that cnu2K −cnu, we get a periodic wave solutionu₂₉x, y, t as3.49.

vFrom the phase portrait, we note that there are two homoclinic orbitsΓ11andΓ12

connected at the saddle pointϕ₉,0. Inϕ, φplane, the expressions of the orbits are given as

φ±

−α

2ϕ−ϕ9² ϕ−ϕ20

ϕ21−ϕ

, 3.74

where

ϕ20 −αϕ₉

2αβ−2α²ϕ²₉

α ,

ϕ₂₁ −αϕ9−

2αβ−2α²ϕ²₉

α .

3.75

Substituting3.74intodϕ/dξ φand integrating them alongΓ11 andΓ12, it follows that

± _ϕ

ϕ20

1

s−ϕ₉²

s−ϕ₂₀

ϕ₂₁−sds

−α 2

_ξ

0

ds,

± _ϕ

ϕ21

1

s−ϕ₉²

s−ϕ₂₀

ϕ₂₁−sds

−α 2

_ξ

0

ds.

3.76

From3.76 and noting thatu ϕξand ξ xy−ct, we get two solitary wave solutionsu30x, y, tas3.51andu31x, y, tas3.52.

3Ifg−g0, we will consider two kinds of orbits.

iFrom the phase portrait, we note that there is a closed orbitΓ13 passing the points ϕ₂₂,0andϕ23,0. Inϕ, φplane, the expressions of the orbit are given as

φ±

−α 2

ϕ₂₃−ϕ

ϕ−ϕ₂₂

ϕ−c₅ ϕ−c₅

, 3.77

where−

3β/α <ϕ22<−2

β/3α <ϕ23 <

β/3α,c5andc5are conjugate complex numbers.

Substituting3.77intodϕ/dξφand integrating it alongΓ13, we have

± _ϕ

ϕ22

1 ϕ₂₃−s

s−ϕ₂₂

s−c₅s−c₅ds

−α 2

_ξ

0

ds. 3.78

From3.78and noting thatuϕξandξxy−ct, we get a periodic wave solutions u₃₂x, y, tas3.53.

iiFrom the phase portrait, we note that there is a closed orbitΓ14passing the points ϕ₂₄,0andϕ₂₅,0. Inϕ, φplane, the expressions of the orbit are given as

φ±

−α 2

ϕ₂₅−ϕ

ϕ−ϕ₂₄

ϕ−c₆ ϕ−c₆

, 3.79

where ϕ24 < −

3β/α < ϕ22 < −2

β/3α < ϕ23 <

β/3α < ϕ25,c6 and c6 are conjugate complex numbers.