Abstract
The integrable Alice–Bob system with the shifted parity and delayed time reversal is presented through the Lax pair for the (1 + 1)-dimensional Boussinesq equation. After introducing an extended Bäcklund transformation, this system shows abundant exact solutions with the auxiliary functions consisting of hyperbolic functions or rational functions. The corresponding soliton structures contain line solitons, breathers, and lumps, all which satisfied the shifted parity and delayed time-reversal symmetry for the states of Alice A and Bob B. In particular, some lower-order circumstances can be expressed through their explicit solutions and their dynamic structures.
1 Introduction
For one (1 + 1)-dimensional model, except for identity transformation, there are the shifted parity and delayed time-reversal transformations for the spatial variable x and time variable t, respectively. In other words,with x0 and t0 being arbitrary constants [1, 2]. However, the Alice–Bob system, which can be successively used to describe two-place physical problems, may be entangled with each other through the following relation:with the state of Alice A ≡ A(x, t) and the Bob’s state B ≡ B(x′, t′); is a suitable operator (such as or ) [3–7]. Usually, this intrinsic Alice–Bob system is non-local since {x′, t′} is far away from {x, t}. However, for one Alice–Bob system, through the transformation, there indeed exist some types of multiple soliton structures with symmetry-breaking solutions according to Lou’s research. In other words, by applying the operator on one solution S, one can find .
For an illustrated model, the (1 + 1)-dimensional Boussinesq equation has the following form:orwhere . Eq. 3 is an integrable equation as it has the following Lax pair:and its adjoint version is as follows:where I is an imaginary unit.
In non-linear science, this equation is one of the important prototypic models. It can be used to study the dynamics of thin inviscid layers with a free surface, the non-linear string, and the propagation of waves in elastic rods and in the continuum limit of lattice dynamics or coupled electrical circuits. Multiple complex soliton solutions through multiple exponential function schemes, interactions between solitons and cnoidal periodic waves using the truncated Painlevé expansion method, and soliton solutions by the extended Kudryashov’s approach can all be presented [8–10].
Except for the former works where the physical quantity is taken directly, we derive the Alice–Bob system of Eq. 3 through its Lax pair and the dark parameterization approach [11–13].
After adopting the symmetry principle through , the Boussinesq Eq. 3 can be induced into the following Alice–Bob system:
The corresponding Lax pairs of Eqs 5, 6areas follows:withand we can obtain the following coupled equations:when σ = 0. After letting w = A+ B and v = A− B, the non-local systems (9) and (10) are a direct result from Eqs 14, 15.
Another method to derive the non-local systems (9) and (10) is the dark parameterization approach [14–17]. For the coupling Boussinesq system,w0 is the usual solution of Eq. 3; when taking w = u + vα (α is a dark parameter) and n = 1, the coupled equations are as follows:andThese equations can directly derive the non-local systems (9) and (10) through u = A+ B and v = A− B.
The rest of this paper is organized as follows: in Section 2, after introducing an extended Bäcklund transformation, the Hirota bilinear form is presented through an undetermined function f, which may contain some soliton solutions for the Alice–Bob systems (9) and (10) of the (1 + 1)-dimensional Boussinesq Eq. 3. Then, the hyperbolic function solution and the rational solution for this system are shown subsequently. In order to illustrate more clearly, three kinds of explicit solutions and their corresponding soliton structures are given for the lower-order circumstances. All of these results satisfy the symmetry of . A short summary is given in Section 3.
2 The symmetry-breaking solutions
We first introduce an extended Bäcklund transformation:with b and c being two constants; f ≡ f (x, t) is an undetermined function and satisfies the following conditions:By substituting Eq. 19 into the non-local systems (9) and (10), we obtain the following bilinear form:where Hirota’s bilinear derivative operators and are written as [18, 19]Eq. 21 also has the following explicit expression:
2.1 The hyperbolic function solutions and their soliton structures
Through the bilinear form (21) or Eq. 23, the function f exists in the following form of the hyperbolic function for the Boussinesq Eq. 3 [4–6].where {ν} = {νi = ±1} and ki(i = 1, 2, ⋅⋅⋅, N) are arbitrary constants, andwhere (i, j = 1, 2, ⋅⋅⋅, N, i ≠ j).
When N = 1, Eq. 24 has the following simple form:After substituting this form into the Bäcklund transformation (19), the non-local solution of Eqs 9, 10 can be derived as follows:
This single-soliton solution satisfies the condition of symmetry . By introducing Eq. 26, the Alice–Bob system is the coupling form of two solitary waves. The two solitary waves move along the X-axis at the speed ; the direction is determined by δ1, and the amplitude and wave width are determined by k1, b, and c, which is also confirmed by Eqs 27, 28. Figure 1 shows this structure when the related parameters are taken as follows:
FIGURE 1
When N = 2, Eq. 24 becomes as follows:with
The corresponding two-soliton solution can be obtained by substituting Eq. 30 with Eqs 31, 32 into Eq. 19. From the perspective of algebra, it is natural to consider the simplification of the function of Eq. 30 by quantifying the double variables of the hyperbolic function into single variables, that is, k1 = ±k2, ω1 = ±ω2. These four cases may produce the corresponding soliton or breather solutions for the Alice–Bob system, respectively. Two typical cases are presented here for N = 2. Figure 2 presents this structure when the related parameters are taken through the real constants as follows:Therefore,
FIGURE 2
It is not difficult to find that Eq. 34 is coupled by two hyperbolic functions similar to Eq. 26, and its image also shows this phenomenon.
On the other hand, by restricting the parameters k1, k2 to the assumed units on the two-soliton solution, the breather can be obtained. For example, by setting the following parametersthe following equation can be derived:Here, the cosine part of Eq. 36 makes the Alice–Bob system periodic, and the corresponding breather structure is obtained, as shown in Figure 3.
FIGURE 3
When N = 3, Eq. 24 has the following more complicated situation:withand
Based on the selecting parameters b, c, k1, k2, δ1, and δ2 of Eqs 33, 35, two kinds of interactions for the solitons can be constructed by considering the following equation:and
For this time, K{i}(i = 0, 1, 2, 3) are expressed as follows:and
The corresponding functions of Eq. 37 are expressed as follows:and
Figure 4 and Figure 5 show two interaction structures of the Alice–Bob systems (9) and (10) through Eqs 42, 43.
FIGURE 4
FIGURE 5
2.2 The rational solutions and lump structures
The Alice–Bob systems (9) and (10) have a series of rational solutions and hence contain the corresponding lump structures. For this purpose, we introduce the following polynomial function:where are constants determined by the powers of the variables x and t [20, 21].
When N = 1, Eq. 44 has the following simple form:Whenthe lump solution of the Alice–Bob systems (9) and (10) has the following rational form:which is obtained through the Bäcklund transformation (19) (Figure 6).
FIGURE 6
When N = 2, Eq. 44 has the following function form:
A pair of lumps of A and B for the Alice–Bob systems (9) and (10) can be shown after the constants are taken as b = c = a0,0 = 1, just as Eq. 46, while
These two pairs of lumps for the Alice–Bob systems (9) and (10) through Eq. 19, Eq. 49, and Eq. 50 are shown in Figure 7.
FIGURE 7
When N = 3, Eq. 44 has the more complicated function form:
The lumps of A and B for the Alice–Bob systems (9) and (10) can also be constructed after the constants b = c = a0,0 = 1; therefore, the following equations are obtained:
These lump structures of the Alice–Bob systems (9) and (10) obtained through Eqs 19, 51, and 52 are shown in Figure 8.
FIGURE 8
3 Summary
In this paper, according to the (1 + 1)-dimensional Boussinesq Eq. 3, the Alice–Bob systems (9) and (10) for this equation are first derived through the Lax pair and the dark parameterization approach. This non-local system owns the bilinear form and may exist in the explicit solution. Therefore, the N-soliton solutions of the Alice–Bob systems (9) and (10) are presented with the aid of an undetermined function f after introducing an extended Bäcklund transformation. Typically, the auxiliary function can be taken as the hyperbolic function or rational function. These two kinds of functions induce the system having solutions that satisfy . The lower-order circumstances for N = 1, 2, 3 are presented through their auxiliary functions, and the symmetry-breaking solutions can be constructed. With the special parameters, the antisymmetric local structures are depicted, which contain line solitons, breathers, and lumps. Whether the induced Alice–Bob systems (9) and (10) of the (1 + 1)-dimensional Boussinesq Eq. 3 or the derived results through the hyperbolic and rational functions satisfy the symmetry of is first shown here for our understanding. We believe that this approach is important to solve the Alice–Bob system for one integrable equation, which may possess rich local structures.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.
Author contributions
PD: writing—original draft and software. Z-YM: supervision and funding acquisition. J-XF: conceptualization and software. W-PC: formal analysis and investigation.
Funding
This work was supported by the National Natural Science Foundation of China (11775104) and the Zhejiang Province Natural Science Foundation of China (2022SJGYZC01).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1.
BenderCMBoettcherS. Real spectra in non-hermitian Hamiltonians HavingPTSymmetry. Phys Rev Lett (1998) 80:5243–6. 10.1103/physrevlett.80.5243
2.
AblowitzMJMusslimaniZH. Integrable nonlocal nonlinear Schrödinger equation. Phys Rev Lett (2013) 110:064105–5. 10.1103/physrevlett.110.064105
3.
LouSY. Alice-Bob systems, symmetry invariant and symmetry breaking soliton solutions. J Math Phys2018;59:083507–20. 10.1063/1.5051989
4.
LouSY. Prohibitions caused by nonlocality for nonlocal Boussinesq-KdV type systems. Stud Appl Math (2019) 142:1–16. 10.1111/sapm.12265
5.
LiHLouSY. Multiple soliton solutions of Alice-Bob Boussinesq equations. Chin Phys Lett (2019) 36:050501–5. 10.1088/0256-307x/36/5/050501
6.
ZhaoQLLouSYJiaM. Solitons and soliton molecules in two nonlocal Alice-Bob Sawada-Kotera systems. Commun Theor Phys (2020) 72:085005–7. 10.1088/1572-9494/ab8a0e
7.
JiaMLouSY. Exact PsTd invariant and PsTd symmetric breaking solutions, symmetry reductions and Bäcklund transformations for an AB-KdV system. Phys Lett A (2018) 382:1157–66. 10.1016/j.physleta.2018.02.036
8.
WazwazAM. Multiple soliton solutions and multiple complex soliton solutions for two distinct Boussinesq equations. Nonlinear Dyn (2016) 85:731–7. 10.1007/s11071-016-2718-0
9.
YangDLouSYYuWF. Interactions between solitons and cnoidal periodic waves of the Boussinesq equation. Commun Theor Phys (2013) 60:387–90. 10.1088/0253-6102/60/4/01
10.
AdemARYildirimYYasarE. Soliton solutions to the non-local Boussinesq equation by multiple exp-function scheme and extended Kudryashov’s approach. Pramana-j Phys (2019) 92:24–11. 10.1007/s12043-018-1679-x
11.
FeiJXMaZYCaoWP. Controllable symmetry breaking solutions for a nonlocal Boussinesq system. Sci Rep-uk (2019) 9:19667–14. 10.1038/s41598-019-56093-8
12.
CaoWPFeiJXLiJY. Symmetry breaking solutions to nonlocal Alice-Bob Kadomtsev-Petviashivili system. Chaos Soliton Fract (2021) 144:110653–8. 10.1016/j.chaos.2021.110653
13.
WuWBLouSY. Exact solutions of an Alice-Bob KP equation. Commun Theor Phys (2019) 71:629–32. 10.1088/0253-6102/71/6/629
14.
LouSY. Dark parameterizations, equivalent partner fields and integrable systems. Commun Theor Phys (2011) 55:743–6. 10.1088/0253-6102/55/5/02
15.
PengLYangXDLouSY. Integrable KP coupling and its exact solution. Commun Theor Phys (2012) 58:1–4. 10.1088/0253-6102/58/1/01
16.
CaoWPFeiJXMaZYLiuQ. Bosonization and new interaction solutions for the coupled Korteweg-de Vries system. Wave Random Complex (2020) 30:130–41. 10.1080/17455030.2018.1489565
17.
FeiJXCaoWPMaZY. Dark parameterization approach to the Benjamin Ono equation. Int J Mod Phys B (2020) 34:2050247–8. 10.1142/s0217979220502471
18.
HirotaR. Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys Rev Lett (1971) 27:1192–4. 10.1103/physrevlett.27.1192
19.
HirotaR. The direct method in soliton theory. Cambridge: Cambridge University Press (2004).
20.
ClarksonPADowieE. Rational solutions of the Boussinesq equation and applications to rogue waves. Trans Math Appl (2017) 1:1–26. 10.1093/imatrm/tnx003
21.
MaYLLiBQ. Analytic rogue wave solutions for a generalized fourth-order Boussinesq equation in fluid mechanics. Math Meth Appl Sci (2019) 42:39–48. 10.1002/mma.5320
Summary
Keywords
(1+1)-dimensional Alice–Bob Boussinesq equation, Lax pair, Bäcklund transformation, PsTd symmetry-breaking solution, hybrid structure
Citation
Dong P, Ma Z-Y, Fei J-X and Cao W-P (2023) The shifted parity and delayed time-reversal symmetry-breaking solutions for the (1+1)-dimensional Alice–Bob Boussinesq equation. Front. Phys. 11:1137999. doi: 10.3389/fphy.2023.1137999
Received
05 January 2023
Accepted
16 February 2023
Published
09 March 2023
Volume
11 - 2023
Edited by
Yunqing Yang, Zhejiang Ocean University, China
Reviewed by
Sakkaravarthi Karuppaiya, Asia Pacific Center for Theoretical Physics (APCTP), Republic of Korea
Alejandro Aceves, Southern Methodist University, United States
Updates
Copyright
© 2023 Dong, Ma, Fei and Cao.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Zheng-Yi Ma, ma-zhengyi@163.com
This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.