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

Authors: Wang Shen; Zheng-Yi Ma; Jin-Xi Fei; Quan-Yong Zhu; Jun-Chao Chen
Journal: Complexity
 (2020)

---

## 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.


---

## Body

## 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(1)iAt+Axx±A2B=0,B=f^A=P^C^A=A∗−x,t(where P^ and C^ 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 P^-C^-T^ symmetries [22]. Using the Bäcklund transformation, some types of P^T^ 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(2)utt+βu2xx+γuxxxx=0,where β is the nonlinear term coefficient and γ 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 P^-C^-T^ 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 Section2, 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 u=A+B/2 into equation (2), the nonlocal AB-BO system is derived as follows:(3)Att+Btt+βAx+Bx2+βA+BAxx+Bxx+γAxxxx+Bxxxx=0.Equation (3) can be split into the coupled equations(4a)Att+β2Ax+Bx2+βAAxx+Bxx+γAxxxx+GA,B=0,(4b)Btt+β2Ax+Bx2+βBAxx+Bxx+γBxxxx−GA,B=0,where(5)B=f^A=Af^,f^∈Θ≡1,P^s,T^d,P^sT^d,AP^s=A−x+x0,t,AT^d=Ax,−t+t0,with x0 and t0 are two arbitrary constants, and GA,B is an arbitrary function of A and B, with f^ invariant. That is, GA,B=f^GA,B=GA,Bf^. Although there are infinite functions satisfying this GA,B for equations (4a) and (4b), we can construct the function GA,B as(6)GA,B=βAx2−Bx2+β2A+BAxx−Bxx,and equations (4a) and (4b) are reduced to the following AB-BO system:(7a)Att+β2Ax+Bx3Ax−Bx+β23A+BAxx+β2A−BBxx+γAxxxx=0,(7b)Btt−β2Ax+BxAx−3Bx+β23B+ABxx−β2A−BAxx+γBxxxx=0.Forf^=1B=A, equations (7a) and (7b) are just the usual local BO equations. For f^≠1B≠A, equations (7a) and (7b) express three types of nonlocal AB-BO systems with three different nonlocalities, the parity nonlocal AB-BO (PNAB-BO) system B=AP^s, the time-reversal nonlocal AB-BO (TNAB-BO) system B=AT^d, and the parity and time-reversal nonlocal AB-BO (PTNAB-BO) B=AP^sT^d system.Obviously, systems (7a) and (7b) are integrable, and their Lax pair can be written as(8a)ψxxx=M0ZMψ,(8b)ψt=3γψxx+3β6γA+B0A−BA+Bψ,with(9a)M=−β4γA+B∂x−3β24γ3/2∫At+Btdx−β8γAx+Bx+λ1,(9b)Z=−β4γA−B∂x−3β24γ3/2∫At−Btdx−β8γAx−Bx+λ2,λ1 and λ2 being arbitrary constants.Now, we introduce an extended Bäcklund transformation(10)A=6γβlnFxx+b1lnFxxx+b2lnFxxt+α,B=6γβlnFxx−b1lnFxxx−b2lnFxxt+α,with b1,b2,α being arbitrary constants, and F≡Fx,t is an undetermined real function of variables x,t and satisfies(11)F=f^F=Ff^.Whenb1=0andb2=0, 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:(12)Dt2+2αβDx2+γDx4F⋅F=0,where Dt2,Dx2, and Dx4 are the bilinear derivative operators defined by [8, 9].(13)DxmDtnF⋅G=∂∂x−∂∂x′m∂∂t−∂∂t′n×Fx,tGx′,t′x′=x,t′=t.According to the properties of bilinear operatorD, equation (12) is equal to(14)FFtt−Ft2+2αβFFxx−Fx2+γFFxxxx−4FxFxxx+3Fxx2=0,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 F be written as a summation of some special functions [20, 21, 23]:(15)F=FN=∑νKνcosh12∑i=1Nviξi,ξi=kix−x02+ωit−t02+ηi0,ωi=δi−2αβki2−γki4,δi2=1,where the summation of ν=ν1,ν2,…,νN, ki,ηi0i=1,2,…,N are arbitrary constants, while(16)Kν=∏i<jNkikj2γki2+2γk22+2αβ−3νiνjγkikj+ωiωj.We notice that whenηi0=0, the invariant condition Fx,t=F−x+x0,−t+t0 of the function F (15) is satisfied.ForN=1, equation (15) possesses the form(17)F=F1=coshξ12,ξ1=k1x−x02+ω1t−t02.We have the single soliton solution(18)A=k1246γβ−b1k1+b2ω1tanhξ12sech2ξ12+α,B=f^A=A−x+x0,−t+t0.Figure1 shows the profile of the single soliton solution to the AB-BO system. The velocity of this solitary wave is equal to −3 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 b1 and b2.Figure 1
The single soliton solution (18) with the parameters α=4,β=γ=−1,δ1=k1=1,andx0=t0=0: (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(c)ForN=2, equation (15) becomes(19)F≡F2=Kcoshξ1+ξ22+K1coshξ1−ξ22,ξi=kix−x02+ωit−t02,i=1,2,where(20)K=k1k22γk12+2γk22+2αβ−3γk1k2+ω1ω2,K1=k1k22γk12+2γk22+2αβ+3γk1k2+ω1ω2.The two-soliton solution is obtained by substituting equations (19) and (20) into equation (10):(21)A=6γβlnF2xx+b1lnF2xxx+b2lnF2xxt+α,B=f^A=A−x+x0,−t+t0.After takingα=4,β=γ=−1,δ1=δ2=k1=1,k2=−4/5,x0=t0=0, 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.Figure 2
The two-soliton solution (21) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(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(22)α=−4,β=γ=k2=−1,δ1=δ2=k1=1,x0=t0=0.Then,F can be written as(23)F=2cos7t+7coshx,and the corresponding t-breather solution is obtained which is shown in Figure 3.Figure 3
Thet-breather solution of equation (23) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)After setting (I is the imagine unit, I2=1),(24)α=−4,β=γ=−1,δ1=δ2=1,k1=3I2,k2=−3I2,x0=t0=0.The functionF can be written as(25)F=41cos3x2+217cosh341t4,and the corresponding x-breather solution is obtained which is shown in Figure 4.Figure 4
Thex-breather solution of equation (25) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=3, the function F of equation (15) is described as(26)F≡F3=Kcosh12ξ1+ξ2+ξ3+K1cosh12ξ1−ξ2−ξ3+K2cosh12ξ1−ξ2+ξ3+K3cosh12ξ1+ξ2−ξ3,where(27a)K=a12−a13−a23−,K1=a12+a13+a23−,K2=a12+a13−a23+,K3=a12−a13+a23+,(27b)ξi=kix−x02+δi−2αβki2−γki4t−t02,aij±=kikj2γki2+2γk22+2αβ±3kikj+ωiωji,j=1,2,3.The three-soliton solution is obtained by substituting equations (26) and (27a) and (27b) into equation (10):(28)A=6γβlnF3xx+b1lnF3xxx+b2lnF3xxt+α,B=f^A=A−x+x0,−t+t0.After settingα=4,β=γ=−1,δ1=δ2=δ3=k1=1,k2=−4/5,k3=6/5,x0=t0=0, the three solitons can be constructed through equation (28), and these related structures are plotted in Figure 5.Figure 5
The three-soliton solution (28) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(c)ForN=4, the function F of equation (15) can be rewritten regularly as(29)F≡F4=Kcosh12ξ1+ξ2+ξ3+ξ4+K1cosh12−ξ1+ξ2+ξ3+ξ4+K2cosh12ξ1−ξ2+ξ3+ξ4+K3cosh12ξ1+ξ2−ξ3+ξ4+K4cosh12ξ1+ξ2+ξ3−ξ4+K23cosh12ξ1−ξ2−ξ3+ξ4+K24cosh12ξ1−ξ2+ξ3−ξ4+K34cosh12ξ1+ξ2−ξ3−ξ4,where(30)K=a12−a13−a14−a23−a24−a34−,K1=a12+a13+a14+a23−a24−a34−,K2=a12+a13−a14−a23+a24+a34−,K3=a12−a13+a14−a23+a24−a34+,K4=a12−a13−a14+a23−a24+a34+,K23=a12+a13+a14−a23−a24+a34+,K24=a12+a13−a14+a23+a24−a34+,K34=a12−a13+a14+a23+a24+a34−,(31)ξi=kix−x02+δi−2αβki2−γki4t−t02,aij±=kikj2γki2+2γk22+2αβ±3kikj+ωiωj,i,j=1,2,3,4.The four-soliton solution is obtained by substituting equations (29) and (30) into equation (10):(32)A=6γβlnF4xx+b1lnF4xxx+b2lnF4xxt+α,B=f^A=A−x+x0,−t+t0.After settingα=4,β=γ=k4=−1,δ1=δ2=δ3=δ4=k1=1,k2=−4/5,k3=6/5,x0=t0=0, the four solitons can be constructed through equation (32), and these related structures are plotted in Figure 6.Figure 6
The four-soliton solution (32) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(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:(33)α=−4,β=γ=k3=−1,δ1=δ2=δ3=δ4=k1=1,k2=12,k4=−12,x0=t0=0.Then,F can be written as(34)F=Kcos47+314t+2K1cos31t4coshx+2K4cos7tcosx2+K24cos−47+314t+K23coshx2+K34cosh3x2.The corresponding second-ordert-breather structures are depicted in Figure 7.Figure 7
The second-ordert-breather solution of equation (34) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)We set the parameters in equation (29) as follows:(35)α=−4,β=−1,γ=−1,δ1=δ2=δ3=δ4=1,k1=I,k2=I2,k3=−I,k4=−I2,x0=t0=0.Then,F can be written as(36)F=Kcosh12+334t+2K1cosh33t4cosx+2K2cosx2cosht4cosh2t−3+K24cosh−12+334t+K23cosx2+K34cos3x2,and the corresponding second-order x-breather solution is obtained which is shown in Figure 8.Figure 8
The second-orderx-breather solution of equation (36) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)


### 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(37)F=FN=∑m=0nn+1/2∑j=0maj,mx−x022jt−t022m−2jwith suitable constants aj,m [39].ForN=1, equation (37) possesses the form(38)F1=a0,0+a0,1t−t022+a1,1x−x022,where(39)a0,1=−4α2β2a0,03γ,a1,1=−2αβa0,03γ.After setting(40)t0=x0=0,α=β=γ=−1,a0,0=1,the solution (10) becomes(41a)A=−16x9+12t2−2x2b13+4t2+2x23−32t3−6x2+4t2b23+4t2+2x23−16t4−72t2+16t2x2−63+60x2+4x43+4t2+2x22,(41b)B=16x9+12t2−2x2b13+4t2+2x23+32t3−6x2+4t2b23+4t2+2x23−16t4−72t2+16t2x2−63+60x2+4x43+4t2+2x22.The invariant conditionFx,t=F−x+x0,−t+t0 of this function (38) is satisfied. Figure 9(a) is a normal first-order lump structure b1=b2=0 for the solution A=B=−4x4+60x2−63/3+2x22 at time t=0. Figures 9(b) and 9(c) are two symmetry-breaking lump structures for the solution A/B=−4x4+60x2−63/3+2x22∓160x9−2x2/3+2x23, with the parameters b1=b2=10 at time t=0. As these solutions are all rational functions, these functions describe the symmetry-breaking lump structures.Figure 9
The one-order symmetry-breaking lump solutions (41a) and (41b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=2, the function F of equation (37) can be rewritten regularly as(42)F2=a0,0+a0,1t2+a1,1x2+a0,2t4+a1,2x2t2+a2,2x4+a0,3t6+a1,3x2t4+a2,3x4t2+a3,3x6,where(43)a0,1=−76α2β2a0,075γ,a0,2=272α4β4a0,01875γ2,a0,3=−α6β6a0,01875γ3,a1,1=2αβa0,015γ,a1,2=48α3β3a0,0125γ2,a1,3=−32α5β5a0,0625γ3,a2,2=4α2β2a0,075γ2,a2,3=−16α4β4a0,0625γ3,a3,3=−8α3β3a0,01875γ3.After setting(44)t0=x0=0,α=β=γ=−1,a0,0=1,we have(45)a0,1=7675,a0,2=2721875,a0,3=641875,a1,1=−215,a1,2=48125,a1,3=32625,a2,2=475,a2,3=16625,a3,3=81875.Figure10 shows the second-order lump structure under the invariant condition Fx,t=F−x+x0,−t+t0 for the solution of AB-BO systems (7a) and (7b).Figure 10
The second-order symmetry-breaking lump solution under the invariant conditionFx,t=F−x+x0,−t+t0 of AB-BO systems (7a) and (7b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=3, the function F of equation (37) is(46)F3=a0,1t2+a1,1x2+a0,2t4+a1,2x2t2+a2,2x4+a0,3t6+a1,3x2t4+a2,3x4t2+a3,3x6+a0,4t8+a1,4x2t6+a2,4x4t4+a3,4x6t2+a4,4x8+a0,5t10+a1,5x2t8+a2,5x4t6+a3,5x6t4+a4,5x8t2+a5,5x10+a0,6t12+a1,6x2t10+a2,6x4t8+a3,6x6t6+a4,6x8t4+a5,6x10t2+a6,6x12+a0,0.The constrained constants are(47)a0,1=−3480α2β2a0,0847γ,a0,2=642288α4β4a0,0717409γ2,a0,3=−4382976α6β6a0,025109315γ3,a0,4=1997568α8β8a0,0175765205γ4,a0,5=−534528α10β10a0,0878826025γ5,a0,6=36864α12β12a0,0878826025γ6,a1,1=−12αβa0,011γ,a1,2=432α3β3a0,09317γ2,a1,3=3456α5β5a0,0717409γ3,a1,4=105984α7β7a0,02282665γ4,a1,5=−525312α9β9a0,0175765205γ5,a1,6=110592α11β11a0,0878826025γ6,a2,2=−60α2β2a0,0847γ2,a2,3=−25920α4β4a0,0717409γ3,a2,4=123264α6β6a0,05021863γ4,a2,5=−672768α8β8a0,0175765205γ5,a2,6=27648α10β10a0,0175765205γ6,a3,3=−96α3β3a0,046585γ3,a3,4=21888α5β5a0,03587045γ4,a3,5=−4608α7β7a0,02282665γ5,a3,6=18432α9β9a0,0175765205γ6,a4,4=432α4β4a0,03587045γ4,a4,5=−79488α6β6a0,0175765205γ5,a4,6=6912α8β8a0,0175765205γ6,a5,5=−576α5β5a0,017935225γ5,a5,6=6912α7β7a0,0878826025γ6,a6,6=576α6β6a0,0878826025γ6.Figure11 shows the third-order lump structures when t0=x0=0,α=β=γ=−1,a0,0=1, which are symmetry breaking.Figure 11
The third-order symmetry-breaking lump solution under the invariant conditionFx,t=F−x+x0,−t+t0 of AB-BO systems (7a) and (7b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)


## 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 F be written as a summation of some special functions [20, 21, 23]:(15)F=FN=∑νKνcosh12∑i=1Nviξi,ξi=kix−x02+ωit−t02+ηi0,ωi=δi−2αβki2−γki4,δi2=1,where the summation of ν=ν1,ν2,…,νN, ki,ηi0i=1,2,…,N are arbitrary constants, while(16)Kν=∏i<jNkikj2γki2+2γk22+2αβ−3νiνjγkikj+ωiωj.We notice that whenηi0=0, the invariant condition Fx,t=F−x+x0,−t+t0 of the function F (15) is satisfied.ForN=1, equation (15) possesses the form(17)F=F1=coshξ12,ξ1=k1x−x02+ω1t−t02.We have the single soliton solution(18)A=k1246γβ−b1k1+b2ω1tanhξ12sech2ξ12+α,B=f^A=A−x+x0,−t+t0.Figure1 shows the profile of the single soliton solution to the AB-BO system. The velocity of this solitary wave is equal to −3 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 b1 and b2.Figure 1
The single soliton solution (18) with the parameters α=4,β=γ=−1,δ1=k1=1,andx0=t0=0: (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(c)ForN=2, equation (15) becomes(19)F≡F2=Kcoshξ1+ξ22+K1coshξ1−ξ22,ξi=kix−x02+ωit−t02,i=1,2,where(20)K=k1k22γk12+2γk22+2αβ−3γk1k2+ω1ω2,K1=k1k22γk12+2γk22+2αβ+3γk1k2+ω1ω2.The two-soliton solution is obtained by substituting equations (19) and (20) into equation (10):(21)A=6γβlnF2xx+b1lnF2xxx+b2lnF2xxt+α,B=f^A=A−x+x0,−t+t0.After takingα=4,β=γ=−1,δ1=δ2=k1=1,k2=−4/5,x0=t0=0, 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.Figure 2
The two-soliton solution (21) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(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(22)α=−4,β=γ=k2=−1,δ1=δ2=k1=1,x0=t0=0.Then,F can be written as(23)F=2cos7t+7coshx,and the corresponding t-breather solution is obtained which is shown in Figure 3.Figure 3
Thet-breather solution of equation (23) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)After setting (I is the imagine unit, I2=1),(24)α=−4,β=γ=−1,δ1=δ2=1,k1=3I2,k2=−3I2,x0=t0=0.The functionF can be written as(25)F=41cos3x2+217cosh341t4,and the corresponding x-breather solution is obtained which is shown in Figure 4.Figure 4
Thex-breather solution of equation (25) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=3, the function F of equation (15) is described as(26)F≡F3=Kcosh12ξ1+ξ2+ξ3+K1cosh12ξ1−ξ2−ξ3+K2cosh12ξ1−ξ2+ξ3+K3cosh12ξ1+ξ2−ξ3,where(27a)K=a12−a13−a23−,K1=a12+a13+a23−,K2=a12+a13−a23+,K3=a12−a13+a23+,(27b)ξi=kix−x02+δi−2αβki2−γki4t−t02,aij±=kikj2γki2+2γk22+2αβ±3kikj+ωiωji,j=1,2,3.The three-soliton solution is obtained by substituting equations (26) and (27a) and (27b) into equation (10):(28)A=6γβlnF3xx+b1lnF3xxx+b2lnF3xxt+α,B=f^A=A−x+x0,−t+t0.After settingα=4,β=γ=−1,δ1=δ2=δ3=k1=1,k2=−4/5,k3=6/5,x0=t0=0, the three solitons can be constructed through equation (28), and these related structures are plotted in Figure 5.Figure 5
The three-soliton solution (28) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(c)ForN=4, the function F of equation (15) can be rewritten regularly as(29)F≡F4=Kcosh12ξ1+ξ2+ξ3+ξ4+K1cosh12−ξ1+ξ2+ξ3+ξ4+K2cosh12ξ1−ξ2+ξ3+ξ4+K3cosh12ξ1+ξ2−ξ3+ξ4+K4cosh12ξ1+ξ2+ξ3−ξ4+K23cosh12ξ1−ξ2−ξ3+ξ4+K24cosh12ξ1−ξ2+ξ3−ξ4+K34cosh12ξ1+ξ2−ξ3−ξ4,where(30)K=a12−a13−a14−a23−a24−a34−,K1=a12+a13+a14+a23−a24−a34−,K2=a12+a13−a14−a23+a24+a34−,K3=a12−a13+a14−a23+a24−a34+,K4=a12−a13−a14+a23−a24+a34+,K23=a12+a13+a14−a23−a24+a34+,K24=a12+a13−a14+a23+a24−a34+,K34=a12−a13+a14+a23+a24+a34−,(31)ξi=kix−x02+δi−2αβki2−γki4t−t02,aij±=kikj2γki2+2γk22+2αβ±3kikj+ωiωj,i,j=1,2,3,4.The four-soliton solution is obtained by substituting equations (29) and (30) into equation (10):(32)A=6γβlnF4xx+b1lnF4xxx+b2lnF4xxt+α,B=f^A=A−x+x0,−t+t0.After settingα=4,β=γ=k4=−1,δ1=δ2=δ3=δ4=k1=1,k2=−4/5,k3=6/5,x0=t0=0, the four solitons can be constructed through equation (32), and these related structures are plotted in Figure 6.Figure 6
The four-soliton solution (32) with (a) b1=b2=0; (b) and (c) b1=b2=10.
(a)(b)(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:(33)α=−4,β=γ=k3=−1,δ1=δ2=δ3=δ4=k1=1,k2=12,k4=−12,x0=t0=0.Then,F can be written as(34)F=Kcos47+314t+2K1cos31t4coshx+2K4cos7tcosx2+K24cos−47+314t+K23coshx2+K34cosh3x2.The corresponding second-ordert-breather structures are depicted in Figure 7.Figure 7
The second-ordert-breather solution of equation (34) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)We set the parameters in equation (29) as follows:(35)α=−4,β=−1,γ=−1,δ1=δ2=δ3=δ4=1,k1=I,k2=I2,k3=−I,k4=−I2,x0=t0=0.Then,F can be written as(36)F=Kcosh12+334t+2K1cosh33t4cosx+2K2cosx2cosht4cosh2t−3+K24cosh−12+334t+K23cosx2+K34cos3x2,and the corresponding second-order x-breather solution is obtained which is shown in Figure 8.Figure 8
The second-orderx-breather solution of equation (36) with (a) b1=b2=0; (b) and (c) b1=b2=5. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)


## 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(37)F=FN=∑m=0nn+1/2∑j=0maj,mx−x022jt−t022m−2jwith suitable constants aj,m [39].ForN=1, equation (37) possesses the form(38)F1=a0,0+a0,1t−t022+a1,1x−x022,where(39)a0,1=−4α2β2a0,03γ,a1,1=−2αβa0,03γ.After setting(40)t0=x0=0,α=β=γ=−1,a0,0=1,the solution (10) becomes(41a)A=−16x9+12t2−2x2b13+4t2+2x23−32t3−6x2+4t2b23+4t2+2x23−16t4−72t2+16t2x2−63+60x2+4x43+4t2+2x22,(41b)B=16x9+12t2−2x2b13+4t2+2x23+32t3−6x2+4t2b23+4t2+2x23−16t4−72t2+16t2x2−63+60x2+4x43+4t2+2x22.The invariant conditionFx,t=F−x+x0,−t+t0 of this function (38) is satisfied. Figure 9(a) is a normal first-order lump structure b1=b2=0 for the solution A=B=−4x4+60x2−63/3+2x22 at time t=0. Figures 9(b) and 9(c) are two symmetry-breaking lump structures for the solution A/B=−4x4+60x2−63/3+2x22∓160x9−2x2/3+2x23, with the parameters b1=b2=10 at time t=0. As these solutions are all rational functions, these functions describe the symmetry-breaking lump structures.Figure 9
The one-order symmetry-breaking lump solutions (41a) and (41b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=2, the function F of equation (37) can be rewritten regularly as(42)F2=a0,0+a0,1t2+a1,1x2+a0,2t4+a1,2x2t2+a2,2x4+a0,3t6+a1,3x2t4+a2,3x4t2+a3,3x6,where(43)a0,1=−76α2β2a0,075γ,a0,2=272α4β4a0,01875γ2,a0,3=−α6β6a0,01875γ3,a1,1=2αβa0,015γ,a1,2=48α3β3a0,0125γ2,a1,3=−32α5β5a0,0625γ3,a2,2=4α2β2a0,075γ2,a2,3=−16α4β4a0,0625γ3,a3,3=−8α3β3a0,01875γ3.After setting(44)t0=x0=0,α=β=γ=−1,a0,0=1,we have(45)a0,1=7675,a0,2=2721875,a0,3=641875,a1,1=−215,a1,2=48125,a1,3=32625,a2,2=475,a2,3=16625,a3,3=81875.Figure10 shows the second-order lump structure under the invariant condition Fx,t=F−x+x0,−t+t0 for the solution of AB-BO systems (7a) and (7b).Figure 10
The second-order symmetry-breaking lump solution under the invariant conditionFx,t=F−x+x0,−t+t0 of AB-BO systems (7a) and (7b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)ForN=3, the function F of equation (37) is(46)F3=a0,1t2+a1,1x2+a0,2t4+a1,2x2t2+a2,2x4+a0,3t6+a1,3x2t4+a2,3x4t2+a3,3x6+a0,4t8+a1,4x2t6+a2,4x4t4+a3,4x6t2+a4,4x8+a0,5t10+a1,5x2t8+a2,5x4t6+a3,5x6t4+a4,5x8t2+a5,5x10+a0,6t12+a1,6x2t10+a2,6x4t8+a3,6x6t6+a4,6x8t4+a5,6x10t2+a6,6x12+a0,0.The constrained constants are(47)a0,1=−3480α2β2a0,0847γ,a0,2=642288α4β4a0,0717409γ2,a0,3=−4382976α6β6a0,025109315γ3,a0,4=1997568α8β8a0,0175765205γ4,a0,5=−534528α10β10a0,0878826025γ5,a0,6=36864α12β12a0,0878826025γ6,a1,1=−12αβa0,011γ,a1,2=432α3β3a0,09317γ2,a1,3=3456α5β5a0,0717409γ3,a1,4=105984α7β7a0,02282665γ4,a1,5=−525312α9β9a0,0175765205γ5,a1,6=110592α11β11a0,0878826025γ6,a2,2=−60α2β2a0,0847γ2,a2,3=−25920α4β4a0,0717409γ3,a2,4=123264α6β6a0,05021863γ4,a2,5=−672768α8β8a0,0175765205γ5,a2,6=27648α10β10a0,0175765205γ6,a3,3=−96α3β3a0,046585γ3,a3,4=21888α5β5a0,03587045γ4,a3,5=−4608α7β7a0,02282665γ5,a3,6=18432α9β9a0,0175765205γ6,a4,4=432α4β4a0,03587045γ4,a4,5=−79488α6β6a0,0175765205γ5,a4,6=6912α8β8a0,0175765205γ6,a5,5=−576α5β5a0,017935225γ5,a5,6=6912α7β7a0,0878826025γ6,a6,6=576α6β6a0,0878826025γ6.Figure11 shows the third-order lump structures when t0=x0=0,α=β=γ=−1,a0,0=1, which are symmetry breaking.Figure 11
The third-order symmetry-breaking lump solution under the invariant conditionFx,t=F−x+x0,−t+t0 of AB-BO systems (7a) and (7b) with (a) b1=b2=0; (b) and (c) b1=b2=10. (d)–(f) The corresponding density plots of (a)–(c), respectively.
(a)(b)(c)(d)(e)(f)


## 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 eventsA and B.



---

*Source: 2370970-2020-05-31.xml*