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.