2. Bifurcation of Phase Portraits

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Volume 2012, Article ID 597431,17pages doi:10.1155/2012/597431

Research Article

Bifurcation of Traveling Wave Solutions for a Two-Component Generalized θ-Equation

Zhenshu Wen

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Correspondence should be addressed to Zhenshu Wen,[email protected] Received 19 October 2012; Accepted 20 November 2012

Academic Editor: Ezzat G. Bakhoum

Copyrightq2012 Zhenshu Wen. 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 study the bifurcation of traveling wave solutions for a two-component generalizedθ-equation.

We show all the explicit bifurcation parametric conditions and all possible phase portraits of the system. Especially, the explicit conditions, under which there exist kinkor antikinksolutions, are given. Additionally, not only solitons and kinkantikinksolutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.

1. Introduction

In 2008, Liu 1 introduced a class of nonlocal dispersive models, that is, θ-equations, as follows:

ut−uxxtuux 1−θuxuxxθuuxxx, x∈R, t >0, 1.1 whereux, tdenotes the velocity field at timetin the spatialxdirection.

Recently, Ni 2 further investigated the cauchy problem for the following two- component generalizedθ-equations:

u_t−u_xxtuu_x−1−θuxu_xx−θuu_xxxσρρ_x0, x∈R, t >0,

ρtθρxu 1−2θρux0, x∈R, t >0, 1.2 whereσtakes 1 or−1. This system includes two componentsux, tandρx, t. The first one describes the horizontal velocity of the fluid, while the other one describes the horizontal deviation of the surface from equilibrium, both are measured in dimensionless units.

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In this paper, we study the bifurcation of traveling wave solutions for the following system:

u_t−u_xxtuu_x− 3

5u_xu_xx−2

5uu_xxxρρ_x0, x∈R, t >0, ρt2

5ρxu1

5ρux0, x∈R, t >0,

1.3

which is a special form of system1.2through takingθ2/5 andσ1, by employing the bifurcation method and qualitative theory of dynamical systems3–7. We give all the explicit bifurcation parametric conditions for various solutions and all possible phase portraints of the system, from which not only solitons and kinkantikinksolutions, but also peakons and periodic cusp waves are obtained.

2. Bifurcation of Phase Portraits

For given constantc, multiplying both sides of the second equation of system1.3byρx, t and substitutingux, t ϕξ, ρψξwithξx−ctinto system1.3, it follows that

−cϕcϕϕϕ−3

5ϕϕ− 2

5ϕϕψψ0,

−cψψ 2

5ψψϕ1

5ψ²ϕ0.

2.1

Integrating system2.1once leads to

−cϕcϕ1 2ϕ²− 1

10 ϕ₂

−2

5ϕϕ1 2ψ²g,

−c 2ψ² 1

5ψ²ϕG,

2.2

where bothgandGare integral constants, respectively.

From the second equation of system2.2, we obtain

ψ² 5G

ϕ−5/2c. 2.3

Substituting2.3into the first equation of system2.2, it leads to

ϕ−5 2c

₂

ϕ−1 4

ϕ− 5

2c ϕ25

4

ϕ−5 2c

ϕ²−2cϕ−2g

5G . 2.4

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By settingϕφ 3/2c,2.4becomes φ−c₂

φ−1 4

φ−c φ₂

5 4

φ−c φ−3

2c 2

−2c

φ−3 2c

−2g

5G

−1 4

φ−c φ₂

5 4

φ³− 7

4c²2g φ 3

4c³5G2cg .

2.5

Lettingyϕ, we obtain a planar system dφ

dξ y, dy

dξ −1/4 φ−c

y² 5/4 φ³−

7/4c²2g

φ 3/4c³5G2cg φ−c₂ ,

2.6

with first integral

H φ, y

1 2

φ−cy²−4

φ−c320c

φ−c2

25c²−40g φ−c

−100G 8

φ−c ,

forφ > c,

2.7

or

H φ, y

1 2

c−φy²−4

c−φ3−20c

c−φ2

25c²−40g c−φ

100G 8

c−φ ,

forφ < c.

2.8

Note that whenG0, systems2.6,2.7, and2.8become, respectively, dφ

dξ y, dy

dξ −1/4y² 5/4

φ²cφ−7/4c²−2g

φ−c ,

2.9

H φ, y

1 2

φ−cy²− 1 8

4

φ−c5/220c

φ−c3/2

25c²−40g

φ−c1/2 ,

forφ > c,

2.10 H

φ, y 1

2

c−φy²− 1 8

4

c−φ_5/2

−20c

c−φ_3/2

25c²−40g

c−φ_1/2 ,

forφ < c.

2.11

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Transformed by dξ φ−c²dτ, system2.6becomes a Hamiltonion system dφ

dτ φ−c₂

y, dy

dτ −1 4

φ−c y²5

4

φ³− 7

4c²2g

φ3

4c³5G2cg .

2.12

Since the first integral of system2.6is the same as that of the Hamiltonian system 2.12, system2.6should have the same topological phase portraits as system2.12except the straight line l : φ c. Therefore, we should be able to obtain the topological phase portraits of system2.6from those of system2.12.

Let

f φ

φ³− 7

4c²2g

φ3

4c³5G2cg. 2.13

It is easy to obtain the two extreme points offφas follows:

φ_±^∗ ±

7c²8g

12 , forg >−7

8c², 2.14

from which we can obtain a critical curve forgas follows:

g₀c −7

8c². 2.15

We obtain two bifurcation curves:

G₁− 1 180

72cg27c³

8g7c²

21c²24g , G₂− 1

180

72cg27c³−

8g7c²

21c²24g ,

2.16

fromfφ^∗₋ 0 andfφ^∗ 0, respectively. Note that wheng < g₀c, obviouslyG₁< G₂. For convenience, we assume thatg∝c²in this paper, then we haveG1∝c³andG2∝c³.

Further, fromG₁0 orG₂0, we can obtain another two critical curves forg, that is, g₁c −1

2c², 2.17

g₂c 5

8c². 2.18

Note that2.18can also be obtained by lettingφ^∗c,c >0 orφ^∗₋c,c <0.

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φ=c

φ=c φ^∗− φ^∗+

φ1

f(φ) φ=c φ^∗+

φ2

f(φ) φ1=φ^∗−

φ=c φ^∗−

φ^∗+

φ2

φ1

φ3

f(φ)

φ1 φ^∗+ φ1 φ1

f(φ) φ=c f(φ) φ=c f(φ)

φ3

φ2

φ^∗− φ^∗− φ^∗+=φ2 φ−^∗ φ+^∗

G < G1 G1< G <0

0< G < G2 G=G2 G > G2

G=G1

Figure 1: The graphics offφwheng > g2c.

Letφ^∗,0be one of the singular points of system2.12, then the characteristic values of the linearized system of system2.12at the singular pointφ^∗,0are

λ_±±1 2

5

φ^∗−c2f φ^∗

. 2.19

From the qualitative theory of dynamical systems, we can determine the property of singular pointφ^∗,0by the sign offφ^∗and whetherφ^∗equals tocor not. However, we also know thatHc, y ∞from2.7and2.8. Therefore,φcis an isolated orbit, dividing φ, y-plane into two parts.

Based on the above analysis, we give the property of the singular points for system 2.12and their relationship withφ^∗₋,φ^∗andcin the following lemma.

Lemma 2.1. Forg > g2c, one hasG1 <0 < G2 and the singular points of system2.12can be described as follows.

aIfG < G1, then there is only one singular point denoted asS1φ1,0 φ^∗₋ < c < φ^∗ < φ1. S₁is a saddle point.

bIf G G₁, then there are two singular points denoted as S₁φ1,0and S₂φ2,0 φ1 φ^∗₋< c < φ^∗< φ2, respectively.S1is a degenerate saddle point andS2is a saddle point.

cIfG₁ < G < 0, then there are three singular points denoted asS₁φ1,0,S₂φ2,0, and S₃φ3,0 φ1 < φ^∗₋ < φ₂< c < φ^∗ < φ₃, respectively.S₁andS₃are saddle points andS₂ is a center.

dIf 0 < G < G₂, then there are three singular points denoted asS₁φ1,0,S₂φ2,0, and S3φ3,0 φ1 < φ^∗₋ < c < φ2 < φ^∗ < φ3, respectively.S1andS3are saddle points andS2

is a center.

eIf G G₂, then there are two singular points denoted as S₁φ1,0and S₂φ2,0 φ1 <

φ^∗₋< c < φ^∗φ2, respectively.S1is a saddle point andS2is a degenerate saddle point.

fIfG > G₂, then there is only one singular point denoted asS₁φ1,0 φ1 < φ^∗₋ < c < φ^∗. S₁is a saddle point.

Proof. Lemma 2.1follows easily from the graphics of the functionfφwhich can be obtained directly and shown inFigure 1.

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For the other cases, the similar analysis can be taken to make the conclusions. We just omit these processes for the ease of simplicity. However, it is worth mentioning that, when g₀c< g < g₂candG₁ < G < G₂ G /0, there exist two saddle points and one center lie on the same side of singular lineφc. Hence, there may exist heteroclinic orbits for system2.6.

We will show the existence of heteroclinic orbits for system2.6in the following analysis.

IfG₁ < G < G₂, we set three solutions offφ 0 beφ_s,φ_m, andφ_b φs < φ_m < φ_b, respectively. Through simple calculation, we can expressφsandφbas the function ofφm, that is,

φs −φm−

8g7c²−3φ²_m

2 ,

φ_b −φm

8g7c²−3φ_m²

2 .

2.20

It follows fromφ_s< φ_m< φ_bthatφ_mmust satisfy condition φ²_m< 8g7c²

12 . 2.21

FromHφs,0 Hφb,0, we obtain the expression ofGas the function ofφm,

G 1

100

9c³24cg−

8g15c²

φ_m−8cφ²_m4φ³_m

2c²−16g−4cφm

4φ²m−8g4cφm−3c³ .

2.22

Substituting2.22intofφm 0, we obtain the expression ofφmfromfφm 0 as follows:

φ_m1 1 6

5c−2

c²−6g

, 2.23

φ_m2 1 6

5c2

c²−6g

, 2.24

φ_m3−

7c²8g

3 , 2.25

φ_m4

7c²8g

3 , 2.26

φ_m5 1 2

−c−2

c²2g

, 2.27

φ_m6 1 2

−c2

c²2g

, 2.28

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Note that from 2.23–2.28, we obtain three critical curves for g, that is, g0c, in 2.12,g₁cin2.15, and

g₃c 1

6c². 2.29

We then check the conditionφ²_m<8g7c²/12i.e.,2.21for the aboveφ_ms one by one and give the results in the following lemma.

Lemma 2.2. Starting from interval−7/8c²,5/8c², one has the following.

1For g ∈ −7/8c²,1/6c² ⊂ −7/8c²,5/8c² and c > 0, φm 1/65c − 2

c²−6g(i.e.,2.23) satisfies2.21.

2For g ∈ −7/8c²,1/6c² ⊂ −7/8c²,5/8c² and c < 0, φm 1/65c 2

c²−6g(i.e.,2.24) satisfies2.21.

3For anyg∈−7/8c²,5/8c²,2.25does not satisfy2.21.

4For anyg∈−7/8c²,5/8c²,2.26does not satisfy2.21.

5For g ∈ −1/2c²,5/8c² ⊂ −7/8c²,5/8c² and c < 0, φm 1/2−c − 2

c²2g(i.e.,2.27) satisfies2.21.

6For g ∈ −1/2c²,5/8c² ⊂ −7/8c²,5/8c² and c > 0, φm 1/2−c 2

c²2g(i.e.,2.28) satisfies2.21.

Proof. Lemma 2.2follows easily from the definitional domain of theφms and general logical reasoning.

FromLemma 2.2, substituting2.23and2.24intofφm 0, respectively, we obtain another two bifurcation curvesdenoted byG^∗₁andG^∗₂forGas follows:

G^∗₁ 4 135

−c³9cg

c²−6g c²−6g

, forg₀c< g < g₃c, c >0,

G^∗₂ 4 135

−c³9cg−

c²−6g c²−6g

, forg₀c< g < g₃c, c <0.

2.30

Similarly, substituting2.27and2.28intofφm 0, we have G_∗0, forg₁c< g < g₃c, c <0

org₁c< g < g₃c, c >0

. 2.31

Note that we have indicated that wheng₀c < g < g₂candG₁ < G < G₂ G /0, there exist two saddle points and one center lying on the same side of singular lineφ c.

Therefore, we obtain the fifth critical curve forgfromG^∗₁0 c >0orG^∗₂0 c <0,

g4c 0. 2.32

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φ=c

φ=c G2

φ

G

G1 c φ

φ=c φ

G1

φ=c φ φ=c φ φ=cφ φ=c

φ G2

o

Figure 2: The phase portraits of system2.6wheng > g2c.

φ=c

φ=c G2

φ G

φ

φ=c φ

G1

φ=c

φ φ

φ=c

φ

o φ=c

φ=c G1

φ φ=c

φ=c φ

G2

c

φ φ

Figure 3: The phase portraits of system2.6whengg2c.

Lemma 2.3. (1) Forg∈g0c, g4c∪g4c, g3c, andGG^∗₁, c >0 (orGG^∗₂, c <0), there exist heteroclinic orbits for system2.6.

(2) For anyg /∈g0c, g4c∪g4c, g3corG /G^∗₁, c >0 andG /G^∗₂, c <0, there exist no heteroclinic orbits for system2.6.

Proof. Lemma 2.3follows easily from the above analysis.

Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system2.6in Figures2,3,4,5,6,7,8, and9under corresponding conditions.

3. Main Results and the Theoretic Derivations of Main Results

In this section, we state our results about solitons, kinkantikinksolutions, peakons, and periodic cusp waves for the first component of system1.3. To relate conveniently, we omit ϕφ 2/3cand the expression of the second component of system1.3in the following theorems.

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Theorem 3.1. For constant wave speedc, integral constantsgandG, one has the following.

1Ifc,g,Gsatisfy one of the following conditions:

ig > g₂c,G₁< G <0 andc /0;

iig₁c< g≤g₂c,G₁< G <0 andc >0;

iiig₁c≤g < g₄c,g₄c< g < g₃c, 0< G < G^∗₁andc >0;

ivg0c< g < g1c,G1< G < G^∗₁andc >0;

then there exist smooth solitons for system1.3, which can be implicitly expressed as c−φ

/ φ₁^∗−φ

−1 c−φ

/ φ₁^∗−φ

1

·

c−φ/φ₁^∗−φ √ αα

c−φ/φ₁^∗−φ−√

αα e^|ξ|, 3.1

where

α c−φ₁ φ₁^∗−φ1

. 3.2

vg > g₂c, 0< G < G₂andc /0;

vig₁c< g≤g₄c, 0< G < G₂andc <0;

viig₁c≤g < g₄c,g₄c< g < g₃c,G^∗₂< G <0 andc <0;

viiig0c< g < g1c,G^∗₂< G < G2andc <0;

then there exist smooth solitons for system1.3, which can be implicitly expressed as φ−c

/ φ−φ^∗₂

−1 φ−c

/ φ−φ^∗₂

1

·

φ−c/φ−φ^∗₂ ββ

φ−c/φ−φ^∗₂−

ββ e^|ξ|, 3.3

where

β φ₂−c

φ₂−φ^∗₂. 3.4

ixg₃c≤g < g₂c,G₁< G <0 andc <0;

xg₀c< g < g₃c,G₁< G < G^∗₂andc <0;

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then there exist smooth solitons for system1.3, which can be implicitly expressed as

g1

_u₁

0

du− φ3−c φ3−φ₃₁^∗

_u₁

0

d n²u 1−γ₁²sn²udu

e^|ξ|, 3.5

where

g₁ 2

φ^∗₃₂−c,

γ₁²k₁²φ^∗₃₂−φ₃ φ^∗₃₁−φ₃, k²₁ φ^∗₃₁−c

φ^∗₃₂−c, snu₁sinφ.

3.6

xig₃c≤g < g₂c, 0< G < G₂andc >0;

xiig₀c< g < g₃c,G^∗₁< G < G₂andc >0;

then there exist smooth solitons for system1.3, which can be implicitly expressed as:

g₂ _u₂

0

du− c−φ₄ φ^∗₄₁−φ4

_u₂

0

d n²u 1−γ₂²sn²udu

e^|ξ|, 3.7

where

g2 2

c−φ₄₂^∗ ,

γ₂²k₂²φ₄−φ^∗₄₂ φ₄−φ^∗₄₁, k²₂ c−φ^∗₄₁

c−φ^∗₄₂, snu₂sinφ.

3.8

Proof. 1From the phase portraits in Figures2–9, we see that whenc,g,Gsatisfy one of the conditions, that is,i,ii,iii, oriv, there exist homoclinic orbits as showed individually in Figures10aand10b. The expressions of the homoclinic orbits can be given as follows:

y± φ−φ1

φ₁^∗−φ

c−φ , φ1≤φ≤φ₁^∗< c, 3.9

whereφ₁andφ^∗₁can be obtained from2.8.

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φ=c G2 φ=c

φ

G

c φ

φ=c φ

G1

φ=c φ φ=c

φ G2

o φ=c

φ=c φ

φ=c φ=c φ

φ φ

φ=cφ φ=c φ

φ=c φ φ

φ=c

φ φ=c G1

Figure 4: The phase portraits of system2.6wheng3c≤g < g2c.

φ=c

G2 φ

G

c φ

φ=c

φ G1

φ=c φ φ=c

φ φ=c

φ=c

φ G2

o φ

φ=c φ=c

φ

φ=c φ=c

φ=c φ=c φ=c

φ φ φ

φ φ

φ=cφ=c G1

φ

φ=φc φ=c

φ φ G^∗₁

G^∗₂

Figure 5: The phase portraits of system2.6wheng4c< g < g3c.

Substituting 3.9 into the first equation of system 2.6, and integrating along the homoclinic orbits, it follows that

_φ^∗

1

φ

√c−sds s−φ₁

φ₁^∗−s |ξ|. 3.10

From3.10, we obtain the solitons3.1along with3.2.

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φ=c

G2 φ

G

c φ

φ=c G1 φ

φ=c φ φ=c

φ

φ φ=c φ=c

φ

o G^∗₁

G^∗₂

φ=c φ

G1

φ=c φ=c φ

φ=c φ

φ φ φ=c

φ=c φ=c

φ

φ G2

φ=c

G2 φ

G

c φ

φ=c

φ G1

φ=c φ

φ=c

φ φ=c

φ=c

φ G2

o φ

φ=c φ=c

φ

φ=c

φ=c φ=c

φ φ φ

φ φ=c

φ=c G1

φ

φ φ

φ=c φ=c

φ φ G^∗₁

G^∗₂ φ=c φ

2When c,g,Gsatisfy one of the conditions, that is,v,vi,vii, orviii, there exist homoclinic orbits as showed individually in Figures8cand8d. The expressions of the homoclinic orbits can be given as follows:

y±

φ₂−φ φ−φ^∗₂

φ−c , c < φ₂^∗≤φ≤φ₂, 3.11 whereφ₂andφ^∗₂can be obtained from2.7.

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φ=c

φ=c G2

φ G

c φ φ=c

φ

G1

φ

φ φ=c

φ=c φ φ=c

φ=c

φ=c φ

φ φ

φ=c

φ=c G1

φ

G^∗₁

G^∗₂

φ φ=c

φ

φ=c

φ o

φ=c

φ φ=c

G2

φ=c

φ G

c φ=c φ

φ φ

φ=c

φ=cφ φ=c

φ=c

φ

φ φ=c φφ=c

φ

φ φ=c

φ=c φ

o

φ=c

φ φ=c G2

φ φ φ=c

φ=c

φ G1

G^∗₂ φ

G1

G^∗₁ G2

φ=c

φ=φc φ

φ₁^∗ φ φ=c φ₁

a

φ₁^∗ φ φ=c φ₁

b

φ φ=c

φ₂ φ₂^∗ c

φ=c φ₂ φ φ^∗₂ d

φ=c φ₃₁^∗ φ^∗₃₂ φ3

φ

e

φ=c φ^∗₄₁ φ^∗₄₂ φ4 φ

f Figure 10: The diﬀerent kinds of homoclinic orbits for system2.6.

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Substituting3.11 into the first equation of system2.6, and integrating along the homoclinic orbits, it follows that

_φ

φ^∗₂

√s−cds φ₂−s

s−φ₂^∗ |ξ|. 3.12

From3.12, we obtain the solitons3.3along with3.4.

3Whenc,g,Gsatisfy one of the conditions, that is,ixorx, there exist homoclinic orbits as showed individually inFigure 8e. The expressions of the homoclinic orbits can be given as follows:

y± φ−φ3

φ^∗₃₁−φ

φ₃₂^∗ −φ

φ−c , c < φ3≤φ≤φ₃₁^∗ < φ^∗₃₂, 3.13 whereφ3,φ^∗₃₁andφ^∗₃₂can be obtained from2.7.

_φ^∗

31

φ

√s−cds s−φ₃

φ^∗₃₁−s

φ₃₂^∗ −s |ξ|. 3.14

From3.14 8, we obtain the solitons3.5along with3.6.

4Whenc,g,Gsatisfy one of the conditions, that is,xiorxii, there exist homoclinic orbits as showed individually inFigure 8f. The expressions of the homoclinic orbits can be given as follows:

y±

φ₄−φ

φ−φ₄₁^∗

φ−φ^∗₄₂

c−φ , φ₄₂^∗ < φ^∗₄₁ ≤φ≤φ₄< c, 3.15 whereφ₄,φ^∗₄₁, andφ^∗₄₂can be obtained from2.8.

_φ

φ₄₁^∗

√c−sds φ₄−s

s−φ^∗₄₁

s−φ^∗₄₂ |ξ|. 3.16

From3.16 8, we obtain the solitons3.7along with3.8.

Theorem 3.2. If constant wave speedc, integral constantsg andGsatisfyg₀c < g < g₄cor g4c< g < g3c, andGG^∗₁ (c >0) orGG^∗₂(c <0), then there exist kink (antikink) solutions for system1.3.

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Proof. We have showed that, wheng0c< g < g4corg4c< g < g3c, andGG^∗₁c >0 orG G^∗₂c <0, there exist heteroclinic orbits for system2.6. The heteroclinic orbits can be expressed as

y±

φ−φ_s φ_b−φ

c−φ , forc >0, 3.17

where

φ_s 1 12

⎛

⎝−5c2

c²−6g−

15

11c²4c

c²−6g24g

⎞⎠,

φ_b 1 12

⎛

⎝−5c2

c²−6g

15

11c²4c

c²−6g24g

⎞⎠,

3.18

which can be obtained by substituting2.23into2.20.

Substituting3.17 into the first equation of system2.6, and integrating along the heteroclinic orbits, it follows that

_φ

φ0

√c−sds s−φs

φb−s ±ξ, 3.19

whereφ₀∈φs, φ_bis the initial value.

From3.19, we have c−φ_s−

c−φ√

c−φs/φb−φs

c−φ_s

c−φ√

c−φs/φb−φs ·

c−φ

c−φ_b√

c−φb/φb−φs

c−φ−

c−φ_b√

c−φb/φb−φs

c−φ_s−

c−φ₀√

c−φs/φb−φs

c−φ_s

c−φ₀√

c−φs/φb−φs ·

c−φ₀

c−φ_b√

c−φb/φb−φs

c−φ₀−

c−φ_b√

c−φb/φb−φse^±ξ.

3.20

The case whenc <0, can be analyzed similarly. We omit it here for the ease of simplicity.

Theorem 3.3. (1) Ifg g₄c,G0 andc /0, then there exist peakons for system1.3, which can be explicitly expressed as

φ 5

2ce^−|x−ct|−3

2c. 3.21

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(2) Ifg4c≤g < g2c,G0 andc /0, then system1.3has periodic cusp waves ux, t φξ−2iT 3

2c, 3.22

wherei0,±1,±2, . . . , ξx−ct∈2i−1T,2i1T, and

φξ 1

4

5c−

25c²−40g

e^|x−ct|1 4

5c

25c²−40g

e^−|x−ct|−3

2c, 3.23

with

T ln

⎛

⎜⎝5c

25c²−40g 2

10g

⎞

⎟⎠. 3.24

Proof. 1Wheng g₄c,G0 andc /0, fromFigure 6, we see that there is a triangle orbit, which can be expressed as

y±

φ3 2c

, for −3

2c≤φ≤cc >0, 3.25 φc, fory≤

√5

2 cc >0. 3.26 Substituting3.25 into the first equation of system2.6, and integrating along the triangle orbits, it follows that

_c

φ

dt

t 3/2c |ξ|. 3.27

From3.27, we obtain peakons3.21.

2Wheng4c≤g < g2c,G0 andc /0, from Figures4and5, we see that there is a semiellipse orbit, which can be expressed as

y±

φ²3cφ 9

4c²−10g, for 1 2

−3c2 10g

≤φ≤cc >0, 3.28

φc, fory≤

5

c²−8g

2 c >0. 3.29 Substituting3.28 into the first equation of system2.6, and integrating along the semiellipse orbits, it follows that

_c

φ

dt

t²3ct 9/4c²−10g |ξ|. 3.30

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From3.30, we obtain periodic cusp waves3.22along with3.23and3.24.

Note that we only show the case whenc > 0, in fact, we can analyze the case when c <0 following the same procedure. We just omit it here.

4. Conclusions

In this paper, by employing the bifurcation method and qualitative theory of dynamical systems, we study the bifurcation of traveling wave solutions for a two-component generalizedθ-equation1.3, show all the explicit parametric conditions and all the phase portraits of system 1.3 determinately. Through the phase portraits, we can investigate various kinds of solutions. Specifically, the implicit expressions of the solitons, kinkantikink solutions for system1.3are given. Besides, we also obtain peakons and periodic cusp waves with explicit expressions for system1.3.

Acknowledgment

This research is supported by the foundation of Huaqiao Universityno. 12BS223.

References

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24, no. 3, pp. 423–440, 2008.

2 L. Ni, “The Cauchy problem for a two-component generalizedθ-equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 5, pp. 1338–1349, 2010.

3 Z. Wen, Z. Liu, and M. Song, “New exact solutions for the classical Drinfel’d-Sokolov-Wilson equation,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2349–2358, 2009.

4 Z. Wen and Z. Liu, “Bifurcation of peakons and periodic cusp waves for the generalization of the Camassa-Holm equation,” Nonlinear Analysis: Real World Applications, vol. 12, no. 3, pp. 1698–1707, 2011.

5 Z. Liu, T. Jiang, P. Qin, and Q. Xu, “Trigonometric function periodic wave solutions and their limit forms for the KdV-like and the PC-like equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 810217, 23 pages, 2011.

6 M. Song and Z. Liu, “Traveling wave solutions for the generalized Zakharov equations,” Mathematical Problems in Engineering, vol. 2012, Article ID 747295, 14 pages, 2012.

7 Z. Wen, “Extension on bifurcations of traveling wave solutions for a two-component Fornberg- Whitham equation,” Abstract and Applied Analysis, vol. 2012, Article ID 704931, 15 pages, 2012.

8 P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, vol. 33 of Die Grundlehren der Mathematischen Wissenschaften, Band 67, Springer, New York, NY, USA, 2nd edition, 1971.

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2. Bifurcation of Phase Portraits

Research Article

Bifurcation of Traveling Wave Solutions for a Two-Component Generalized θ-Equation

Zhenshu Wen

1. Introduction

2. Bifurcation of Phase Portraits

3. Main Results and the Theoretic Derivations of Main Results

4. Conclusions

Acknowledgment

References

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