By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the above waves are determined.

1. Introduction

Recently, by using an algebraic method, Hon and Fan [1] studied the following coupled Higgs field equation: 𝑢𝑡𝑡−𝑢𝑥𝑥−𝛼𝑢+𝛽|𝑢|2𝑢−2𝑢𝑣=0,𝑣𝑡𝑡+𝑣𝑥𝑥−𝛽|𝑢|2𝑥𝑥=0.(1.1) The Higgs field equation [2] describes a system of conserved scalar nucleons interacting with neutral scalar mesons. Here, real constant 𝑣 represents a complex scalar nucleon field and 𝑢(𝑥,𝑡) a real scalar meson field. Equation (1.1) is the coupled nonlinear Klein-Gordon equation for 𝛼<0,𝛽<0 and the coupled Higgs field equation for 𝛼>0, 𝛽>0. The existence of N-soliton solutions for (1.1) has been shown by the Hirota bilinear method [3].

It is very important to consider the bifurcation behavior for the traveling wave solutions of (1.1). In this paper, we consider (1.1) and its traveling wave solutions in the form of 𝑢(𝑥,𝑡)=𝜙(𝜉)𝑒𝑖𝜂(𝜉),𝑣(𝑥,𝑡)=𝑣(𝜉),𝜉=𝑥−𝑐𝑡.(1.2) Substitute (1.2) into (1.1) and for 𝑐2−1≠0 reduce system (1.1) to the following system of ordinary differential equations:𝑐2𝜙−1î…žî…žâˆ’î€·ğ‘2𝜙𝜂−12−𝛼𝜙+𝛽𝜙3−2𝜙𝑣=0,2ğœ™î…žğœ‚î…ž+ğœ™ğœ‚î…žî…žî€·ğ‘=0,2𝑣+1î…žî…žî€·ğœ™âˆ’ğ›½2=0,(1.3) where “” is the derivative with respect to 𝜉. Integrating second equation of (1.3) once and integrating third equation of (1.3) twice, respectively, we have ğœ‚î…ž=𝑔2𝜙2,𝑣=𝛽𝜙2+𝑔1𝑐2,+1(1.4) where 𝑔2≠0,𝑔1 are integral constants. Substituting (1.4) into first equation of (1.3), we have 𝑐2𝜙−1î…žî…žâˆ’î€·ğ‘2𝑔−122𝜙3−𝛼+2𝑔1𝑐2𝛽𝑐+1𝜙+2−1𝑐2𝜙+13=0.(1.5) Equation (1.5) is equivalent to the two-dimensional systems as follows: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦𝜙𝑑𝜉=ğ‘Ž3+𝑏𝜙+𝑒𝜙−3(1.6) with the first integral 𝑦21=ğ‘Ž2𝜙4+𝑏𝜙2−𝑒𝜙−2,𝑦+ℎ(1.7)𝐻(𝜙,𝑦)=2ğ‘Žâˆ’12𝜙4−𝑏𝜙2+𝑒𝜙−2=ℎ,(1.8) where ğ‘Ž=−𝛽/(𝑐2+1),𝑏=−(𝛼(𝑐2+1)+2𝑔1)/𝛽(𝑐2−1),𝑒=−𝑔22(𝑐2+1)/𝛽≠0, ğ‘Žğ‘’>0.

System (1.6) is a 3-parameter planar dynamical system depending on the parameter group (ğ‘Ž,𝑏,𝑒). For a fixed ğ‘Ž, we will investigate the bifurcations of phase portraits of (1.6) in the phase plane (𝜙,𝑦) as the parameters 𝑏,𝑒 are changed. Here we are considering a physical model where only bounded traveling waves are meaningful. So we only pay attention to the bounded solutions of (1.6).

Suppose that 𝜙(𝜉) is a continuous solution of (1.6) for 𝜉∈(−∞,∞) and limğœ‰â†’âˆžğœ™(𝜉)=ğ‘Ž1, limğœ‰â†’âˆ’âˆžğœ™(𝜉)=ğ‘Ž2. Recall that (i) 𝜙(𝑥,𝑡) is called a solitary wave solution if ğ‘Ž1=ğ‘Ž2; (ii) 𝜙(𝑥,𝑡) is called a kink or antikink solution if ğ‘Ž1â‰ ğ‘Ž2. Usually, a solitary wave solution of (1.6) corresponds to a homoclinic orbit of (1.6); a kink (or antikink) wave solution (1.6) corresponds to a heteroclinic orbit (or the so-called connecting orbit) of (1.6). Similarly, a periodic orbit of (1.6) corresponds to a periodically traveling wave solution of (1.6). Thus, to investigate all possible bifurcations of solitary waves and periodic waves of (1.6), we need to find all periodic annuli and homoclinic orbits of (1.6), which depend on the system parameters. The bifurcation theory of dynamical systems (see [4–11]) plays an important role in our study.

The paper is organized as follows. In Section 2, we discuss bifurcations of phase portraits of (1.6), where explicit parametric conditions will be derived. In Section 3, all explicit parametric representations of bounded traveling wave solutions are given. Section 4 contains the concluding remarks.

2. Bifurcations of Phase Portraits of (1.6)

In this section, we study all possible periodic annuluses defined by the vector fields of (1.6) when the parameters 𝑏,𝑒 are varied.

Let 𝑑𝜉=𝜙3𝑑𝜁. Then, except on the straight lines 𝜙=0, the system (1.6) has the same topological phase portraits as the following system: 𝑑𝜙𝑑𝜁=𝜙3𝑦,𝑑𝑦𝜙𝑑𝜁=ğ‘Ž6+𝑏𝜙4.+𝑒(2.1) Now, the straight lines 𝜙=0 is an integral invariant straight line of (2.1).

Denote that 𝑓(𝜙)=𝜙6+𝑏𝜙4+𝑒,ğ‘“î…ž(𝜙)=2𝜙33𝜙2.+2𝑏(2.2) When 𝜙=𝜙±√=±−2𝑏/3, ğ‘“î…ž(𝜙±)=0. We have 𝑓𝜙±=4𝑏327+𝑒,(2.3) which implies the relations in the (𝑏,𝑒)-parameter plane 𝐿∶𝑒=−4𝑏3.27(2.4)

Thus, we have the following.(i)If 𝑓(𝜙±)<0, 𝑓(0)>0, there exist 4 equilibrium points of (2.1): 𝜙1<𝜙2≤0<𝜙3<𝜙4.(ii)If 𝑓(𝜙±)<0, 𝑓(0)<0, there exist 2 equilibrium points of (2.1): 𝜙1≤0<𝜙2.(iii)If 𝑓(𝜙±)>0, 𝑓(0)>0, there exist no equilibrium points of (2.1).

Let 𝑀(𝜙𝑒,𝑦𝑒) be the coefficient matrix of the linearized system of (2.1) at an equilibrium point (𝜙𝑒,𝑦𝑒). Then, we have 𝐽𝜙𝑒𝑀𝜙,0=det𝑒,0=ğ‘Žğœ™3ğ‘’ğ‘“î…žî€·ğœ™ğ‘’î€¸=−2ğ‘Žğœ™6𝑒3𝜙2𝑒.+2𝑏(2.5) By the theory of planar dynamical systems, we know that for an equilibrium point of a planar integrable system, if 𝐽<0, then the equilibrium point is a saddle point; if 𝐽>0 and Trace(𝑀(𝜙𝑒,𝑦𝑒))=0, then it is a center point; if 𝐽>0 and (Trace(𝑀(𝜙𝑒,𝑦𝑒))2−4𝐽(𝜙𝑒,𝑦𝑒)>0 then it is a node; if 𝐽=0 and the index of the equilibrium point is 0, then it is a cusp; otherwise, it is a high-order equilibrium point.

For the function defined by (1.8), we denote that â„Žğ‘–î€·ğœ™=𝐻𝑖1,0=−23𝑏𝜙2𝑖−𝑒𝜙𝑖−2,𝑖=1-4.(2.6)

We next use the above statements to consider the bifurcations of the phase portraits of (2.1). In the (𝑏,𝑒)-parameter plane, the curves 𝐿 and the straight line 𝑒=0 partition it into 4 regions shown in Figure 1.

We use Figures 2 and 3 to show the bifurcations of the phase portraits of (2.1). Notice that for ğ‘Ž>0, 𝑒<0, ⋃(𝑏,𝑒)∈(𝐼𝐼𝐼)(𝐼𝑉) or for ğ‘Ž<0, 𝑒>0, ⋃(𝑏,𝑒)∈(𝐼)(𝐼𝐼), and we have ğ‘Žğ‘’<0, So that we would not give the phase portrait of (2.1) for these cases.

Case 1 (ğ‘Ž>0). We use Figure 2 to show the bifurcations of the phase portraits of (2.1).

Case 2 (ğ‘Ž<0). We use Figure 3 to show the bifurcations of the phase portraits of (2.1).

3. Exact Explicit Parametric Representations of Traveling Wave Solutions of (1.6)

In this section, we give all exact explicit parametric representations of solitary wave solutions and periodic wave solutions. Denote that 𝑠𝑛(𝑥,𝑘) is the Jacobian elliptic functions with the modulus 𝑘 and ∏(𝜑,𝛼2,𝑘) is Legendre's incomplete elliptic integral of the third kind (see [12]).

(1) Suppose that ğ‘Ž>0, (𝑏,𝑒)∈(𝐼𝐼). Notice that 𝐻(𝜙1,0)=−(1/2)𝜙41−𝑏𝜙21+𝑒𝜙1−2=ℎ1, corresponding to 𝐻(𝜙,𝑦)=ℎ1 defined by (1.8), and we see from (1.6) that the arch curve connects 𝐴(𝜙1,0) (see Figure 2(b)). The arch curve has the algebraic equation 𝑦21=ğ‘Ž2𝜙4+𝑏𝜙2−𝑒𝜙−2+ℎ1𝜙=ğ‘Ž2−𝜓3212+𝑏+𝜓3𝜙−2,(3.1) where 𝜓3>𝜓2>0>𝜓1 satisfies the equation𝜓3+𝑏𝜓2+𝑒=0.(3.2)

By using the first equations of (1.6) and (3.1), we obtain the parametric representation of (1.6), a smooth solitary wave solution of valley type and a smooth solitary wave solution of peak type as follows: îƒŽğœ™(𝜉)=±−2𝑏+𝜓33+2𝑏+2𝜓3tanh2î‚™ğ‘Žî‚€3𝑏+2𝜓3𝜉.(3.3)

Thus, (1.1) has the following solitary wave solution of valley type and a solitary wave solution of peak type as follows: 𝑢1=±−2𝑏+𝜓33+2𝑏+2𝜓3tanh2î‚™ğ‘Žî‚€3𝑏+2𝜓3𝜉𝑒𝑖𝜂1(𝜉),𝑣1=−2𝛽𝑏+𝜓3−b+(3/2)𝜓3tanh2î”ğ‘Žî€·ğ‘+(3/2)𝜓3𝜉+𝑔1c2,𝜂+11=𝑔2𝜓3âŽ¡âŽ¢âŽ¢âŽ£îƒŽğœ‰+−1ğ‘Žî€·ğ‘+𝜓3⎛⎜⎜⎝arctan−𝑏+(3/2)𝜓3𝑏+𝜓3tanh2î”ğ‘Žî€·ğ‘+𝜓3î€¸ğœ‰âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦.(3.4)

(2) Suppose that â‹ƒğ‘Ž<0,(𝑏,𝑒)∈(𝐼𝐼𝐼)(𝐼𝑉). Notice that 𝐻(𝜙1,0)=𝐻(𝜙2=−𝜙1,0)=−(1/2)𝜙41−𝑏𝜙21+𝑒𝜙1−2=ℎ1, corresponding to 𝐻(𝜙,𝑦)=ℎ, ℎ∈(−∞,ℎ1) defined by (1.8), and system (1.6) has two families of periodic solutions enclosing the center 𝐴+(𝜙1,0) and 𝐴−(−𝜙1,0), respectively. These orbits determine uncountably infinite many periodic wave solutions of (1.1) (see Figures 3(a) and 3(b)). These orbits have the algebraic equation îƒŽğ‘¦=Â±ğ‘Žğœ™212𝜙6+𝑏𝜙4−𝑒+â„Žğœ™2.(3.5) Integrating them along the periodic orbits, it follows that 𝜙𝑑𝜙√−𝜙6−2𝑏𝜙4−2â„Žğœ™2+2𝑒=Â±âˆ’ğ‘Ž2𝜉.(3.6) Substituting 𝜙2=𝜓 into (3.6), we have 𝑑𝜓𝜓𝑀−𝜓𝜓−𝜓𝑙𝜓−𝜓𝑚1=±2î‚™âˆ’ğ‘Ž2𝜉,(3.7) where 𝜓𝑀>𝜓𝑙>0>𝜓𝑚. From (3.7), we have î„¶î„µî„µâŽ·ğœ™=±𝜓𝑙𝜓𝑀−𝜓𝑚−𝜓𝑚𝜓𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1𝜓𝑙𝜓𝑀−𝜓𝑚𝜓1−𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1,(3.8) where Ω1√=(1/4)âˆ’ğ‘Ž(𝜓𝑀−𝜓𝑚)/2,𝑘21=(𝜓𝑀−𝜓𝑙)/(𝜓𝑀−𝜓𝑚).

Thus, (1.1) has the following uncountably infinite many periodic wave solutions as follows: 𝑢2⎷=±𝜓𝑙𝜓𝑀−𝜓𝑚−𝜓𝑚𝜓𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1𝜓𝑙𝜓𝑀−𝜓𝑚𝜓1−𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1𝑒𝑖𝜂2(𝜉),𝑣2=1𝑐2𝛽𝜓+1𝑙𝜓𝑀−𝜓𝑚−𝜓𝑚𝜓𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1𝜓𝑙𝜓𝑀−𝜓𝑚𝜓1−𝑀−𝜓𝑙𝑠𝑛2Ω1𝜉,𝑘1+𝑔1,𝜂2=𝑔2𝛼2Ω1𝛼2−𝛼21𝜑,𝛼2,𝑘1+𝛼21𝜉,(3.9)

where 𝛼21=𝜓𝑀−𝜓𝑙, 𝛼2=𝜓𝑚(𝜓𝑀−𝜓𝑙)/𝜓𝑙(𝜓𝑀−𝜓𝑚), 𝜑=ğ‘Žğ‘šğœ‰.

4. Conclusion

In this paper, we have considered all traveling wave solutions for the coupled Higgs field equation (1.1) in its parameter space, by using the method of dynamical systems. We obtain all parametric representations for solitary wave solutions and uncountably infinite many periodic wave solutions of (1.1) in different parameter regions of the parameter space.


This research was supported by NNSF of China (11061010) and the Foundation of Guangxi Key Lab of Trusted Software.