Controllable symmetry breaking solutions for a nonlocal Boussinesq system

The generalized Boussinesq equation is a useful model to describe the water wave. In this paper, with the coupled Alice-Bob (AB) systems, the nonlocal Boussinesq system can be obtained via the parity and time reversal symmetry reduction. By introducing an extended Bäcklund transformation, the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions for a nonlocal Boussinesq system are obtained through the derived Hirota bilinear form. The residual symmetry and finite symmetry transformation of the nonlocal AB-Boussinesq system are also studied.

systems 12 . The multisoliton solutions for the nonlocal KdV and Boussinesq systems were investigated and the possible prohibitions on multisoliton solutions were also discussed. Now, we focus our attention on the following generalized Boussinesq equation tt xx xx xxxx 2 which was first introduced by Boussinesq in 1871 to describe the propagation of long waves in shallow water 13 . This Boussinesq Eq. (1) is a soliton equation solvable by inverse scattering which arises in several other physical applications including one-dimensional nonlinear lattice-waves, vibrations in a nonlinear string and ion sound waves in a plasma [14][15][16][17] . Also, the soliton-cnoidal wave interaction solutions of the Boussinesq Eq. (1) was described when α γ β = = − =− 1, 3 18 . The above Boussinesq equation with minus fourth-order term had multiple soliton solutions, whereas the model with the plus fourth-order term had multiple complex soliton solutions when the second-order derivative u xx was deleted and β = ±1 19 . Rational solutions of this equation and applications to rogue waves was shown when α β γ = − = =− 1, 1 3 20 . The bilinear transformation method was proposed to find the rogue wave solutions for the above generalized Eq. (1) 21 . Two exponential-type integrators were proposed and analyzed for the "good" Boussinesq equation, i.e. α β γ = = − = 1, 1 with rough initial data 22 . The Lie symmetry analysis was applied and some special solitons such as dark, singular and periodic solitons was devoted 23 .
The next several sections, we focus on the AB-Boussinesq system of the generalized Eq. (1). In Section 2, a nonlocal AB-Boussinesq system is constructed and its bilinear form is written through an extended Bäcklund transformation. In Section 3, the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions are presented through the derived Hirota bilinear form. Starting from a modified Bäcklund transformation, the residual symmetry of the system is studied by introducing two auxiliary variables in Section 4. Therefore, the finite symmetry transformation is obtained through solving the initial value problem. Some conclusions are given in the final section.

Methods
A nonlocal AB-Boussinesq system and its bilinear form. In this section, the model of AB system is applied into the Boussinesq equation. Based on the principle of the AB system referred in 1,2 , when substituting = + u A B ( ) 1 2 into (1), the nonlocal AB-Boussinesq system is Now, we introduce an extended Bäcklund transformation where D t 2 , D x 2 and D x 4 are the bilinear derivative operators defined by 24,25 x m t n m n According to the properties of bilinear operator D, Eq. (6) is equal to

Symmetry breaking rogue wave, soliton and breather solutions of the AB-Boussinesq system
In this section, we turn our attention to the Hirota bilinear form (6) of the nonlocal AB-Boussinesq system (4) to derive the symmetry breaking rogue wave, symmetry breaking soliton and symmetry breaking breather solutions.
Symmetry breaking rogue wave solutions. In nonlinear science, especially for some nonlinear integrable systems which as one of the essential topics, the theoretical study of rogue waves has gotten more and more attention in recent years. Rogue waves, also known as freak waves, monster waves, extreme waves, or hundred year waves, are relatively large and spontaneous ocean surface waves, which are a serious threat even to large ships and ocean liners 26,27 . These phenomena are ubiquitous in nature and can appear in a variety of different contexts such as reported in liquid Helium, nonlinear optics and microwave cavities 28 .
For describing the symmetry breaking rogue wave solutions, we seek solutions of Eq. (4) in the form in X 2 and T 2 described by  (6), we derive the following polynomials T  T  T  1  1875  1875  1875  1875 17 1875  www.nature.com/scientificreports www.nature.com/scientificreports/ The following four sets of figures (Figs. 1, 2, 3 and 4) are presented for the purposes of illustration the solutions u 1 , u 2 and u 3 . Figure 1 shows three ranks of the symmetry breaking rogue wave solutions

Example 3.1 Symmetry breaking line soliton solution
We take t anh( ) 6 tanh ( ) (17b)    www.nature.com/scientificreports www.nature.com/scientificreports/ Therefore, the interaction of the triple symmetry breaking soliton solution of Eq. (4) can be expressed as 1, 4 and = b 10 at different times = − t 2, 1 and 3, respectively. From these presented results of Examples 3.1-3.3, we have a different conclusion from the previous, that is "the multisoliton solutions with odd numbers and the multisoliton solutions with even numbers but with pursuant interactions are prohibited" 12 .

Residual symmetry of the AB-Boussinesq system
Recently, it was found that the residue in Painlevé truncated expansion corresponds to a nonlocal symmetry for the Painlevé integrable system and then such type of symmetry was referred to residual symmetry. This residual symmetry was localized to Lie point symmetry by introducing suitable prolonged system 30 . From the leading term analysis of the AB-Boussinesq system (4), the Bäcklund transformation is extended as follows x x The enlarged AB-Boussinesq system of Eq. (4) according to Eqs. (4), (29), (33) and (34) given by