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 variable x 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.
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(where the operators and 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 -- principle with (the parity), (time reversal), and (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 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 -- 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 expressionplays 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 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 formwhere 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 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.
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 into equation (3), the AB-KP initial equation iswhich can be split into the coupled equationswhere B is related to A through ( expresses parity with a shift of the space variable x and time reversal with a delay), and is an arbitrary function of A and B, but should be invariant. That is, . Although there are infinitely many functions satisfying this, we take a nontrivial function asat 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:by taking the reduction condition and letting the arbitrary constants with
Now, we introduce an extended Bäcklund transformation:where are arbitrary constants and is a new function of variables , and t, satisfying the invariant condition
When , 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:where and are the Hirota bilinear derivative operators defined by [7, 8]
According to the properties of the Hirota bilinear operator, equation (12) readswhich 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]:where , with , and , and are undetermined constants, while
For , equation (15) takes the form
However, the invariant condition (11) of this function f (17) is not satisfied for the constants , and 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 when , in which the function f of equation (15) iswhere
Furthermore, one can verify that, for any odd (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 , equation (15) becomeswhere
By fixing the real parameters,the invariant condition in equation (11) is satisfied. Correspondingly, by writinga symmetry breaking two-soliton solution of equations (7a) and (7b) are expressed as
Figure 1 shows the symmetry breaking two-soliton structure of solution (24) with the parameters being taken as
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Meanwhile, Figure 1(a) describes a standard two-soliton structure for solution (24) at time . 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 at time . Obviously, Figure 1(c) depicts a reversal structure of Figure 1(b) by the solution B which is 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.
For , the function f of equation (15) can be rewritten regularly aswhere
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 with . 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) iswherewith
If setting , and , 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 for the solution at time . Figures 2(b) and 2(c) are two symmetry breaking four-soliton structures for the solution A and B, respectively, with the selected parameters at time .
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3.2. Symmetry Breaking Breather Solutions
When taking ( is a real constant and I is the imaginary unit, ), a symmetry breaking breather solution of the AB-KP systems (7a) and (7b) can readwith the ansatz functionfrom 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 constantsequation (34) becomesaccording to Euler’s formula. This function indicates that the solution has two wave components, that is, a regular solitary wave with the propagating speed 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 for the solution at time . Figures 3(b) and 3(c) are two symmetry breaking breather structures for the solution , with the selected parameters at time . 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].
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In the abovementioned situation, when taking the constantsequation (34) has the expression
Figure 4 is a density plot of the breathers described according to equation (38) under the parameter selection (37). That is, when the parameter also takes the imaginary unit I, the -periodic symmetry breaking breathers of the AB-KP systems (7a) and (7b) are formed.
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The abovementioned idea can be extended to solution (30). After setting the parametersthe -periodic and -periodic second-order breather solutions can be derived, which are symmetry breaking (Figures 5 and 6, respectively).
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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 puttinginto the bilinear function (20) and setting the limit , the function f can be obtained:
When lettingequation (41) can be arranged inwhere are all real parameters to be determined. Note that the function f in equation (43) is positive if the parameter . Combining equation (43) with equations (11) and (12), the constraining relations of these parameters arewhere all the denominators are nonzero.
This time, solution (10) becomeswhere the functions and h are given as follows:with 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 and h of the three variables .
If the constants are taken as , equation (43) becomesaccording 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 for the solution at time . Figures 7(b) and 7(c) are two symmetry breaking lump structures for the solution , with the selected parameters , at time . As these solutions are all rational functions, the property of these functions describes symmetry breaking lump structures.
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Furthermore, if we take , andafter setting the limit the function f (26) can be simplified into
Figure 8 shows the second-order lump structures when , which are symmetry breaking.
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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 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 of A and B (which should be invariant) is diverse, although we take 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 symmetry of this paper could be generalized to other nonlinear systems by taking the specific elements of the larger symmetry group [15].
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
The project was supported by the National Natural Science Foundation of China (nos. 11775104 and 11705077) and Scientific Research Foundation of the First-Class Discipline of Zhejiang Province (B) (no. 201601).