# Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

Authors: Hong-Yu Wu; Jin-Xi Fei; Zheng-Yi Ma; Jun-Chao Chen; Wen-Xiu Ma
Journal: Complexity
 (2020)

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## Abstract

The Kadomtsev–Petviashvili equation is one of the well-studied models of nonlinear waves in dispersive media and in multicomponent plasmas. In this paper, the coupled Alice-Bob system of the Kadomtsev–Petviashvili equation is first constructed via the parity with a shift of the space variablex and time reversal with a delay. By introducing an extended Bäcklund transformation, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions for this system are presented through the established Hirota bilinear form. According to the corresponding constants in the involved ansatz function, a few fascinating symmetry breaking structures of the presented explicit solutions are shown.


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## Body

## 1. Introduction

The localized excitations in nonlinear evolution equations have been studied widely, which were originated from many scientific fields, such as fluid dynamics, plasma physics, superconducting physics, condensed matter physics, and optical problems. Explicitly, the inverse scattering method [1], the Darboux transformation and the Bäcklund transformation [2, 3], the Painlevé analysis approach [4–6], the Hirota bilinear method [7, 8], and the generalized bilinear method [9] are among important approaches for studying these structures, especially solitary waves and solitons.Owing to the idea of the parity-time reversal (PT) symmetry, the nonlinear Schrödinger (NLS) equation(1)iAt+Axx±A2B=0,B=f^A=P^C^A=A∗−x,t,(where the operators P^ and C^ are the usual parity and charge conjugation) was introduced and investigated [10]. Based on this, the revolutionary works, which named the Alice-Bob (AB) systems to describe two-place physical problems, were made by Lou recently [11, 12]. The technical approach originated from the so-called P^-T^-C^ principle with P^ (the parity), T^ (time reversal), and C^ (charge conjugation) [11–25]. From this, a general Nth Darboux transformation for the AB-mKdV equation was constructed [13]. By using this Darboux transformation, some types of P^T^ symmetry breaking solutions including soliton and rogue wave solutions were explicitly obtained. Combined with their Hirota bilinear forms, prohibitions caused by nonlocality for nonlocal Boussinesq-KdV type systems were investigated [14]. The two/four-place nonlocal Kadomtsev–Petviashvili (KP) equation were also explicitly solved for special types of multiple soliton solutions via a P^-T^-C^ symmetric-antisymmetric separation approach [15]. From the viewpoint of physical phenomena in climate disasters, a special approximate solution was applied to theoretically capture the salient features of two correlated dipole blocking events in atmospheric dynamical systems and the original two-vortex interaction was given to describe two correlated dipole blocking events with a lifetime through the models established from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a b-plane [21, 22]. Also, a concrete AB-KdV system established from the nonlinear inviscid dissipative and barotropic vorticity equation in a β-plane channel was applied to the two correlated monople blocking events, which were responsible for the snow disaster in the winter of 2007/2008 that happened in Southern China [18]. Meanwhile, the expression(2)f=∑νKνcosh12∑i=1Nνiξi,plays a crucial role in constructing analytical group invariant multisoliton solutions of the AB systems, including the KdV-KP-Toda type, mKdV-sG type, NLS type, and discrete H1 type AB systems [11–16, 18].In this paper, we consider the KP equation (3) as an illustrative example, which is one of the well-studied models of nonlinear waves in dispersive media [26, 27] and in multicomponent plasmas [28]. In the immovable laboratory coordinate frame, it can be presented in the form(3)ut+cux+αuux+βuxxxx+c2uyy=0,where c is the velocity of long linear perturbations and α and β are the nonlinear and dispersive coefficients which are determined by specific types of wave and medium properties.The rest of this paper is organized as follows. In Section2, an AB-KP system is constructed based on equation (3) and its Hirota bilinear form is presented through an extended Bäcklund transformation. In Section 3, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions are generated through the established Hirota bilinear form, according to the corresponding constants of the involved ansatz function. Some conclusions are given in the final section.


## 2. An AB-KP System and Its Bäcklund Transformation and Bilinear Form

Based on the principle of the AB system in Refs. [11, 12], after substituting u=1/2A+B into equation (3), the AB-KP initial equation is(4)Axt+Bxt+12αAx+Bx2+c+12αA+BAxx+Bxx+βAxxxx+Bxxxx+c2Ayy+Byy=0,which can be split into the coupled equations(5a)Axt+α4Ax+Bx2+cAxx+α4A+BAxx+Bxx+βAxxxx+c2Ayy+GA,B=0,(5b)Bxt+α4Ax+Bx2+cBxx+α4A+BAxx+Bxx+βBxxxx+c2Byy−GA,B=0,where B is related to A through B=P^sxT^dA=A−x+x0,y,−t+t0 (P^sxT^d expresses parity with a shift of the space variable x and time reversal with a delay), and GA,B is an arbitrary function of A and B, but should be P^sxT^d invariant. That is, GA,B=P^sxT^dGA,B. Although there are infinitely many functions satisfying this, we take a nontrivial function GA,B as(6)GA,B=α2Ax2+AAxx−Bx2−BBxx,at present, and equation (5) is reduced to the following AB-KP system:(7a)Axt+α4Ax+Bx3Ax−Bx+α4A−BBxx+α43A+B+cAxx+βAxxxx+c2Ayy=0,(7b)Bxt−α4Ax+BxAx−3Bx−α4A−BAxx+α4A+3B+cBxx+βBxxxx+c2Byy=0.In fact, this AB-KP system can also be derived as a special reduction of the coupled KP system:(8a)At+c1A+c2Bx+c3A+c4Bxxx+c5A+c6BAx+c7A+c8BBxx+c9Ayy=0,(8b)Bt+c1B+c2Ax+c3B+c4Axxx+c5B+c6ABx+c7B+c8AAxx+c9Byy=0,by taking the reduction condition B=P^sxT^dA=A−x+x0,y,−t+t0 and letting the arbitrary constants cii=1,2,…,9 with(9)c1=c,c2=c4=0,c3=β,c5=3α4,c6=c7=α4,c8=−α4,c9=c2.Now, we introduce an extended Bäcklund transformation:(10)A=12βαlnfxx+b1lnfxxx+b2lnfxxt,B=12βαlnfxx−b1lnfxxx−b2lnfxxt,where b1andb2 are arbitrary constants and f≡fx,y,t is a new function of variables x,y, and t, satisfying the invariant condition(11)fx,y,t=P^sxT^dfx,y,t=f−x+x0,y,−t+t0.Whenb1=0andb2=0, equation (10) becomes the standard Bäcklund transformation of equation (3). Substituting the transformation equation (10) into equation (7), we obtain a bilinear form of equation (7) as follows:(12)DxDt+cDx2+βDx4+c2Dy2f⋅f=0,whereDx4 and Dy2 are the Hirota bilinear derivative operators defined by [7, 8](13)DxmDynDtlf⋅g=∂∂x−∂∂x′m∂∂y−∂∂y′n∂∂t−∂∂t′l×fx,y,tgx′,y′,t′x′=x,y=y′,t′=t.According to the properties of the Hirota bilinear operator, equation (12) reads(14)2ffxt−fxft+2cffxx−fx2+2βffxxxx−4fxfxxx+3fxx2+cffyy−fy2=0,which is also the Hirota bilinear form of equation (3).As we know, the Hirota bilinear method is direct and effective for constructing exact solutions, in which a given nonlinear equation is converted to a bilinear form through an appropriate transformation. With different types of ansatz for the auxiliary function, a variety of soliton, rational, and periodic solutions of the nonlinear equation can be derived.


## 3. Symmetry Breaking Soliton, Breather, and Lump Solutions

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


### 3.1. Symmetry Breaking Soliton Solutions

Based on the bilinear form (12), we can first determine symmetry breaking soliton solutions through the Bäcklund transformation (10) of the AB-KP systems (7a) and (7b) with the function f being written as a summation of some special functions [11–16, 18]:(15)f=fN=∑νKνcosh12∑i=1Nνiξi,ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,where ν=ν1,ν2,…,νN, with vi=±1, and ki,pi,ηi0i=1,2,…,N,c,x0, and t0 are undetermined constants, while(16)Kν=∏i<jNcpi−pj2−6βki−νiνjkj2.ForN=1, equation (15) takes the form(17)f=f1=coshξ12,ξ1=k1x−x02+p1y−c+k12β+cp122t−t02+η10.However, the invariant condition (11) of this function f (17) is not satisfied for the constants k1,p1,c,x0,t0, and η10 being not all zero. It means that it is impossible to get any nontrivial symmetry breaking single soliton solution of equation (12) through (10).The same circumstance happens whenN=3, in which the function f of equation (15) is(18)f=f3=Kcosh12ξ1+ξ2+ξ3+K1cosh12ξ1−ξ2−ξ3+K2cosh12ξ1−ξ2+ξ3+K3cosh12ξ1+ξ2−ξ3,where(19a)K=a12−a13−a23−,K1=a12+a13+a23−,K2=a12+a13−a23+,K3=a12−a13+a23+,(19b)ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,aij±=cpi−pj2−6βki±kj2,i,j=1,2,3.Furthermore, one can verify that, for any oddN=2n−1 (n is a positive integer), the function f (15) does not satisfy the invariant condition in equation (11). In other words, symmetry breaking soliton solutions of odd orders for the AB-KP systems (7a) and (7b) are prohibited.ForN=2, equation (15) becomes(20)f=f2=Kcoshξ1+ξ22+K1coshξ1−ξ22,where(21)K=cp1−p22−6βk1−k22,K1=cp1−p22−6βk1+k22.By fixing the real parameters,(22)k2=±k1,p2=−p1,η20=∓η10,the invariant condition in equation (11) is satisfied. Correspondingly, by writing(23)f=f2=cp12coshk1x−x02−c+k12β+cp122t−t02+cp12−6βk12coshk1p1y+η10,a symmetry breaking two-soliton solution of equations (7a) and (7b) are expressed as(24)A=12βαlnf2xx+b1lnf2xxx+b2lnf2xxt,B=12βαlnf2xx−b1lnf2xxx−b2lnf2xxt.Figure1 shows the symmetry breaking two-soliton structure of solution (24) with the parameters being taken as(25)α=k1=p1=1,β=16,c=4,x0=t0=η10=0.Figure 1
The symmetry breaking two-soliton solution (24) of the AB-KP systems (7a) and (7b), with the selecting parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Meanwhile, Figure1(a) describes a standard two-soliton structure b1=b2=0 for solution (24) at time t=0. Under this special condition, the solution A coincides with the solution B exactly. Figures 1(b) and 1(c) are two symmetry breaking two-soliton structures for solution (24) with the selected parameters b1=b2=10 at time t=0. Obviously, Figure 1(c) depicts a reversal structure of Figure 1(b) by the solution B which is P^sxT^d symmetry of the solution A for the AB-KP systems (7a) and (7b). This corresponds to the phenomenon that the shifted parity and delayed time reversal are applied to describe two-place events [11, 12]. These structures are realized via the symbolic computation software Maple efficiently.ForN=4, the function f of equation (15) can be rewritten regularly as(26)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(27)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−,ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,aij±=cpi−pj2−6βki±kj2,i,j=1,2,3,4.After finishing some detailed analysis, there are two independent real parameter selections of the symmetry breaking four-soliton solution for (7a) and (7b), which are (28)k3=±k1,k4=±k2,p3=−p1,p4=−p2,η30=∓η10,η40=∓η20,with B=P^sT^dA=A−x+x0,y=y,−t+t0. Based on these restrictions in (28), we have(29)a23±=a14±,a34±=a12±,K3=K1,K4=K2.At this time, the symmetry breaking four-soliton solution of the AB-KP systems (7a) and (7b) is(30)A=12βαlnf4xx+b1lnf4xxx+b2lnf4xxt,B=12βαlnf4xx−b1lnf4xxx−b2lnf4xxt,where(31)f4=Kcoshζ1++ζ2++K1coshζ1−+ζ2++coshζ1−−ζ2++K2coshζ1++ζ2−+coshζ1+−ζ2−+K23coshζ1−−ζ2−+K24coshζ1+−ζ2++K34coshζ1−+ζ2−,with(32)ζ1±=12ξ1±ξ3,ζ2±=12ξ2±ξ4.If settingα=k1=p1=1,β=1/6,c=−5,k2=−p2=6/5, and x0=t0=η10=η20=0, we can depict the abovementioned symmetry breaking four-soliton structure in (Figure 2). The similar situation is as follows: Figure 2(a) is the standard four-soliton structure b1=b2=0 for the solution A=B at time t=0. Figures 2(b) and 2(c) are two symmetry breaking four-soliton structures for the solution A and B, respectively, with the selected parameters b1=b2=10 at time t=0.Figure 2
The symmetry breaking four-soliton solution (30) of the AB-KP systems (7a) and (7b), with the selecting parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)


### 3.2. Symmetry Breaking Breather Solutions

When takingp1=p0I (p0 is a real constant and I is the imaginary unit, I2=−1), a symmetry breaking breather solution of the AB-KP systems (7a) and (7b) can read(33)A=12βαlnfbxx+b1lnfbxxx+b2lnfbxxt,B=P^sxT^dA=A−x+x0,y,−t+t0,with the ansatz function(34)f=fb=−cp02coshk1x−x02−c+k12β−cp022t−t02+−cp02−6βk12coshk1p0Iy+η10,from equation (23).By some constraints to the parameters in this solution, a family of analytical breather solutions can be generated. For example, when taking the real constants(35)c=−4,α=k1=1,β=16,p0=2,x0=t0=η10=0,equation (34) becomes(36)f=fb=4coshx−256t+15cos2y,according to Euler’s formula. This function indicates that the solution has two wave components, that is, a regular solitary wave with the propagating speed −25/6 and a periodic wave with period π. Figure 3 is a density plot of the breathers defined by equation (36) with the parameters in (35). Figure 3(a) is the standard first-order breather structure b1=b2=0 for the solution A=B=815cos2ycoshx+4/4coshx+15cos2y2 at time t=0. Figures 3(b) and 3(c) are two symmetry breaking breather structures for the solution A,B=815cos2ycoshx+4/4coshx+15cos2y2±95sinhx32+415cos2ycoshx−15cos22y/4coshx+15cos2y3, with the selected parameters b1=b2=10 at time t=0. As these solutions combine the trigonometric cosine function with hyperbolic sine/cosine functions, the property of these functions describes the symmetry breaking breather structures [29, 30].Figure 3
The density plot of they-periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)In the abovementioned situation, when taking the constants(37)c=−4,α=1,β=16,k1=I,p0=2,x0=t0=η10=0,equation (34) has the expression(38)f=fb=16cosx−236t+17cosh2y.Figure4 is a density plot of the breathers described according to equation (38) under the parameter selection (37). That is, when the parameter k1 also takes the imaginary unit I, the x-periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b) are formed.Figure 4
The density plot of thex-periodic symmetry breaking breather of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)The abovementioned idea can be extended to solution (30). After setting the parameters(39)c=−5,α=k1=1,β=16,k2=12,p1=I,p2=−2I,x0=t0=η10=η20=0,c=−4,α=1,β=16,k1=p1=−p2=I,k2=−2I,x0=t0=η10=η20=0,the y-periodic and x-periodic second-order breather solutions can be derived, which are symmetry breaking (Figures 5 and 6, respectively).Figure 5
They-periodic symmetry breaking density plot of the second-order breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Figure 6
Thex-periodic symmetry breaking density plot of the second-order breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)


### 3.3. Symmetry Breaking Lump Solutions

As we know, the lump solution is expressed by the rational function which is localized in all directions in the space. Based on the long-wave limit idea of generating lump solutions to nonlinear equations, we derive this kind of solutions of the AB-KP systems (7a) and (7b) by taking a long-wave limit. After putting(40)k1=δ1ϵ1,k2=δ2ϵ2,η10=η20=Iπ,into the bilinear function (20) and setting the limit ϵ1⟶0,ϵ2⟶0, the function f can be obtained:(41)f=fl=δ1δ2x−x02+p1y−c1+p122t−t02x−x02+p2y−c1+p222t−t02+24δ1δ2βcp1−p22.When letting(42)p1=a2+a5Iδ1,p2=a2−a5Iδ2,δ1=a1+a4I,δ2=a1−a4I,equation (41) can be arranged in(43)f=fl=g2+h2+a7,g=a1x−x02+a2y+a3t−t02,h=a4x−x02+a5y+a6t−t02,where ai1≤i≤7 are all real parameters to be determined. Note that the function f in equation (43) is positive if the parameter a7>0. Combining equation (43) with equations (11) and (12), the constraining relations of these parameters are(44)a3=a1ca22−2a422a42,a5=−a1a2a4,a6=ca22−2a422a4,a7=−6a42βa12+a42a22c,where all the denominators are nonzero.This time, solution (10) becomes(45a)A=12βαlng2+h2−6a42βa12+a42a22cxx+b1lng2+h2−6a42βa12+a42a22cxxx+b2lng2+h2−6a42βa12+a42a22cxxt,(45b)B=12βαlng2+h2−6a42βa12+a42a22cxx−b1lng2+h2−6a42βa12+a42a22cxxx−b2lng2+h2−6a42βa12+a42a22cxxt,where the functions g and h are given as follows:(46)g=a1x−x02+a2y+a1ca22−2a422a42t−t02,h=a4x−x02−a1a2a4y+ca22−2a422a4t−t02,with c,a1,a2,a4,x0,andt0 being some free real constants. Obviously, this set of solution (45a) and (45b) represents solitary waves in the form of rational structures, which is a family of two-wave solutions with different velocities and directions due to the linear functions g and h of the three variables x,y,andt.If the constants are taken asa1=a4=−1,a2=c=α=1,β=1/6,andx0=t0=0, equation (43) becomes(47)f=fl=x−y−t22+x+y−t22−2,according to equation (44). Figure 7 is a density plot of the lumps by equation (36) under the abovementioned parameter selection. Figure 7(a) is a normal first-order lump structure b1=b2=0 for the solution A=B=−4x2−y2+1/x2+y2−12 at time t=0. Figures 7(b) and 7(c) are two symmetry breaking lump structures for the solution A,B=−4x2−y2+1/x2+y2−12±20xx2+3x−3y2/x2+y2−13, with the selected parameters b1=b2=10, at time t=0. As these solutions are all rational functions, the property of these functions describes symmetry breaking lump structures.Figure 7
The density plot of the symmetry breaking lumps of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Furthermore, if we takeki=δiϵi,ηi0=πI,i=1,2,3,4, and(48)p1=1+I,p2=1−I,p3=−p1,p4=−p2,δ3=δ1,δ4=δ2,after setting the limit ϵi⟶0,i=1,2,3,4, the function f (26) can be simplified into(49)f=fl=c2x−x024+4y4+4c4t−t024−4c3x−x023t−t02+8c4x−x022t−t022−4c2x−x02t−t022c3t−t022−2cy2−3β−4c32cy2+3βt−t022−36cβy2+81β2.Figure8 shows the second-order lump structures when c=−4,α=1,β=1/6,andx0=t0=η10=η20=0, which are symmetry breaking.Figure 8
The symmetry breaking density plot of the second-order lumps of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=5, at time t=0.
(a)
(b)
(c)


## 3.1. Symmetry Breaking Soliton Solutions

Based on the bilinear form (12), we can first determine symmetry breaking soliton solutions through the Bäcklund transformation (10) of the AB-KP systems (7a) and (7b) with the function f being written as a summation of some special functions [11–16, 18]:(15)f=fN=∑νKνcosh12∑i=1Nνiξi,ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,where ν=ν1,ν2,…,νN, with vi=±1, and ki,pi,ηi0i=1,2,…,N,c,x0, and t0 are undetermined constants, while(16)Kν=∏i<jNcpi−pj2−6βki−νiνjkj2.ForN=1, equation (15) takes the form(17)f=f1=coshξ12,ξ1=k1x−x02+p1y−c+k12β+cp122t−t02+η10.However, the invariant condition (11) of this function f (17) is not satisfied for the constants k1,p1,c,x0,t0, and η10 being not all zero. It means that it is impossible to get any nontrivial symmetry breaking single soliton solution of equation (12) through (10).The same circumstance happens whenN=3, in which the function f of equation (15) is(18)f=f3=Kcosh12ξ1+ξ2+ξ3+K1cosh12ξ1−ξ2−ξ3+K2cosh12ξ1−ξ2+ξ3+K3cosh12ξ1+ξ2−ξ3,where(19a)K=a12−a13−a23−,K1=a12+a13+a23−,K2=a12+a13−a23+,K3=a12−a13+a23+,(19b)ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,aij±=cpi−pj2−6βki±kj2,i,j=1,2,3.Furthermore, one can verify that, for any oddN=2n−1 (n is a positive integer), the function f (15) does not satisfy the invariant condition in equation (11). In other words, symmetry breaking soliton solutions of odd orders for the AB-KP systems (7a) and (7b) are prohibited.ForN=2, equation (15) becomes(20)f=f2=Kcoshξ1+ξ22+K1coshξ1−ξ22,where(21)K=cp1−p22−6βk1−k22,K1=cp1−p22−6βk1+k22.By fixing the real parameters,(22)k2=±k1,p2=−p1,η20=∓η10,the invariant condition in equation (11) is satisfied. Correspondingly, by writing(23)f=f2=cp12coshk1x−x02−c+k12β+cp122t−t02+cp12−6βk12coshk1p1y+η10,a symmetry breaking two-soliton solution of equations (7a) and (7b) are expressed as(24)A=12βαlnf2xx+b1lnf2xxx+b2lnf2xxt,B=12βαlnf2xx−b1lnf2xxx−b2lnf2xxt.Figure1 shows the symmetry breaking two-soliton structure of solution (24) with the parameters being taken as(25)α=k1=p1=1,β=16,c=4,x0=t0=η10=0.Figure 1
The symmetry breaking two-soliton solution (24) of the AB-KP systems (7a) and (7b), with the selecting parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Meanwhile, Figure1(a) describes a standard two-soliton structure b1=b2=0 for solution (24) at time t=0. Under this special condition, the solution A coincides with the solution B exactly. Figures 1(b) and 1(c) are two symmetry breaking two-soliton structures for solution (24) with the selected parameters b1=b2=10 at time t=0. Obviously, Figure 1(c) depicts a reversal structure of Figure 1(b) by the solution B which is P^sxT^d symmetry of the solution A for the AB-KP systems (7a) and (7b). This corresponds to the phenomenon that the shifted parity and delayed time reversal are applied to describe two-place events [11, 12]. These structures are realized via the symbolic computation software Maple efficiently.ForN=4, the function f of equation (15) can be rewritten regularly as(26)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(27)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−,ξi=kix−x02+piy−c+ki2β+cpi22t−t02+ηi0,aij±=cpi−pj2−6βki±kj2,i,j=1,2,3,4.After finishing some detailed analysis, there are two independent real parameter selections of the symmetry breaking four-soliton solution for (7a) and (7b), which are (28)k3=±k1,k4=±k2,p3=−p1,p4=−p2,η30=∓η10,η40=∓η20,with B=P^sT^dA=A−x+x0,y=y,−t+t0. Based on these restrictions in (28), we have(29)a23±=a14±,a34±=a12±,K3=K1,K4=K2.At this time, the symmetry breaking four-soliton solution of the AB-KP systems (7a) and (7b) is(30)A=12βαlnf4xx+b1lnf4xxx+b2lnf4xxt,B=12βαlnf4xx−b1lnf4xxx−b2lnf4xxt,where(31)f4=Kcoshζ1++ζ2++K1coshζ1−+ζ2++coshζ1−−ζ2++K2coshζ1++ζ2−+coshζ1+−ζ2−+K23coshζ1−−ζ2−+K24coshζ1+−ζ2++K34coshζ1−+ζ2−,with(32)ζ1±=12ξ1±ξ3,ζ2±=12ξ2±ξ4.If settingα=k1=p1=1,β=1/6,c=−5,k2=−p2=6/5, and x0=t0=η10=η20=0, we can depict the abovementioned symmetry breaking four-soliton structure in (Figure 2). The similar situation is as follows: Figure 2(a) is the standard four-soliton structure b1=b2=0 for the solution A=B at time t=0. Figures 2(b) and 2(c) are two symmetry breaking four-soliton structures for the solution A and B, respectively, with the selected parameters b1=b2=10 at time t=0.Figure 2
The symmetry breaking four-soliton solution (30) of the AB-KP systems (7a) and (7b), with the selecting parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)


## 3.2. Symmetry Breaking Breather Solutions

When takingp1=p0I (p0 is a real constant and I is the imaginary unit, I2=−1), a symmetry breaking breather solution of the AB-KP systems (7a) and (7b) can read(33)A=12βαlnfbxx+b1lnfbxxx+b2lnfbxxt,B=P^sxT^dA=A−x+x0,y,−t+t0,with the ansatz function(34)f=fb=−cp02coshk1x−x02−c+k12β−cp022t−t02+−cp02−6βk12coshk1p0Iy+η10,from equation (23).By some constraints to the parameters in this solution, a family of analytical breather solutions can be generated. For example, when taking the real constants(35)c=−4,α=k1=1,β=16,p0=2,x0=t0=η10=0,equation (34) becomes(36)f=fb=4coshx−256t+15cos2y,according to Euler’s formula. This function indicates that the solution has two wave components, that is, a regular solitary wave with the propagating speed −25/6 and a periodic wave with period π. Figure 3 is a density plot of the breathers defined by equation (36) with the parameters in (35). Figure 3(a) is the standard first-order breather structure b1=b2=0 for the solution A=B=815cos2ycoshx+4/4coshx+15cos2y2 at time t=0. Figures 3(b) and 3(c) are two symmetry breaking breather structures for the solution A,B=815cos2ycoshx+4/4coshx+15cos2y2±95sinhx32+415cos2ycoshx−15cos22y/4coshx+15cos2y3, with the selected parameters b1=b2=10 at time t=0. As these solutions combine the trigonometric cosine function with hyperbolic sine/cosine functions, the property of these functions describes the symmetry breaking breather structures [29, 30].Figure 3
The density plot of they-periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)In the abovementioned situation, when taking the constants(37)c=−4,α=1,β=16,k1=I,p0=2,x0=t0=η10=0,equation (34) has the expression(38)f=fb=16cosx−236t+17cosh2y.Figure4 is a density plot of the breathers described according to equation (38) under the parameter selection (37). That is, when the parameter k1 also takes the imaginary unit I, the x-periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b) are formed.Figure 4
The density plot of thex-periodic symmetry breaking breather of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)The abovementioned idea can be extended to solution (30). After setting the parameters(39)c=−5,α=k1=1,β=16,k2=12,p1=I,p2=−2I,x0=t0=η10=η20=0,c=−4,α=1,β=16,k1=p1=−p2=I,k2=−2I,x0=t0=η10=η20=0,the y-periodic and x-periodic second-order breather solutions can be derived, which are symmetry breaking (Figures 5 and 6, respectively).Figure 5
They-periodic symmetry breaking density plot of the second-order breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Figure 6
Thex-periodic symmetry breaking density plot of the second-order breathers of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)


## 3.3. Symmetry Breaking Lump Solutions

As we know, the lump solution is expressed by the rational function which is localized in all directions in the space. Based on the long-wave limit idea of generating lump solutions to nonlinear equations, we derive this kind of solutions of the AB-KP systems (7a) and (7b) by taking a long-wave limit. After putting(40)k1=δ1ϵ1,k2=δ2ϵ2,η10=η20=Iπ,into the bilinear function (20) and setting the limit ϵ1⟶0,ϵ2⟶0, the function f can be obtained:(41)f=fl=δ1δ2x−x02+p1y−c1+p122t−t02x−x02+p2y−c1+p222t−t02+24δ1δ2βcp1−p22.When letting(42)p1=a2+a5Iδ1,p2=a2−a5Iδ2,δ1=a1+a4I,δ2=a1−a4I,equation (41) can be arranged in(43)f=fl=g2+h2+a7,g=a1x−x02+a2y+a3t−t02,h=a4x−x02+a5y+a6t−t02,where ai1≤i≤7 are all real parameters to be determined. Note that the function f in equation (43) is positive if the parameter a7>0. Combining equation (43) with equations (11) and (12), the constraining relations of these parameters are(44)a3=a1ca22−2a422a42,a5=−a1a2a4,a6=ca22−2a422a4,a7=−6a42βa12+a42a22c,where all the denominators are nonzero.This time, solution (10) becomes(45a)A=12βαlng2+h2−6a42βa12+a42a22cxx+b1lng2+h2−6a42βa12+a42a22cxxx+b2lng2+h2−6a42βa12+a42a22cxxt,(45b)B=12βαlng2+h2−6a42βa12+a42a22cxx−b1lng2+h2−6a42βa12+a42a22cxxx−b2lng2+h2−6a42βa12+a42a22cxxt,where the functions g and h are given as follows:(46)g=a1x−x02+a2y+a1ca22−2a422a42t−t02,h=a4x−x02−a1a2a4y+ca22−2a422a4t−t02,with c,a1,a2,a4,x0,andt0 being some free real constants. Obviously, this set of solution (45a) and (45b) represents solitary waves in the form of rational structures, which is a family of two-wave solutions with different velocities and directions due to the linear functions g and h of the three variables x,y,andt.If the constants are taken asa1=a4=−1,a2=c=α=1,β=1/6,andx0=t0=0, equation (43) becomes(47)f=fl=x−y−t22+x+y−t22−2,according to equation (44). Figure 7 is a density plot of the lumps by equation (36) under the abovementioned parameter selection. Figure 7(a) is a normal first-order lump structure b1=b2=0 for the solution A=B=−4x2−y2+1/x2+y2−12 at time t=0. Figures 7(b) and 7(c) are two symmetry breaking lump structures for the solution A,B=−4x2−y2+1/x2+y2−12±20xx2+3x−3y2/x2+y2−13, with the selected parameters b1=b2=10, at time t=0. As these solutions are all rational functions, the property of these functions describes symmetry breaking lump structures.Figure 7
The density plot of the symmetry breaking lumps of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=10, at time t=0.
(a)
(b)
(c)Furthermore, if we takeki=δiϵi,ηi0=πI,i=1,2,3,4, and(48)p1=1+I,p2=1−I,p3=−p1,p4=−p2,δ3=δ1,δ4=δ2,after setting the limit ϵi⟶0,i=1,2,3,4, the function f (26) can be simplified into(49)f=fl=c2x−x024+4y4+4c4t−t024−4c3x−x023t−t02+8c4x−x022t−t022−4c2x−x02t−t022c3t−t022−2cy2−3β−4c32cy2+3βt−t022−36cβy2+81β2.Figure8 shows the second-order lump structures when c=−4,α=1,β=1/6,andx0=t0=η10=η20=0, which are symmetry breaking.Figure 8
The symmetry breaking density plot of the second-order lumps of the AB-KP systems (7a) and (7b), with the selected parameters b1 and b2 are (a) b1=b2=0 and (b) and (c) b1=b2=5, at time t=0.
(a)
(b)
(c)


## 4. Conclusion

As everyone knows, the two-place correlated physical events widely exist in the field of natural science, and the discussed AB physics (two-place physics) has a profound influence on other scientific fields. In this work, by establishing a special AB-KP system via the parity with a shift of the space variablex and time reversal with a delay, some group invariant solutions, such as symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions have been presented through introducing an extended Bäcklund transformation and the established Hirota bilinear form. At the same time, the corresponding symmetry breaking structures of these explicit solutions are depicted according to the ansatz functions.In fact, these are the following few open problems. Firstly, we may investigate more local and nonlocal symmetry breaking structures, such as the cnoidal wave and rogue wave through expression (2). Secondly, the arbitrary function GA,B of A and B (which should be P^sxT^d invariant) is diverse, although we take GA,B=α/2Ax2+AAxx−Bx2−BBxx in this paper. Thirdly, algebraic structures involving the related Lie point symmetry and Lie-Bäcklund symmetry reductions, and Bäcklund transformations determined by residual symmetries may be discussed mathematically for the AB-KP systems (7a) and (7b). Finally, the P^sxT^d symmetry of this paper could be generalized to other nonlinear systems by taking the specific elements of the larger P^sT^dC^ symmetry group [15].



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