The shifted parity and delayed time-reversal symmetry-breaking solutions for the (1+1)-dimensional Alice–Bob Boussinesq equation

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.


Introduction
For one (1 + 1)-dimensional model, except for identity transformation, there are the shifted parityP s and delayed time-reversalT d transformations for the spatial variable x and time variable t, respectively. In other words, with x 0 and t 0 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′);f is a suitable operator (such asf 2 1 orf ∈ Θ ≡ {1,P s ,T d ,P sTd }) [3][4][5][6][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 theP sTd 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 operatorP sTd on one solution S, one can findP sTd S ≠ S.
For an illustrated model, the ( or where w tt ≡ z 2 zt 2 w, w ix ≡ z i zx i w. 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][9][10].
Except for the former works where the physical quantity A+B 2 is taken directly, we derive the Alice-Bob system of Eq. 3 through its Lax pair and the dark parameterization approach [11][12][13].
After adopting thef P s xT d t symmetry principle through B f A Af, the Boussinesq Eq. 3 can be induced into the following Alice-Bob system: The corresponding Lax pairs of Eqs 5, 6areas follows: and 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][15][16][17]. For the coupling Boussinesq system, w 0 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: and These equations can directly derive the non-local systems (9) and (10) 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 B P sTd A. A short summary is given in Section 3.

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 D 2 t , D 2 x and D 4 x are written as [18,19] Eq. 21 also has the following explicit expression: Frontiers in Physics frontiersin.org

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][5][6].
where {]} = {] i = ±1} and k i (i = 1, 2, ···, N) are arbitrary constants, and where (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 ofP sTd symmetry B f A A(−x, −t). 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 δ 1 , and the amplitude and wave width are determined by k 1 , b, and c, which is also confirmed by Eqs 27, 28. Figure 1 shows this structure when the related parameters are taken as follows: When N = 2, Eq. 24 becomes as follows: 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, k 1 = ±k 2 , ω 1 = ±ω 2 . 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, 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 k 1 , k 2 to the assumed units on the two-soliton solution, the breather can be obtained. For example, by setting the following parameters b 1, c 1, k 1 I, k 2 I, δ 1 1, the 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. When N = 3, Eq. 24 has the following more complicated situation: and Based on the selecting parameters b, c, k 1 , k 2 , δ 1 , and δ 2 of Eqs 33, 35, two kinds of interactions for the solitons can be constructed by considering the following equation:    Figure 4 and Figure 5 show two interaction structures of the Alice-Bob systems (9) and (10) through Eqs 42, 43.

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: