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

Department of Photoelectric Engineering, Lishui University, Lishui 323000, China Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620-5700, USA Institute of Nonlinear Analysis and Department of Mathematics, Lishui University, Lishui 323000, China Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia School of Mathematics, South China University of Technology, Guangzhou 510640, China College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X2046, Mmabatho 2735, South Africa


Introduction
e 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][5][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 (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]. e technical approach originated from the so-called P-T-C principle with P (the parity), T (time reversal), and C (charge conjugation) [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. From this, a general Nth Darboux transformation for the AB-mKdV equation was constructed [13]. By using this Darboux transformation, some types of PT 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]. e 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 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 H 1 type AB systems [11][12][13][14][15][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 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. e rest of this paper is organized as follows. In Section 2, 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.

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 � 1/2(A + B) into equation (3), the AB-KP initial equation is which can be split into the coupled equations at present, and equation (5) is reduced to the following AB-KP system: In fact, this AB-KP system can also be derived as a special reduction of the coupled KP system: and letting the arbitrary constants c i (i � 1, 2, . . . , 9) with 2 Complexity Now, we introduce an extended Bäcklund transformation: where b 1 and b 2 are arbitrary constants and f ≡ f(x, y, t) is a new function of variables x, y, and t, satisfying the invariant condition When b 1 � 0 and b 2 � 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: whereD 4 x and D 2 y are the Hirota bilinear derivative operators defined by [7,8] According to the properties of the Hirota bilinear operator, equation (12) reads 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.

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.

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][12][13][14][15][16]18]: where For N � 1, equation (15) takes the form However, the invariant condition (11) of this function f (17) is not satisfied for the constants k 1 , p 1 , c, x 0 , t 0 , and η 10 being not all zero. It means that it is impossible to get any nontrivial symmetry breaking single soliton solution of equation (12) through (10). e same circumstance happens when N � 3, in which the function f of equation (15) is Furthermore, one can verify that, for any odd N � 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.
For N � 2, equation (15) becomes where By fixing the real parameters, the invariant condition in equation (11) is satisfied. Correspondingly, by writing a symmetry breaking two-soliton solution of equations (7a) and (7b) are expressed as (24) Figure 1 shows the symmetry breaking two-soliton structure of solution (24) with the parameters being taken as Meanwhile, Figure 1(a) describes a standard two-soliton structure (b 1 � b 2 � 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 b 1 � b 2 � 10 at time t � 0. Obviously, Figure 1(c) depicts a reversal structure of Figure 1 is corresponds to the phenomenon that the shifted parity and delayed time reversal are applied to describe two-place events [11,12]. ese structures are realized via the symbolic computation software Maple efficiently.
For N � 4, the function f of equation (15) can be rewritten regularly as 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 . Based on these restrictions in (28), we have At this time, the symmetry breaking four-soliton solution of the AB-KP systems (7a) and (7b) is Complexity 5 where with If setting α � k 1 � p 1 � 1, β � 1/6, c � − 5, k 2 � − p 2 � 6/5, and x 0 � t 0 � η 10 � η 20 � 0, we can depict the abovementioned symmetry breaking four-soliton structure in (Figure 2). e similar situation is as follows: Figure 2

Symmetry Breaking Breather Solutions.
When taking p 1 � p 0 I (p 0 is a real constant and I is the imaginary unit, I 2 � − 1), a symmetry breaking breather solution of the AB-KP systems (7a) and (7b) can read with the ansatz function 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 equation (34) becomes according to Euler's formula. is 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 cos(2y)) 2 at time t � 0. Figures 3(b) and 3(c) are two symmetry breaking breather structures for the so- cos(2y)cosh x − 15 cos 2 (2y))/ (4 cosh x + �� 15 √ cos(2y)) 3 , with the selected parameters b 1 � b 2 � 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].
In the abovementioned situation, when taking the constants equation (34) has the expression Figure 4 is a density plot of the breathers described according to equation (38) under the parameter selection (37). at is, when the parameter k 1 also takes the imaginary unit I, the x-periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b) are formed. e abovementioned idea can be extended to solution (30). After setting the parameters 6 Complexity   Complexity the y-periodic and x-periodic second-order breather solutions can be derived, which are symmetry breaking ( Figures 5 and 6, respectively).

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

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 variable x 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 G (A, B) of A and B (which should be P x s T d invariant) is diverse, although we take G(A, B) � (α/2)(A 2 x + AA xx − B 2 x − BB xx ) in this paper. irdly, 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 x s T d symmetry of this paper could be generalized to other nonlinear systems by taking the specific elements of the larger P s T d C symmetry group [15].  Data Availability e 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
e authors declare that they have no conflicts of interest.