# Bifurcation of Traveling Wave Solutions for a Two-Component Generalizedθ-Equation

Authors: Zhenshu Wen
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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 kink (or antikink) solutions, are given. Additionally, not only solitons and kink (antikink) solutions, but also peakons and periodic cusp waves with explicit expressions, are obtained.


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## Body

## 1. Introduction

In 2008, Liu [1] introduced a class of nonlocal dispersive models, that is, θ-equations, as follows:
(1.1)ut-uxxt+uux=(1-θ)uxuxx+θuuxxx,x∈ℝ,t>0,
where u(x,t) denotes the velocity field at time t in the spatial x direction.Recently, Ni [2] further investigated the cauchy problem for the following two-component generalized θ-equations:
(1.2)ut-uxxt+uux-(1-θ)uxuxx-θuuxxx+σρρx=0,x∈ℝ,t>0,ρt+θρxu+(1-2θ)ρux=0,x∈ℝ,t>0,
where σ takes 1 or -1. This system includes two components u(x,t) and ρ(x,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.In this paper, we study the bifurcation of traveling wave solutions for the following system:(1.3)ut-uxxt+uux-35uxuxx-25uuxxx+ρρx=0,x∈ℝ,t>0,ρt+25ρxu+15ρux=0,x∈ℝ,t>0,
which is a special form of system (1.2) through taking θ=2/5 and σ=1, by employing the bifurcation method and qualitative theory of dynamical systems [3–7]. 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 kink (antikink) solutions, 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 system (1.3) by ρ(x,t) and substituting u(x,t)=φ(ξ),ρ=ψ(ξ) with ξ=x-ct into system (1.3), it follows that
(2.1)-cφ′+cφ′′′+φφ′-35φ′φ′′-25φφ′′′+ψψ′=0,-cψψ′+25ψψ′φ+15ψ2φ′=0.Integrating system (2.1) once leads to
(2.2)-cφ+cφ′′+12φ2-110(φ′)2-25φφ′′+12ψ2=g,-c2ψ2+15ψ2φ=G,
where both g and G are integral constants, respectively.From the second equation of system (2.2), we obtain
(2.3)ψ2=5Gφ-(5/2)c.Substituting (2.3) into the first equation of system (2.2), it leads to
(2.4)(φ-52c)2φ′′=-14(φ-52c)(φ′)2+54[(φ-52c)(φ2-2cφ-2g)+5G].By settingφ=ϕ+(3/2)c, (2.4) becomes
(2.5)(ϕ-c)2ϕ′′=-14(ϕ-c)(ϕ′)2+54[(ϕ-c)((ϕ-32c)2-2c(ϕ-32c)-2g)+5G]=-14(ϕ-c)(ϕ′)2+54[ϕ3-(74c2+2g)ϕ+34c3+5G+2cg].Lettingy=φ′, we obtain a planar system
(2.6)dϕdξ=y,dydξ=-(1/4)(ϕ-c)y2+(5/4)[ϕ3-((7/4)c2+2g)ϕ+(3/4)c3+5G+2cg](ϕ-c)2,
with first integral
(2.7)H(ϕ,y)=12ϕ-cy2-4(ϕ-c)3+20c(ϕ-c)2+(25c2-40g)(ϕ-c)-100G8ϕ-c,forϕ>c,
or
(2.8)H(ϕ,y)=12c-ϕy2-4(c-ϕ)3-20c(c-ϕ)2+(25c2-40g)(c-ϕ)+100G8c-ϕ,forϕ<c.Note that whenG=0, systems (2.6), (2.7), and (2.8) become, respectively,
(2.9)dϕdξ=y,dydξ=-(1/4)y2+(5/4)(ϕ2+cϕ-(7/4)c2-2g)ϕ-c,(2.10)H(ϕ,y)=12ϕ-cy2-18(4(ϕ-c)5/2+20c(ϕ-c)3/2+(25c2-40g)(ϕ-c)1/2),forϕ>c,(2.11)H(ϕ,y)=12c-ϕy2-18(4(c-ϕ)5/2-20c(c-ϕ)3/2+(25c2-40g)(c-ϕ)1/2),forϕ<c.Transformed bydξ=(ϕ-c)2dτ, system (2.6) becomes a Hamiltonion system
(2.12)dϕdτ=(ϕ-c)2y,dydτ=-14(ϕ-c)y2+54[ϕ3-(74c2+2g)ϕ+34c3+5G+2cg].Since the first integral of system (2.6) is the same as that of the Hamiltonian system (2.12), system (2.6) should have the same topological phase portraits as system (2.12) except the straight line l:ϕ=c. Therefore, we should be able to obtain the topological phase portraits of system (2.6) from those of system (2.12).Let(2.13)f(ϕ)=ϕ3-(74c2+2g)ϕ+34c3+5G+2cg.It is easy to obtain the two extreme points off(ϕ) as follows:
(2.14)ϕ±*=±7c2+8g12,forg>-78c2,
from which we can obtain a critical curve for g as follows:
(2.15)g0(c)=-78c2.We obtain two bifurcation curves:(2.16)G1=-1180[72cg+27c3+(8g+7c2)21c2+24g],G2=-1180[72cg+27c3-(8g+7c2)21c2+24g],
from f(ϕ-*)=0 and f(ϕ+*)=0, respectively. Note that when g<g0(c), obviously G1<G2. For convenience, we assume that g∝c2 in this paper, then we have G1∝c3 and G2∝c3.Further, fromG1=0 or G2=0, we can obtain another two critical curves for g, that is,
(2.17)g1(c)=-12c2,(2.18)g2(c)=58c2.Note that (2.18) can also be obtained by letting ϕ+*=c, c>0 or ϕ-*=c, c<0.Let(ϕ*,0) be one of the singular points of system (2.12), then the characteristic values of the linearized system of system (2.12) at the singular point (ϕ*,0) are
(2.19)λ±=±125(ϕ*-c)2f′(ϕ*).From the qualitative theory of dynamical systems, we can determine the property of singular point(ϕ*,0) by the sign of f′(ϕ*) and whether ϕ* equals to c or not. However, we also know that H(c,y)=∞ from (2.7) and (2.8). Therefore, ϕ=c is 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 (2.12) and their relationship with ϕ-*, ϕ+* and c in the following lemma.Lemma 2.1.
Forg>g2(c), one has G1<0<G2 and the singular points of system (2.12) can be described as follows.(a)
IfG<G1, then there is only one singular point denoted as S1(ϕ1,0)(ϕ-*<c<ϕ+*<ϕ1). S1 is a saddle point.(b)
IfG=G1, then there are two singular points denoted as S1(ϕ1,0) and S2(ϕ2,0)(ϕ1=ϕ-*<c<ϕ+*<ϕ2), respectively. S1 is a degenerate saddle point and S2 is a saddle point.(c)
IfG1<G<0, then there are three singular points denoted as S1(ϕ1,0), S2(ϕ2,0), and S3(ϕ3,0)(ϕ1<ϕ-*<ϕ2<c<ϕ+*<ϕ3), respectively. S1 and S3 are saddle points and S2 is a center.(d)
If0<G<G2, then there are three singular points denoted as S1(ϕ1,0), S2(ϕ2,0), and S3(ϕ3,0)(ϕ1<ϕ-*<c<ϕ2<ϕ+*<ϕ3), respectively. S1 and S3 are saddle points and S2 is a center.(e)
IfG=G2, then there are two singular points denoted as S1(ϕ1,0) and S2(ϕ2,0)(ϕ1<ϕ-*<c<ϕ+*=ϕ2), respectively. S1 is a saddle point and S2 is a degenerate saddle point.(f)
IfG>G2, then there is only one singular point denoted as S1(ϕ1,0)(ϕ1<ϕ-*<c<ϕ+*). S1 is a saddle point.Proof.
Lemma2.1 follows easily from the graphics of the function f(ϕ) which can be obtained directly and shown in Figure 1.Figure 1
The graphics off(ϕ) when g>g2(c).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, wheng0(c)<g<g2(c) and G1<G<G2(G≠0), there exist two saddle points and one center lie on the same side of singular line ϕ=c. Hence, there may exist heteroclinic orbits for system (2.6). We will show the existence of heteroclinic orbits for system (2.6) in the following analysis.IfG1<G<G2, we set three solutions of f(ϕ)=0 be ϕs, ϕm, and ϕb (ϕs<ϕm<ϕb), respectively. Through simple calculation, we can express ϕs and ϕb as the function of ϕm, that is,
(2.20)ϕs=-ϕm-8g+7c2-3ϕm22,ϕb=-ϕm+8g+7c2-3ϕm22.It follows fromϕs<ϕm<ϕb that ϕm must satisfy condition
(2.21)ϕm2<8g+7c212.
From H(ϕs,0)=H(ϕb,0), we obtain the expression of G as the function of ϕm,
(2.22)G=1100[4ϕm2-8g+4cϕm-3c39c3+24cg-(8g+15c2)ϕm-8cϕm2+4ϕm3+(2c2-16g-4cϕm)4ϕm2-8g+4cϕm-3c3].Substituting (2.22) into f(ϕm)=0, we obtain the expression of ϕm from f(ϕm)=0 as follows:
(2.23)ϕm1=16(5c-2c2-6g),(2.24)ϕm2=16(5c+2c2-6g),(2.25)ϕm3=-7c2+8g3,(2.26)ϕm4=7c2+8g3,(2.27)ϕm5=12(-c-2c2+2g),(2.28)ϕm6=12(-c+2c2+2g),Note that from (2.23)–(2.28), we obtain three critical curves for g, that is, g0(c), in (2.12), g1(c) in (2.15), and
(2.29)g3(c)=16c2.We then check the conditionϕm2<(8g+7c2)/12 (i.e., (2.21)) for the above ϕms one by one and give the results in the following lemma.Lemma 2.2.
Starting from interval(-(7/8)c2,(5/8)c2), one has the following.(1)
Forg∈(-(7/8)c2,(1/6)c2)⊂(-(7/8)c2,(5/8)c2) and c>0, ϕm=(1/6)(5c-2c2-6g) (i.e., (2.23)) satisfies (2.21).(2)
Forg∈(-(7/8)c2,(1/6)c2)⊂(-(7/8)c2,(5/8)c2) and c<0, ϕm=(1/6)(5c+2c2-6g) (i.e., (2.24)) satisfies (2.21).(3)
For anyg∈(-(7/8)c2,(5/8)c2), (2.25) does not satisfy (2.21).(4)
For anyg∈(-(7/8)c2,(5/8)c2), (2.26) does not satisfy (2.21).(5)
Forg∈(-(1/2)c2,(5/8)c2)⊂(-(7/8)c2,(5/8)c2) and c<0, ϕm=(1/2)(-c-2c2+2g) (i.e., (2.27)) satisfies (2.21).(6)
Forg∈(-(1/2)c2,(5/8)c2)⊂(-(7/8)c2,(5/8)c2) and c>0, ϕm=(1/2)(-c+2c2+2g) (i.e., (2.28)) satisfies (2.21).Proof.
Lemma2.2 follows easily from the definitional domain of the ϕms and general logical reasoning.From Lemma2.2, substituting (2.23) and (2.24) into f(ϕm)=0, respectively, we obtain another two bifurcation curves (denoted by G1* and G2*) for G as follows:
(2.30)G1*=4135(-c3+9cg+(c2-6g)c2-6g),forg0(c)<g<g3(c),c>0,G2*=4135(-c3+9cg-(c2-6g)c2-6g),forg0(c)<g<g3(c),c<0.Similarly, substituting (2.27) and (2.28) into f(ϕm)=0, we have
(2.31)G*=0,forg1(c)<g<g3(c),c<0(org1(c)<g<g3(c),c>0).Note that we have indicated that wheng0(c)<g<g2(c) and G1<G<G2(G≠0), there exist two saddle points and one center lying on the same side of singular line ϕ=c. Therefore, we obtain the fifth critical curve for g from G1*=0(c>0) or G2*=0(c<0),
(2.32)g4(c)=0.Lemma 2.3.
(1) Forg∈(g0(c),g4(c))∪(g4(c),g3(c)), and G=G1*,c>0 (or G=G2*,c<0), there exist heteroclinic orbits for system (2.6).
(2) For anyg∉(g0(c),g4(c))∪(g4(c),g3(c)) or G≠G1*,c>0 and G≠G2*,c<0, there exist no heteroclinic orbits for system (2.6).Proof.
Lemma2.3 follows easily from the above analysis.Therefore, based on the above analysis, we obtain the bifurcation of phase portraits of system (2.6) in Figures 2, 3, 4, 5, 6, 7, 8, and 9 under corresponding conditions.Figure 2
The phase portraits of system (2.6) when g>g2(c).Figure 3
The phase portraits of system (2.6) when g=g2(c).Figure 4
The phase portraits of system (2.6) when g3(c)≤g<g2(c).Figure 5
The phase portraits of system (2.6) when g4(c)<g<g3(c).Figure 6
The phase portraits of system (2.6) when g=g4(c).Figure 7
The phase portraits of system (2.6) when g1(c)<g<g4(c).Figure 8
The phase portraits of system (2.6) when g=g1(c).Figure 9
The phase portraits of system (2.6) when g0(c)<g<g1(c).


## 3. Main Results and the Theoretic Derivations of Main Results

In this section, we state our results about solitons, kink (antikink) solutions, peakons, and periodic cusp waves for the first component of system (1.3). To relate conveniently, we omit φ=ϕ+(2/3)c and the expression of the second component of system (1.3) in the following theorems.Theorem 3.1.
For constant wave speedc, integral constants g and G, one has the following.(1)
Ifc, g, G satisfy one of the following conditions:
(i)
g>g2(c), G1<G<0 and c≠0;(ii)
g1(c)<g≤g2(c), G1<G<0 and c>0;(iii)
g1(c)≤g<g4(c), g4(c)<g<g3(c), 0<G<G1* and c>0;(iv)
g0(c)<g<g1(c), G1<G<G1* and c>0;then there exist smooth solitons for system (1.3), which can be implicitly expressed as
(3.1)(c-ϕ)/(ϕ1*-ϕ)-1(c-ϕ)/(ϕ1*-ϕ)+1·((c-ϕ)/(ϕ1*-ϕ)+α)α((c-ϕ)/(ϕ1*-ϕ)-α)α=e|ξ|,where(3.2)α=c-ϕ1ϕ1*-ϕ1.(2)
Ifc, g, G satisfy one of the following conditions:
(v)
g>g2(c), 0<G<G2 and c≠0;(vi)
g1(c)<g≤g4(c), 0<G<G2 and c<0;(vii)
g1(c)≤g<g4(c), g4(c)<g<g3(c), G2*<G<0 and c<0;(viii)
g0(c)<g<g1(c), G2*<G<G2 and c<0;then there exist smooth solitons for system (1.3), which can be implicitly expressed as
(3.3)(ϕ-c)/(ϕ-ϕ2*)-1(ϕ-c)/(ϕ-ϕ2*)+1·((ϕ-c)/(ϕ-ϕ2*)+β)β((ϕ-c)/(ϕ-ϕ2*)-β)β=e|ξ|,where(3.4)β=ϕ2-cϕ2-ϕ2*.(3)
Ifc, g, G satisfy one of the following conditions:
(ix)
g3(c)≤g<g2(c), G1<G<0 and c<0;(x)
g0(c)<g<g3(c), G1<G<G2* and c<0;then there exist smooth solitons for system (1.3), which can be implicitly expressed as
(3.5)g1(∫0u1‍du-ϕ3-cϕ3-ϕ31*∫0u1‍dn2u1-γ12sn2udu)=e|ξ|,where(3.6)g1=2ϕ32*-c,γ12=k12ϕ32*-ϕ3ϕ31*-ϕ3,k12=ϕ31*-cϕ32*-c,snu1=sinϕ.(4)
Ifc, g, G satisfy one of the following conditions:
(xi)
g3(c)≤g<g2(c), 0<G<G2 and c>0;(xii)
g0(c)<g<g3(c), G1*<G<G2 and c>0;then there exist smooth solitons for system (1.3), which can be implicitly expressed as:
(3.7)g2(∫0u2‍du-c-ϕ4ϕ41*-ϕ4∫0u2‍dn2u1-γ22sn2udu)=e|ξ|,where(3.8)g2=2c-ϕ42*,γ22=k22ϕ4-ϕ42*ϕ4-ϕ41*,k22=c-ϕ41*c-ϕ42*,snu2=sinϕ.Proof.
(1) From the phase portraits in Figures2–9, we see that when c, g, G satisfy one of the conditions, that is, (i), (ii), (iii), or (iv), there exist homoclinic orbits as showed individually in Figures 10(a) and 10(b). The expressions of the homoclinic orbits can be given as follows:
(3.9)y=±(ϕ-ϕ1)ϕ1*-ϕc-ϕ,ϕ1≤ϕ≤ϕ1*<c,
where ϕ1 and ϕ1* can be obtained from (2.8).
Substituting (3.9) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that
(3.10)∫ϕϕ1*‍c-sds(s-ϕ1)ϕ1*-s=|ξ|.
From (3.10), we obtain the solitons (3.1) along with (3.2).
(2) Whenc, g, G satisfy one of the conditions, that is, (v), (vi), (vii), or (viii), there exist homoclinic orbits as showed individually in Figures 8(c) and 8(d). The expressions of the homoclinic orbits can be given as follows:
(3.11)y=±(ϕ2-ϕ)ϕ-ϕ2*ϕ-c,c<ϕ2*≤ϕ≤ϕ2,
where ϕ2 and ϕ2* can be obtained from (2.7).
Substituting (3.11) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that
(3.12)∫ϕ2*ϕ‍s-cds(ϕ2-s)s-ϕ2*=|ξ|.
From (3.12), we obtain the solitons (3.3) along with (3.4).
(3) Whenc, g, G satisfy one of the conditions, that is, (ix) or (x), there exist homoclinic orbits as showed individually in Figure 8(e). The expressions of the homoclinic orbits can be given as follows:
(3.13)y=±(ϕ-ϕ3)(ϕ31*-ϕ)(ϕ32*-ϕ)ϕ-c,c<ϕ3≤ϕ≤ϕ31*<ϕ32*,
where ϕ3, ϕ31* and ϕ32* can be obtained from (2.7).
Substituting (3.13) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that
(3.14)∫ϕϕ31*‍s-cds(s-ϕ3)(ϕ31*-s)(ϕ32*-s)=|ξ|.
From (3.14) [8], we obtain the solitons (3.5) along with (3.6).
(4) Whenc, g, G satisfy one of the conditions, that is, (xi) or (xii), there exist homoclinic orbits as showed individually in Figure 8(f). The expressions of the homoclinic orbits can be given as follows:
(3.15)y=±(ϕ4-ϕ)(ϕ-ϕ41*)(ϕ-ϕ42*)c-ϕ,ϕ42*<ϕ41*≤ϕ≤ϕ4<c,
where ϕ4, ϕ41*, and ϕ42* can be obtained from (2.8).
Substituting (3.15) into the first equation of system (2.6), and integrating along the homoclinic orbits, it follows that
(3.16)∫ϕ41*ϕ‍c-sds(ϕ4-s)(s-ϕ41*)(s-ϕ42*)=|ξ|.
From (3.16) [8], we obtain the solitons (3.7) along with (3.8).The different kinds of homoclinic orbits for system (2.6).
(a)
(b)
(c)
(d)
(e)
(f)Theorem 3.2.
If constant wave speedc, integral constants g and G satisfy g0(c)<g<g4(c) or g4(c)<g<g3(c), and G=G1* (c>0) or G=G2* (c<0), then there exist kink (antikink) solutions for system (1.3).Proof.
We have showed that, wheng0(c)<g<g4(c) or g4(c)<g<g3(c), and G=G1* (c>0) or G=G2* (c<0), there exist heteroclinic orbits for system (2.6). The heteroclinic orbits can be expressed as
(3.17)y=±(ϕ-ϕs)(ϕb-ϕ)c-ϕ,forc>0,
where
(3.18)ϕs=112(-5c+2c2-6g-15(11c2+4cc2-6g+24g)),ϕb=112(-5c+2c2-6g+15(11c2+4cc2-6g+24g)),
which can be obtained by substituting (2.23) into (2.20).
Substituting (3.17) into the first equation of system (2.6), and integrating along the heteroclinic orbits, it follows that
(3.19)∫ϕ0ϕ‍c-sds(s-ϕs)(ϕb-s)=±ξ,
where ϕ0∈(ϕs,ϕb) is the initial value.
From (3.19), we have
(3.20)(c-ϕs-c-ϕ)c-ϕs/(ϕb-ϕs)(c-ϕs+c-ϕ)c-ϕs/(ϕb-ϕs)·(c-ϕ+c-ϕb)c-ϕb/(ϕb-ϕs)(c-ϕ-c-ϕb)c-ϕb/(ϕb-ϕs)=(c-ϕs-c-ϕ0)c-ϕs/(ϕb-ϕs)(c-ϕs+c-ϕ0)c-ϕs/(ϕb-ϕs)·(c-ϕ0+c-ϕb)c-ϕb/(ϕb-ϕs)(c-ϕ0-c-ϕb)c-ϕb/(ϕb-ϕs)e±ξ.
The case when c<0, can be analyzed similarly. We omit it here for the ease of simplicity.Theorem 3.3.
(1) Ifg=g4(c), G=0 and c≠0, then there exist peakons for system (1.3), which can be explicitly expressed as
(3.21)ϕ=52ce-|x-ct|-32c.
(2) Ifg4(c)≤g<g2(c), G=0 and c≠0, then system (1.3) has periodic cusp waves
(3.22)u(x,t)=ϕ(ξ-2iT)+32c,
where i=0,±1,±2,…,ξ=x-ct∈[(2i-1)T,(2i+1)T], and
(3.23)ϕ(ξ)=14(5c-25c2-40g)e|x-ct|+14(5c+25c2-40g)e-|x-ct|-32c,
with
(3.24)T=ln(5c+25c2-40g210g).Proof.
(1) Wheng=g4(c), G=0 and c≠0, from Figure 6, we see that there is a triangle orbit, which can be expressed as
(3.25)y=±(ϕ+32c),for-32c≤ϕ≤c(c>0),(3.26)ϕ=c,for|y|≤52c(c>0).
Substituting (3.25) into the first equation of system (2.6), and integrating along the triangle orbits, it follows that
(3.27)∫ϕc‍dtt+(3/2)c=|ξ|.
From (3.27), we obtain peakons (3.21).
(2) Wheng4(c)≤g<g2(c), G=0 and c≠0, from Figures 4 and 5, we see that there is a semiellipse orbit, which can be expressed as
(3.28)y=±ϕ2+3cϕ+94c2-10g,for12(-3c+210g)≤ϕ≤c(c>0),(3.29)ϕ=c,for|y|≤5(c2-8g)2(c>0).
Substituting (3.28) into the first equation of system (2.6), and integrating along the semiellipse orbits, it follows that
(3.30)∫ϕc‍dtt2+3ct+(9/4)c2-10g=|ξ|.
From (3.30), we obtain periodic cusp waves (3.22) along with (3.23) and (3.24).
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-equation (1.3), 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, kink (antikink) solutions for system (1.3) are given. Besides, we also obtain peakons and periodic cusp waves with explicit expressions for system (1.3).



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