Complexity
Volume 2020 (2020), Article ID 2370970, 12 pages
https://doi.org/10.1155/2020/2370970

Research Article

Abundant Symmetry-Breaking Solutions of the Nonlocal Alice–Bob Benjamin–Ono System

Wang Shen,¹ Zheng-Yi Ma,1,2 Jin-Xi Fei,³ Quan-Yong Zhu,² and Jun-Chao Chen²

¹Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
²Institute of Nonlinear Analysis and Department of Mathematics, Lishui University, Lishui 323000, China
³Department of Photoelectric Engineering, Lishui University, Lishui 323000, China

Correspondence should be addressed to Zheng-Yi Ma; ma-zhengyi@163.com

Received 22 February 2020; Accepted 9 May 2020; Published 31 May 2020

Academic Editor: Eric Campos

Copyright © 2020 Wang Shen et al. 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.

Abstract.

The Benjamin–Ono equation is a useful model to describe the long internal gravity waves in deep stratified fluids. In this paper, the nonlocal Alice–Bob Benjamin–Ono system is induced via the parity and time-reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry-breaking soliton, breather, and lump solutions for this system are obtained through the derived Hirota bilinear form. By taking suitable constants in the involved ansatz functions, abundant fascinating symmetry-breaking structures of the related explicit solutions are shown.

1. Introduction

In the recent years, studying the local excitations in the nonlinear evolution equations (NEEs) has become great significance since the complex nonlinear phenomena related to the NEEs involve in fluid dynamics, plasma physics, superconducting physics, condensed matter physics, and optical problems [1–6]. In fact, researchers have discovered many powerful methods for studying these aspects, such as the Hirota bilinear method [7–9], the inverse scattering method [10, 11], the Painlevé analysis approach [12–14], the Bäcklund transformation [15], and the Darboux transformation [16–18]. Furthermore, the investigation of the solitary waves and solitons taking one or more of the above approaches for the NEEs has become more and more important and attractive.

Meanwhile, one of the proposed two-place nonlocal models, the nonlinear Schrödinger (NLS) equation

(where and are the usual parity and charge conjugation operators) had been investigated [19]. Recently, Lou proposed the Alice–Bob (AB) systems to describe two-place physical problems [20, 21]. The parity, time reversal, charge conjugation, and their suitable combinations were conserved for most of the above problems [20–30]. However, these AB symmetries exist in various physical models, although they are not directly used to solve the nonlinear physical systems, especially the -- symmetries [22]. Using the Bäcklund transformation, some types of symmetry-breaking solutions including soliton and rogue wave solutions were explicitly obtained. In addition to nonlocal nonlinear Schrödinger equation (1), there are many other types of two-place nonlocal models, such as the nonlocal modified KdV systems [20] and the nonlocal Boussinesq-KdV systems [23].

In this work, we consider the (1 + 1)-dimensional Benjamin–Ono (BO) equation

where is the nonlinear term coefficient and is a dispersion coefficient. The BO equation is one of the most important nonlinear equations that describes one-dimensional internal waves in deep water [31]. Ono had developed the Benjamin theory to obtain a species of the NEEs [32]. The two/four-place nonlocal Benjamin–Ono equation was explicitly solved for special types of multiple soliton solutions via -- symmetric-antisymmetric separation approach [22]. Many authors also studied different properties for equation (2), such as the Bäcklund transformation, the existence of quasi-periodic solutions and the nontrivial time-periodic solutions, infinitely many conservation laws, and other integrable properties [33–38].

The outline of this paper is as follows: in Section 2, the AB-BO system and its Lax pair are introduced, and its bilinear form is written through an extended Bäcklund transformation. In Section 3, the symmetry-breaking soliton, breather, and lump solutions are presented through the derived Hirota bilinear form. According to the taken constants in the involved ansatz functions, some sets of the fascinating symmetry-breaking structures of the related explicit solutions are shown, correspondingly. Summary and conclusions are given in the last section.

2. The AB-BO System and Its Lax Pair, Bäcklund Transformation, and Bilinear Form

Based on the principle of the AB system [20, 21], after substituting into equation (2), the nonlocal AB-BO system is derived as follows:

Equation (3) can be split into the coupled equations

where

with and are two arbitrary constants, and is an arbitrary function of and , with invariant. That is, . Although there are infinite functions satisfying this for equations (4a) and (4b), we can construct the function as

and equations (4a) and (4b) are reduced to the following AB-BO system:

For , equations (7a) and (7b) are just the usual local BO equations. For , equations (7a) and (7b) express three types of nonlocal AB-BO systems with three different nonlocalities, the parity nonlocal AB-BO (PNAB-BO) system , the time-reversal nonlocal AB-BO (TNAB-BO) system , and the parity and time-reversal nonlocal AB-BO (PTNAB-BO) system.

Obviously, systems (7a) and (7b) are integrable, and their Lax pair can be written as

with

and being arbitrary constants.

Now, we introduce an extended Bäcklund transformation

with being arbitrary constants, and is an undetermined real function of variables and satisfies

When , equation (10) becomes one normal Bäcklund transformation of equation (2). Substituting equation (10) into equations (7a) and (7b), the bilinear form can be written as follows:

where , and are the bilinear derivative operators defined by [8, 9].

According to the properties of bilinear operator , equation (12) is equal to

which is a bilinear form of equation (2).

3. Symmetry-Breaking Soliton, Breather, and Lump Solutions to the AB-BO System

In this section, we turn our attention to the Hirota bilinear form (12) of AB-BO systems (7a) and (7b) to derive the symmetry-breaking soliton, symmetry-breaking breather, and symmetry-breaking lump solutions.

3.1. Symmetry-Breaking Soliton and Breather Solutions to the AB-BO System

Based on the bilinear form (12), we can first determine the symmetry-breaking soliton and breather solutions through the Bäcklund transformation (10) of AB-BO systems (7a) and (7b) with the function be written as a summation of some special functions [20, 21, 23]:

where the summation of , are arbitrary constants, while

We notice that when , the invariant condition of the function (15) is satisfied.

For , equation (15) possesses the form

We have the single soliton solution

Figure 1 shows the profile of the single soliton solution to the AB-BO system. The velocity of this solitary wave is equal to after our choice of the free parameters. At the same time, we also know that the amplitude of the solitary wave increases with the increase of parameters and .

(a)

(b)

(c)

Figure 1: The single soliton solution (18) with the parameters : (a) ; (b) and (c) .

For , equation (15) becomes

where

The two-soliton solution is obtained by substituting equations (19) and (20) into equation (10):

After taking , the two-soliton is generated by equation (21). The corresponding structures are plotted in Figure 2. Figure 2(a) shows the wave shape, wave velocity, and amplitude are unchanged after two solitons’ head-on collision.

(a)

(b)

(c)

Figure 2: The two-soliton solution (21) with (a) ; (b) and (c) .

In earlier works [33], by some constraints to the parameters on the two solitons, a family of analytical breather solutions can be obtained. Inspired by this technique, we give the breather solution to equation (19) by setting

Then, can be written as

and the corresponding -breather solution is obtained which is shown in Figure 3.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: The -breather solution of equation (23) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

After setting ( is the imagine unit, ),

The function can be written as

and the corresponding -breather solution is obtained which is shown in Figure 4.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4: The -breather solution of equation (25) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

For , the function of equation (15) is described as

where

The three-soliton solution is obtained by substituting equations (26) and (27a) and (27b) into equation (10):

After setting , the three solitons can be constructed through equation (28), and these related structures are plotted in Figure 5.

(a)

(b)

(c)

Figure 5: The three-soliton solution (28) with (a) ; (b) and (c) .

For , the function of equation (15) can be rewritten regularly as

where

The four-soliton solution is obtained by substituting equations (29) and (30) into equation (10):

After setting , the four solitons can be constructed through equation (32), and these related structures are plotted in Figure 6.

(a)

(b)

(c)

Figure 6: The four-soliton solution (32) with (a) ; (b) and (c) .

Similar to the two-soliton solution, we also give the second-order breather solution. In this case, we set the parameters in equation (29) as follows:

Then, can be written as

The corresponding second-order -breather structures are depicted in Figure 7.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The second-order -breather solution of equation (34) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

We set the parameters in equation (29) as follows:

Then, can be written as

and the corresponding second-order -breather solution is obtained which is shown in Figure 8.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: The second-order -breather solution of equation (36) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

3.2. Symmetry-Breaking Lump Solutions to the AB-BO System

Based on the idea of generating the lump solution for one nonlinear equation, we derive this kind of solutions of AB-BO systems (7a) and (7b) with the function F written as

with suitable constants [39].

For , equation (37) possesses the form

where

After setting

the solution (10) becomes

The invariant condition of this function (38) is satisfied. Figure 9(a) is a normal first-order lump structure for the solution at time . Figures 9(b) and 9(c) are two symmetry-breaking lump structures for the solution , with the parameters at time . As these solutions are all rational functions, these functions describe the symmetry-breaking lump structures.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: The one-order symmetry-breaking lump solutions (41a) and (41b) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

For , the function F of equation (37) can be rewritten regularly as

where

After setting

we have

Figure 10 shows the second-order lump structure under the invariant condition for the solution of AB-BO systems (7a) and (7b).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10: The second-order symmetry-breaking lump solution under the invariant condition of AB-BO systems (7a) and (7b) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

For , the function of equation (37) is

The constrained constants are

Figure 11 shows the third-order lump structures when , which are symmetry breaking.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 11: The third-order symmetry-breaking lump solution under the invariant condition of AB-BO systems (7a) and (7b) with (a) ; (b) and (c) . (d)–(f) The corresponding density plots of (a)–(c), respectively.

4. Summary and Conclusion

It is believed that the two-place correlated physical events widely exist in the field of natural science, and discussing AB physics has a profound influence on other scientific fields. In this article, we studied the nonlocal BO equation coupled with an AB system. First of all, one established a special AB-BO system via the parity with a shift of the space variable and time reversal with a delay. At the same time, with the derived extended Bäcklund transformation and the corresponding Hirota bilinear form, the symmetry-breaking soliton, symmetry-breaking breather, and symmetry-breaking lump solutions were presented. Finally, by choosing special parameters, these solutions of the AB-BO system were discussed in detail. The coefficients with shifted parity and delayed time reversal in the nonlocal AB-BO system were discussed, from which the abundant symmetry-breaking solutions were illustrated by changing parameters of events and .

Data Availability

The data used to support the findings of this study are included within the article. For more details, the data are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11775104 and 11705077), the Natural Science Foundation of Zhejiang Province (no. LY14A010005), and the Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) (no. 201601).

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