## Study of Integrability and Exact Solutions for Nonlinear Evolution Equations

View this Special IssueResearch Article | Open Access

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

**Academic Editor:**Weiguo Rui

#### 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.

#### 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], which describes a system of conserved scalar nucleons interacting with neutral scalar mesons in particle physics. Here and are constants, and the function represents a real scalar meson field and 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 and and the Higgs equations for and . Much attention has been paid to investigate exact explicit solutions and integrable properties of (1). The symmetry reductions, the homoclinic orbits, -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 Section 2, 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 one 1–4, the parameter (the nome), and (pure imaginary) are defined by [40] Here, are the complete elliptic integrals of the first kind: is an odd function while the other three are even functions. The zeros of , , , and are at , , , and , respectively, and and are integers. Since , , and (, ) are related by a phase shift of , 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: where for simplicity we have used the notations 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: where and , where and . 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 into (1) and integrating with respect to yield the bilinear forms where is a constant, is an integration constant, and 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 suppose and in (9) and (10) as suitable combination of different theta functions. To obtain new doubly periodic wave solutions, we make a new ansatz, where the parameters , , , and are constants to be determined and the period in the spatial direction and the period 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 , , and to be zero and get where the theta constants are defined by

Due to the linear independence of theta functions, we set the coefficients of , , and in (12) to be zero and obtain

Similarly, substituting the ansatz (11) into (10) and applying identities for Hirota derivatives of theta functions yield For the real part and the imaginary part of (15), setting the coefficients of the terms , , and to be zero yields an algebraic system as follows:

Solving the algebraic system given by (14) and (16) with respect to the variables , , , , , , , and , one obtains a set of nontrivial solutions: where the parameters , , and are arbitrary constants.

Therefore, we obtain a new doubly periodic wave solution of (1): where , , and are arbitrary constants, and the parameters , , , , and 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 or 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: where is an arbitrary constant, and the parameters , , , and are given by which indicates that the period in the spatial direction and the period in the temporal direction are related by In (21) and (22), the complete elliptic integrals and are defined by

From (21), it is easy to check that which implies that the solution is periodic in the -direction with a period and the -direction with a period .

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 is periodic in the -direction and the -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.

**(a) In -direction**

**(b) In -direction**

**(c) Evolution plot of**

**(d) Contour plot**

**(a) In -direction**

**(b) In -direction**

**(c) Evolution plot of**

**(d) Contour plot**

When the modulus , . And if , . 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 When the modulus and , one obtains a new periodic-solitary wave solution as follows: where the parameters , and are given by

With proper selections of the values of , , , and , the interactions of periodic solitary waves (27) are shown in Figures 3 and 4. The solution displays the feature of a dark soliton in the -direction; the cosine function causes periodic modulation and thus it is periodic in the -direction.

**(a) In -direction**

**(b) In -direction**

**(c) Evolution plot of**

**(d) Contour plot**

**(a) In -direction**

**(b) In -direction**

**(c) Evolution plot of**

**(d) Contour plot**

With the aid of the computer algebra software** Maple**, 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.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This work was supported by the Natural Science Foundation of China under Grant no. 11201290.

#### References

- M. J. Ablowitz and P. A. Clarkson,
*Solitons, Nonlinear Evolution Equations and Inverse Scattering*, vol. 149 of*London Mathematical Society Lecture Note Series*, Cambridge University Press, Cambridge, UK, 1991. View at: Publisher Site | MathSciNet - R. Hirota,
*The Direct Method in Soliton Theory*, vol. 155 of*Cambridge Tracts in Mathematics*, Cambridge University Press, Cambridge, UK, 2004. View at: Publisher Site | MathSciNet - A. Nakamura, “A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution,”
*Journal of the Physical Society of Japan*, vol. 47, no. 5, pp. 1701–1705, 1979. View at: Publisher Site | Google Scholar | MathSciNet - F. Gesztesy and H. Holden,
*Soliton Equations and Their Algebro-Geometric Solutions*, vol. 79 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 2003. View at: Publisher Site | MathSciNet - X. G. Geng, Y. T. Wu, and C. W. Cao, “Quasi-periodic solutions of the modified Kadomtsev-Petviashvili equation,”
*Journal of Physics A: Mathematical and General*, vol. 32, no. 20, pp. 3733–3742, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. G. Geng, “Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations,”
*Journal of Physics A: Mathematical and General*, vol. 36, no. 9, pp. 2289–2303, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. P. Boyd, “Theta functions, Gaussian series, and spatially periodic solutions of the Korteweg-de Vries equation,”
*Journal of Mathematical Physics*, vol. 23, no. 3, pp. 375–387, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. V. Porubov, “Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid layer,”
*Physics Letters A*, vol. 221, no. 6, pp. 391–394, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. K. Liu, Z. T. Fu, S. D. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,”
*Physics Letters A*, vol. 289, no. 1-2, pp. 69–74, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Q. Xu and Z. B. Li, “Applications of Jacobi elliptic function expansion method for nonlinear differential-difference equations,”
*Communications in Theoretical Physics*, vol. 43, no. 3, pp. 385–388, 2005. View at: Publisher Site | Google Scholar | MathSciNet - G. Q. Xu, “New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation,”
*Applied Mathematics and Computation*, vol. 217, no. 12, pp. 5967–5971, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Y. Lou and G. J. Ni, “The relations among a special type of solutions in some $(D+1)$-dimensional nonlinear equations,”
*Journal of Mathematical Physics*, vol. 30, no. 7, pp. 1614–1620, 1989. View at: Publisher Site | Google Scholar | MathSciNet - E. G. Fan, “Extended tanh-function method and its applications to nonlinear equations,”
*Physics Letters A*, vol. 277, no. 4-5, pp. 212–218, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Y. Lou, H. C. Hu, and X. Y. Tang, “Interactions among periodic waves and solitary waves of the $\left(N+1\right)$-dimensional sine-Gordon field,”
*Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, vol. 71, no. 3, Article ID 036604, 2005. View at: Publisher Site | Google Scholar - G. Q. Xu, “An elliptic equation method and its applications in nonlinear evolution equations,”
*Chaos, Solitons and Fractals*, vol. 29, no. 4, pp. 942–947, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. G. Rui, B. He, and Y. Long, “The binary $F$-expansion method and its application for solving the $(n+1)$-dimensional sine-Gordon equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 4, pp. 1245–1258, 2009. View at: Publisher Site | Google Scholar | MathSciNet - W. G. Rui, C. Chen, X. S. Yang, and Y. Long, “Some new soliton-like solutions and periodic wave solutions with loop or without loop to a generalized KdV equation,”
*Applied Mathematics and Computation*, vol. 217, no. 4, pp. 1666–1677, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Lei and S. Y. Lou, “Interactions among periodic waves and solitary waves of the $\left(2+1\right)$-dimensional Konopelchenko-Dubrovsky equation,”
*Chinese Physics Letters*, vol. 30, no. 6, Article ID 060202, 2013. View at: Google Scholar - M. Tajiri, “On $N$-soliton solutions of coupled Higgs field equation,”
*Journal of the Physical Society of Japan*, vol. 52, no. 7, pp. 2277–2280, 1983. View at: Publisher Site | Google Scholar | MathSciNet - X. B. Hu, B. L. Guo, and H. W. Tam, “Homoclinic orbits for the coupled Schrödinger-Boussinesq equation and coupled higgs equation,”
*Journal of the Physical Society of Japan*, vol. 72, no. 1, pp. 189–190, 2003. View at: Publisher Site | Google Scholar - Y. C. Hon and E. G. Fan, “A series of exact solutions for coupled Higgs field equation and coupled Schrödinger-Boussinesq equation,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 71, no. 7-8, pp. 3501–3508, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Mu and Z. Y. Qin, “Rogue waves for the coupled Schrödinger-Boussinesq equation and the coupled Higgs equation,”
*Journal of the Physical Society of Japan*, vol. 81, no. 8, Article ID 084001, 2012. View at: Publisher Site | Google Scholar - A. M. Wazwaz, “Abundant soliton and periodic wave solutions for the coupled Higgs field equation, the Maccari system and the Hirota-Maccari system,”
*Physica Scripta*, vol. 85, no. 6, Article ID 065011, 2012. View at: Publisher Site | Google Scholar - B. Talukdar, S. K. Ghosh, A. Saha, and D. Pal, “Solutions of the coupled Higgs field equations,”
*Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, vol. 88, no. 1, Article ID 015201, 2013. View at: Publisher Site | Google Scholar - X. B. Hu and P. A. Clarkson, “Rational solutions of a differential-difference KdV equation, the Toda equation and the discrete KdV equation,”
*Journal of Physics A: Mathematical and General*, vol. 28, no. 17, pp. 5009–5016, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. B. Hu and Z. N. Zhu, “A Bäcklund transformation and nonlinear superposition formula for the Belov-Chaltikian lattice,”
*Journal of Physics A: Mathematical and General*, vol. 31, no. 20, pp. 4755–4761, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. R. Gilson, X. B. Hu, W. X. Ma, and H. W. Tam, “Two integrable differential-difference equations derived from the discrete BKP equation and their related equations,”
*Physica D: Nonlinear Phenomena*, vol. 175, no. 3-4, pp. 177–184, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. J. Zhang and D. Y. Chen, “Negatons, positions, rational-like solutions and conservation laws of the Korteweg-de Vries equation with loss and non-uniformity terms,”
*Journal of Physics A: Mathematical and General*, vol. 37, no. 3, pp. 851–865, 2004. View at: Publisher Site | Google Scholar | MathSciNet - E. G. Fan and Y. C. Hon, “Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii's breaking soliton equation in $(2+1)$ dimensions,”
*Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, vol. 78, no. 3, Article ID 036607, 2008. View at: Publisher Site | Google Scholar | MathSciNet - W. X. Ma and E. G. Fan, “Linear superposition principle applying to Hirota bilinear equations,”
*Computers & Mathematics with Applications*, vol. 61, no. 4, pp. 950–959, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. X. Ma and Z. N. Zhu, “Solving the $(3+1)$-dimensional generalized KP and BKP equations by the multiple exp-function algorithm,”
*Applied Mathematics and Computation*, vol. 218, no. 24, pp. 11871–11879, 2012. View at: Publisher Site | Google Scholar | MathSciNet - Y. Zhang, Y. Song, L. Cheng, J. Y. Ge, and W. W. Wei, “Exact solutions and Painlevé analysis of a new $(2+1)$-dimensional generalized KdV equation,”
*Nonlinear Dynamics*, vol. 68, no. 4, pp. 445–458, 2012. View at: Publisher Site | Google Scholar | MathSciNet - Y. H. Wang and Y. Chen, “Binary Bell polynomial manipulations on the integrability of a generalized $(2+1)$-dimensional Korteweg-de Vries equation,”
*Journal of Mathematical Analysis and Applications*, vol. 400, no. 2, pp. 624–634, 2013. View at: Publisher Site | Google Scholar | MathSciNet - G. Q. Xu, “Painlevé integrability of generalized fifth-order KdV equation with variable coefficients: exact solutions and their interactions,”
*Chinese Physics B*, vol. 22, no. 5, Article ID 050203, 2013. View at: Publisher Site | Google Scholar - G. Q. Xu and X. Z. Huang, “New variable separation solutions for two nonlinear evolution equations in higher dimensions,”
*Chinese Physics Letters*, vol. 30, no. 3, Article ID 130202, 2013. View at: Publisher Site | Google Scholar - K. W. Chow, “A class of exact, periodic solutions of nonlinear envelope equations,”
*Journal of Mathematical Physics*, vol. 36, no. 8, pp. 4125–4137, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. W. Chow, “A class of doubly periodic waves for nonlinear evolution equations,”
*Wave Motion*, vol. 35, no. 1, pp. 71–90, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. H. Li, S. Y. Lou, and K. W. Chow, “Doubly periodic patterns of modulated hydrodynamic waves: exact solutions of the Davey-Stewartson system,”
*Acta Mechanica Sinica*, vol. 27, no. 5, pp. 620–626, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. G. Fan, K. W. Chow, and J. H. Li, “On doubly periodic standing wave solutions of the coupled Higgs field equation,”
*Studies in Applied Mathematics*, vol. 128, no. 1, pp. 86–105, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. F. Lawden,
*Elliptic Functions and Applications*, vol. 80 of*Applied Mathematical Sciences*, Springer-Verlag, New York, NY ,USA, 1989. View at: MathSciNet

#### Copyright

Copyright © 2014 Gui-qiong Xu. 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.