# New Types of Doubly Periodic Standing Wave Solutions for the Coupled Higgs Field Equation

Authors: Gui-qiong Xu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Based on the Hirota bilinear method and theta function identities, we obtain a new type of doubly periodic standing wave solutions for a coupled Higgs field equation. The Jacobi elliptic function expression and long wave limits of the periodic solutions are also presented. By selecting appropriate parameter values, we analyze the interaction properties of periodic-periodic waves and periodic-solitary waves by some figures.


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## Body

## 1. Introduction

As is well known, the investigation of the explicit exact solutions to nonlinear evolution equations (NLEEs) plays an important role in the study of nonlinear physical phenomena [1, 2]. In the past decades, much effort has been spent on the construction of various forms of periodic wave solutions for NLEEs, and many powerful methods have been developed such as the algebrogeometrical approach [3, 4], nonlinearization approach of Lax pairs [5, 6], Weierstrass elliptic function expansion method [7, 8], Jacobi elliptic function expansion method [9–11], and subsidiary ordinary differential equation methods [12–18].In this paper, we will focus on a coupled Higgs field equation with important physical interests [19],
(1)utt-uxx-βu+δ|u|2u-2uv=0,vtt+vxx-δ(|u|2)xx=0,
which describes a system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. Here β and δ are constants, and the function v=v(x,t) represents a real scalar meson field and u=u(x,t) a complex scalar nucleon field. Equation (1) is related to some nonlinear models with physical interests. Equation (1) is the coupled nonlinear Klein-Gordon equations for β<0 and δ<0 and the Higgs equations for β>0 and δ>0. Much attention has been paid to investigate exact explicit solutions and integrable properties of (1). The symmetry reductions, the homoclinic orbits, N-soliton solutions, rogue wave solutions, Jacobi periodic solutions, and other types of travelling wave solutions have been presented [19–24].The Hirota bilinear method is a powerful tool for constructing various exact solutions for NLEEs, which include soliton, negaton, rogue waves, rational solutions, and quasiperiodic solutions [25–35]. Recently, by means of Hirota bilinear method and theta function identities [36–38], Fan et al. obtained a class of doubly periodic standing wave solutions of (1) [39], which was expressed as rational functions of elliptic/theta functions of different moduli. A significant portion of these solutions represents travelling wave, that is, those which will remain steady in an appropriate frame of reference. Physically, the envelope of these oscillations is bounded by a pattern periodic in both time and space. The focus of this work is to investigate new types of doubly periodic standing wave solutions for (1).This paper is organized as follows. In Section2, we briefly illustrate some properties of theta functions and Jacobi elliptic functions. In Section 3, we construct a new kind of doubly periodic wave solutions for the coupled Higgs field equation. In Section 4, for the obtained periodic solution, we derive its Jacobi elliptic function representation and analyze interaction properties by some figures. Some conclusions are given in Section 5.


## 2. The Theta and Jacobi Elliptic Functions

The main tools used in this paper are Hirota operators and theta function formulas, which will be discussed here, to fix the notations and make our presentation self-contained. More formulas for the theta functions can be found in [36, 37].The Riemann theta functions of genus oneϑn(x,τ)  (n=1–4), the parameter q (the nome), and τ (pure imaginary) are defined by [40]
(2)ϑ1(x,τ)=2∑n=0∞‍(-1)nq(n+1/2)2sin⁡(2n+1)x,ϑ2(x,τ)=2∑n=0∞‍q(n+1/2)2cos⁡(2n+1)x,ϑ3(x,τ)=1+2∑n=1∞‍qn2cos⁡2nx,ϑ4(x,τ)=1+2∑n=1∞‍(-1)nqn2cos⁡2nx,0<q<1,q=exp⁡(iπτ),q=exp⁡(-πK′K).
Here, K,K′ are the complete elliptic integrals of the first kind:
(3)K=∫0π/2‍(1-k2sin2ξ)-1/2dξ,K1=∫0π/2‍(1-k12sin2ξ)-1/2dξ.ϑ1 is an odd function while the other three are even functions. The zeros of ϑ1, ϑ2, ϑ3, and ϑ4 are at Mπ+Nπτ, (M+(1/2))π+Nπτ, (M+(1/2))π+(N+(1/2))πτ, and Mπ+(N+(1/2))πτ, respectively, and M and N are integers. Since ϑ1, ϑ2, and (ϑ3, ϑ4) are related by a phase shift of π/2, there are roughly two groups of theta functions.There exists a large class of bilinear identities involving products of theta functions, some of which are listed here:(4)Dx2ϑ1(x,τ)·ϑ2(x,τ)=(b2+ϑ24(0,τ))ϑ1(x,τ)ϑ2(x,τ),Dx2ϑ1(x,τ)·ϑ4(x,τ)=(b2-ϑ44(0,τ))ϑ1(x,τ)ϑ4(x,τ),Dx2ϑ2(x,τ)·ϑ3(x,τ)=(b2-ϑ44(0,τ))ϑ2(x,τ)ϑ3(x,τ),Dx2ϑ1(x,τ)·ϑ3(x,τ)=b3ϑ1(x,τ)ϑ3(x,τ),Dx2ϑ2(x,τ)·ϑ4(x,τ)=b3ϑ2(x,τ)ϑ4(x,τ),Dx2ϑ2(x,τ)·ϑ2(x,τ)=b2ϑ22(x,τ)-b1ϑ42(x,τ),Dx2ϑ4(x,τ)·ϑ4(x,τ)=b1ϑ22(x,τ)+b2ϑ42(x,τ),
where for simplicity we have used the notations
(5)b1=2ϑ22(0,τ)ϑ42(0,τ),b2=2ϑ3′′(0,τ)ϑ3(0,τ),b3=ϑ2′′(0,τ)ϑ2(0,τ)+ϑ4′′(0,τ)ϑ4(0,τ),
and the formulas (4) can be derived from product identities of theta functions; the details can be found in [36, 37].There are close connections between theta functions and elliptic functions as follows:(6)ϑ1(γx,τ)ϑ4(γx,τ)=k1/2sn(λx,k)≡S,ϑ2(γx,τ)ϑ4(γx,τ)=k1/2(1-k2)-1/4cn(λx,k)≡C,ϑ3(γx,τ)ϑ4(γx,τ)=(1-k2)-1/4dn(λx,k)≡D,
where λ=γϑ32(0,τ) and k=ϑ22(0,τ)/ϑ32(0,τ),
(7)ϑ1(ωt,τ1)ϑ4(ωt,τ1)=k11/2sn(ρt,k1)≡S1,ϑ2(ωt,τ1)ϑ4(ωt,τ1)=k11/2(1-k12)-1/4cn(ρt,k1)≡C1,ϑ3(ωt,τ1)ϑ4(ωt,τ1)=(1-k12)-1/4dn(ρt,k1)≡D1,
where ρ=ωϑ32(0,τ1) and k1=ϑ22(0,τ1)/ϑ32(0,τ1). It is clearly shown that arguments of the theta and elliptic functions are related by a scale factor.


## 3. A New Class of Doubly Periodic Wave Solutions

In this section, we construct a new class of doubly periodic wave solutions by Hirota bilinear method [2]. For (1), substituting the following transformation
(8)u(x,t)=gf,v(x,t)=v0+2(ln⁡f)xx
into (1) and integrating with respect to x yield the bilinear forms
(9)(Dt2+Dx2-Λ)f·f=δg·g⋆,(10)[Dt2-Dx2-(β+2v0+Λ)]g·f=0,
where v0 is a constant, Λ=Λ(t) is an integration constant, and D is the well-known Hirota bilinear operator. Equation (10) is slightly different from the results given in [39] by adding one integration constant term.The crucial step to derive doubly periodic wave solutions is to supposef and g in (9) and (10) as suitable combination of different theta functions. To obtain new doubly periodic wave solutions, we make a new ansatz,
(11)g=η[ϑ3(ωt,τ1)ϑ2(γx,τ)+iϑ1(ωt,τ1)ϑ4(γx,τ)]exp⁡⁡(ipt),f=ϑ4(ωt,τ1)ϑ4(γx,τ)+ϑ2(ωt,τ1)ϑ2(γx,τ),
where the parameters ω, γ, η, and p are constants to be determined and the period τ in the spatial direction and the period τ1 in the temporal direction are purely imaginary constants.Inserting the ansatz (11) into (9), together with the theta function identities given in Section 2, we set the coefficients of the terms ϑ22(γx,τ), ϑ42(γx,τ), and ϑ2(γx,τ)ϑ4(γx,τ) to be zero and get
(12)(γ2b2+ω2b~2-Λ-δη2b~4)ϑ22(ωt,τ1)+(γ2b1-ω2b~1-δη2b~5)ϑ42(ωt,τ1)=0,(γ2b1-ω2b~1-δη2b~5)ϑ22(ωt,τ1)-(γ2b2+ω2b~2-Λ-δη2b~4)ϑ42(ωt,τ1)=0,(γ2b3+ω2b~3-Λ)ϑ2(ωt,τ1)ϑ4(ωt,τ1)=0,
where the theta constants b~j(j=1,…,5) are defined by
(13)b~1=2ϑ22(0,τ1)ϑ42(0,τ1),b~2=2ϑ3′′(0,τ1)ϑ3(0,τ1),b~3=ϑ2′′(0,τ1)ϑ2(0,τ1)+ϑ4′′(0,τ1)ϑ4(0,τ1),b~4=ϑ22(0,τ1)ϑ32(0,τ1),b~5=ϑ42(0,τ1)ϑ32(0,τ1).Due to the linear independence of theta functions, we set the coefficients ofϑ22(ωt,τ1), ϑ42(ωt,τ1), and ϑ2(ωt,τ1)ϑ4(ωt,τ1) in (12) to be zero and obtain
(14)γ2b1-ω2b~1-δη2b~5=0,γ2b2+ω2b~2-Λ-δη2b~4=0,γ2b3+ω2b~3-Λ=0.Similarly, substituting the ansatz (11) into (10) and applying identities for Hirota derivatives of theta functions yield
(15)(ω2(b~2-ϑ44(0,τ1))-γ2b2-(β+2v0+Λ+p2))×ϑ2(ωt,τ1)ϑ3(ωt,τ1)ϑ22(γx,τ)+(γ2b1-2pωϑ42(0,τ1))ϑ2(ωt,τ1)ϑ3(ωt,τ1)ϑ42(γx,τ)+(ω2(b~2+ϑ24(0,τ1))-γ2b3-2pωϑ22(0,τ1)-(β+2v0+Λ+p2))ϑ3×(ωt,τ1)ϑ4(ωt,τ1)ϑ2(γx,τ)ϑ4(γx,τ)+i[-(γ2b1-2pωϑ42(0,τ1))ϑ1(ωt,τ1)ϑ4(ωt,τ1)ϑ22(γx,τ)+(ω2(b~2+ϑ24(0,τ1))-γ2b3-2pωϑ22(0,τ1)-(β+2v0+Λ+p2)(b~2+ϑ24(0,τ1)))×ϑ1(ωt,τ1)ϑ2(ωt,τ1)ϑ2(γx,τ)ϑ4(γx,τ)+(ω2(b~2-ϑ44(0,τ1))-γ2b2-(β+2v0+Λ+p2))×ϑ1(ωt,τ1)ϑ4(ωt,τ1)ϑ42(γx,τ)(γ2b1-2pωϑ42(0,τ1))]=0.
For the real part and the imaginary part of (15), setting the coefficients of the terms ϑ22(γx,τ), ϑ42(γx,τ), and ϑ2(γx,τ)ϑ4(γx,τ) to be zero yields an algebraic system as follows:
(16)γ2b1-2pωϑ42(0,τ1)=0,ω2(b~2-ϑ44(0,τ1))-γ2b2-(β+2v0+Λ+p2)=0,ω2(b~2+ϑ24(0,τ1))-γ2b3-2pωϑ22(0,τ1)-(β+2v0+Λ+p2)=0.Solving the algebraic system given by (14) and (16) with respect to the variables v0, η, p, γ, ω, τ, τ1, and Λ, one obtains a set of nontrivial solutions:
(17)ω2=(b1b~4-b~5(b2-b3)b~1b~4+(b~2-b~3)b~5)γ2,p=b1γ22ωϑ42(0,τ1),Λ=b3γ2+b~3ω2,η2=γ2(b2-b3)+ω2(b~2-b~3)δb~4,v0=ω2(b~2-ϑ44(0,τ1))2-b2γ2+β+Λ+p22,
where the parameters γ, τ, and τ1 are arbitrary constants.Therefore, we obtain a new doubly periodic wave solution of (1):
(18)u(x,t)=η[ϑ3(ωt,τ1)ϑ2(γx,τ)+iϑ1(ωt,τ1)ϑ4(γx,τ)]exp⁡⁡(ipt)ϑ4(ωt,τ1)ϑ4(γx,τ)+ϑ2(ωt,τ1)ϑ2(γx,τ),v(x,t)=v0+2(ln⁡[ϑ4(ωt,τ1)ϑ4(γx,τ)+ϑ2(ωt,τ1)ϑ2(γx,τ)])xx,
where γ, τ, and τ1 are arbitrary constants, and the parameters ω, p, η, v0, and Λ are given by (17). To the author’s knowledge, the solution (18) is firstly reported here.In fact, the coupled Higgs field equation (1) admits abundant families of doubly periodic wave solutions. For example, we can suppose the solutions of the bilinear equations (9) and (10) as
(19)g=η[ϑ1(ωt,τ1)ϑ3(γx,τ)+iϑ2(ωt,τ1)ϑ4(γx,τ)]exp⁡⁡(ipt),f=ϑ4(ωt,τ1)ϑ4(γx,τ)+ϑ3(ωt,τ1)ϑ3(γx,τ),
or
(20)g=η[ϑ3(ωt,τ1)ϑ4(γx,τ)+iϑ1(ωt,τ1)ϑ2(γx,τ)]exp⁡⁡(ipt),f=ϑ4(ωt,τ1)ϑ2(γx,τ)+ϑ2(ωt,τ1)ϑ4(γx,τ),
and so on. For the sake of simplicity, the tedious computations are omitted here. It is noted that these two types of periodic solutions have also two independent periods in the spatial and temporal directions.


## 4. Jacobi Elliptic Function Expressions and Long Wave Limit

In order to analyze the periodic property by some figures, we may first convert solution (18) into Jacobi elliptic function expressions. Together with (6) and (7), solution (18) can be expressed as rational forms of Jacobi elliptic functions:
(21)u(x,t)=±λ2(1-2k2)+ρ2(1-2k12)δk1D1C+iS11+C1Cexp⁡(ipt),v(x,t)=v0+2λ2(1-k2-EK)+2λ2[k1-k2(C2-C12)+(2k2-1)CC1](1+C1C)2,
where λ is an arbitrary constant, and the parameters ρ, p, Λ, and v0 are given by
(22)ρ2=λ2[2k2-1+2kk1(1-k2)1/2(1-k12)-1/2],p=λ2ρk(1-k2)1/2(1-k12)-1/2,Λ=λ2(1-2EK)+ρ2(1-2E1K1),v0=ρ2((1-k12)2-E1K1)-λ2(1-k2-EK)-β+p2+Λ2,
which indicates that the period k in the spatial direction and the period k1 in the temporal direction are related by
(23)2k2-1+2kk1(1-k2)1/2(1-k12)-1/2>0.
In (21) and (22), the complete elliptic integrals E and E1 are defined by
(24)E=∫0π/2‍1-k2sin2ξdξ,E1=∫0π/2‍1-k12sin2ξdξ.From (21), it is easy to check that
(25)u(x,t)=u(x+4Kλ,t),|u(x,t)|=|u(x,t+4K1ρ)|,v(x,t)=v(x+4Kλ,t),v(x,t)=v(x,t+4K1ρ),
which implies that the solution (|u|,v) is periodic in the x-direction with a period 4K/λ and the t-direction with a period 4K1/ρ.By selecting appropriate parameter values in (21), the interactions of doubly periodic waves are shown in Figures 1 and 2. It is clearly seen that (|u|,v) is periodic in the x-direction and the t-direction. For the solution in [39], the periodic waves are both bell shaped in the spatial and temporal directions. However, with regard to solution (21), the periodic waves are of different shapes in the spatial and temporal directions.Doubly periodic wave solutionu(x,t) given by (21) with δ=0.2, β=0.4, λ=0.2, k=0.8, and k1=0.2.
(a)
Inx-direction
(b)
Int-direction
(c)
Evolution plot of|u|
(d)
Contour plotDoubly periodic wave solutionv(x,t) given by (21) with δ=0.2, β=0.4, λ=0.2, k=0.8, and k1=0.2.
(a)
Inx-direction
(b)
Int-direction
(c)
Evolution plot ofv
(d)
Contour plotWhen the modulusk→1, (snx,cnx,dnx)→(tanhx,sechx,sechx). And if k→0, (snx,cnx,dnx)→(sinx,cos⁡x,1). Therefore, the long wave limits of the periodic wave solutions can be readily obtained. A long wave limit of the solution (21) can be taken by assuming
(26)k(1-k12)1/4=m,0<m<1.
When the modulus k→0 and k1→1, one obtains a new periodic-solitary wave solution as follows:
(27)u=η[mcos⁡(λx)sech⁡(ρt)+itanh⁡(ρt)1+mcos⁡⁡(λx)sech⁡(ρt)]exp⁡⁡(ipt),v=-β2-λ2(5m4-6m2+2)2(2m2-1)-2mλ2sech⁡(ρt)(msech⁡(ρt)+cos⁡⁡(λx))(1+mcos⁡⁡(λx)sech⁡(ρt))2,
where the parameters ρ,p, and η are given by
(28)ρ=λ2m2-1,p=λm22m2-1,η=λ2(1-m2)δ.With proper selections of the values ofβ, δ, λ, and m, the interactions of periodic solitary waves (27) are shown in Figures 3 and 4. The solution (|u|,v) displays the feature of a dark soliton in the t-direction; the cosine function causes periodic modulation and thus it is periodic in the x-direction.The periodic-solitary wave solutionu(x,t) given by (27) with δ=0.4, β=2.0, λ=1.2, and m=0.8.
(a)
Inx-direction
(b)
Int-direction
(c)
Evolution plot of|u|
(d)
Contour plotThe periodic-solitary wave solutionv(x,t) given by (27) with δ=0.4, β=2.0, λ=1.2, and m=0.8.
(a)
Inx-direction
(b)
Int-direction
(c)
Evolution plot ofv
(d)
Contour plotWith the aid of the computer algebra softwareMaple, the validity of the new solutions (18) and (27) are verified by putting them back into the original systems (1).


## 5. Conclusions

The combination of the Hirota bilinear method and theta function identities is demonstrated to be a powerful tool in finding periodic waves for the coupled Higgs field equation. As a result, we have derived a new kind of doubly periodic standing wave solutions for the coupled Higgs field equation, which is different from those of the known solutions reported in the literature. The interaction properties of periodic-periodic waves and periodic-solitary waves are analyzed by some figures.The key of the combination method is that the solutions are supposed as rational expressions of elliptic functions of different moduli, which should be applicable to other nonlinear evolution equations or systems with bilinear forms in mathematical physics. The doubly periodic solutions will prove to be beneficial and instructive in modeling and understanding nonlinear phenomenon.



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