Bifurcations of Traveling Wave Solutions for the Coupled Higgs Field Equation

Abstract

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: 𝑢_𝑡𝑡−𝑢_𝑥𝑥−𝛼𝑢+𝛽|𝑢|²𝑢−2𝑢𝑣=0,𝑣_𝑡𝑡+𝑣_𝑥𝑥−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)𝑒^{𝑖𝜂(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:𝑐²−1𝜙^−𝑐²−2𝜙1_𝜙𝜂^²−𝛼𝜙+𝛽𝜙³−2𝜙𝑣=0,𝑐𝜂^+𝜙𝜂^=0,₂+1𝑣^−𝛽𝜙²^=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 𝜂^=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 𝑐²−1𝜙^−𝑐²−1𝑔²²𝜙³−𝛼+2𝑔¹𝑐²+1𝜙+𝛽𝑐²−1𝑐²+1𝜙³=0.(1.5) Equation (1.5) is equivalent to the two-dimensional systems as follows: 𝑑𝜙𝑑𝜉=𝑦,𝑑𝑑𝑦𝜉=𝑎𝜙³+𝑏𝜙+𝑒𝜙⁻³(1.6) with the first integral 𝑦²=𝑎12𝜙⁴𝐻(𝜙,𝑦)=𝑦₂+𝑏𝜙²−𝑒𝜙⁻²+ℎ,(1.7)𝑎−21𝜙⁴−𝑏𝜙²+𝑒𝜙⁻²=ℎ,(1.8) where 𝑎=−𝛽/(𝑐²+1),𝑏=−(𝛼(𝑐²+1)+2𝑔₁)/𝛽(𝑐²−1),𝑒=−𝑔²²(𝑐²+1)/𝛽≠0, 𝑎𝑒>0.

System (1.6) is a 3-parameter planar dynamical system depending on the parameter group (𝑎,𝑏,𝑒). For a fixed 𝑎, 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 lim_𝜉→∞𝜙(𝜉)=𝑎₁, lim_{𝜉→−∞}𝜙(𝜉)=𝑎₂. Recall that (i) 𝜙(𝑥,𝑡) is called a solitary wave solution if 𝑎₁=𝑎₂; (ii) 𝜙(𝑥,𝑡) is called a kink or antikink solution if 𝑎₁≠𝑎₂. 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,𝑑𝑦𝑑𝜁=𝑎𝜙⁶+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,𝑓^(𝜙)=2𝜙³3𝜙²+2𝑏.(2.2) When 𝜙=𝜙_±=±√−2𝑏/3, 𝑓^(𝜙_±)=0. We have 𝑓𝜙_±=4𝑏³27+𝑒,(2.3) which implies the relations in the (𝑏,𝑒)-parameter plane 𝐿∶𝑒=−4𝑏³27.(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(𝜙_𝑒,𝑦_𝑒) be the coefficient matrix of the linearized system of (2.1) at an equilibrium point (𝜙_𝑒,𝑦_𝑒). Then, we have 𝐽𝜙_𝑒,0=det𝑀𝜙_𝑒,0=𝑎𝜙^3𝑒𝑓^𝜙_𝑒=−2𝑎𝜙^6𝑒3𝜙^2𝑒+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(𝜙_𝑒,𝑦_𝑒))=0, then it is a center point; if 𝐽>0 and (Trace(𝑀(𝜙_𝑒,𝑦_𝑒))²−4𝐽(𝜙_𝑒,𝑦_𝑒)>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 ℎ_𝑖=𝐻𝜙_𝑖,0=−123𝑏𝜙^2𝑖−𝑒𝜙^𝑖−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.

Figure 1 

The bifurcation set of (1.6) in (𝑏,𝑒)-parameter plane.

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

(a) (𝑏,𝑒)∈(𝐼)

(b) (𝑏,𝑒)∈(𝐼𝐼)

Figure 2 

The phase portraits of (2.1) for 𝑎>0.

(a) (𝑏,𝑒)∈(𝐼𝐼𝐼)

(b) (𝑏,𝑒)∈(𝐼𝑉)

Figure 3 

The phase portraits of (2.1) for 𝑎<0.

Case 1 (𝑎>0). We use Figure 2 to show the bifurcations of the phase portraits of (2.1).

Case 2 (𝑎<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 𝑎>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=𝑎12𝜙⁴=𝑎𝜙₂+𝑏𝜙²−𝑒𝜙⁻²+ℎ₁−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: 𝜙(𝜉)=±−2𝑏+𝜓₃+2𝑏+32𝜓³tanh²𝑎𝑏+32𝜓³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=±−2𝑏+𝜓₃+2𝑏+32𝜓³tanh²𝑎𝑏+32𝜓³𝜉𝑒^{𝑖𝜂1(𝜉)}𝑣₁,=−2𝛽𝑏+𝜓³−b+(3/2)𝜓₃tanh²𝑎𝑏+(3/2)𝜓₃𝜉+𝑔₁𝜂c²+1₁,=𝑔²𝜓₃⎡⎢⎢⎣𝜉+−1𝑎𝑏+𝜓₃arctan⎛⎜⎜⎝−𝑏+(3/2)𝜓³𝑏+𝜓₃tanh²𝑎𝑏+𝜓₃𝜉⎞⎟⎟⎠⎤⎥⎥⎦.(3.4)

(2) Suppose that 𝑎<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 𝑦=±𝑎𝜙²21𝜙⁶+𝑏𝜙⁴−𝑒+ℎ𝜙².(3.5) Integrating them along the periodic orbits, it follows that √−𝜙⁶−2𝑏𝜙𝜙𝑑₄𝜙−2ℎ𝜙²+2𝑒=±−𝑎2𝜉.(3.6) Substituting 𝜙²=𝜓 into (3.6), we have 𝑑𝜓𝜓_𝑀−𝜓𝜓−𝜓_𝑙𝜓−𝜓_𝑚=±12−𝑎2𝜉,(3.7) where 𝜓_𝑀>𝜓_𝑙>0>𝜓_𝑚. From (3.7), we have 𝜙=±⎷𝜓_𝑙𝜓_𝑀−𝜓_𝑚−𝜓_𝑚𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁𝜓_𝑙𝜓_𝑀−𝜓_𝑚1−𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁,(3.8) where Ω₁=(1/4)√−𝑎(𝜓_𝑀−𝜓_𝑚)/2,𝑘²¹=(𝜓_𝑀−𝜓_𝑙)/(𝜓_𝑀−𝜓_𝑚).

Thus, (1.1) has the following uncountably infinite many periodic wave solutions as follows: 𝑢₂=±⎷𝜓_𝑙𝜓_𝑀−𝜓_𝑚−𝜓_𝑚𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁𝜓_𝑙𝜓_𝑀−𝜓_𝑚1−𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁𝑒^{𝑖𝜂2(𝜉)}𝑣₂,=1𝑐²+1𝛽𝜓^𝑙𝜓_𝑀−𝜓_𝑚−𝜓_𝑚𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁𝜂₂𝜓_𝑙𝜓_𝑀−𝜓_𝑚1−𝜓_𝑀−𝜓_𝑙𝑠𝑛²Ω₁𝜉,𝑘₁+𝑔¹,=𝑔²𝛼²Ω₁𝛼²−𝛼²¹𝜑,𝛼²,𝑘₁+𝛼²¹𝜉,(3.9)

where 𝛼²¹=𝜓_𝑀−𝜓_𝑙, 𝛼²=𝜓_𝑚(𝜓_𝑀−𝜓_𝑙)/𝜓_𝑙(𝜓_𝑀−𝜓_𝑚), 𝜑=𝑎𝑚𝜉.

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.

Acknowledgment

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