⇦Chinese Physics Letters, 2019, Vol. 36, No. 5, Article code 050501 Multiple Soliton Solutions of Alice–Bob Boussinesq Equations * Hui Li (李辉)1, S. Y. Lou (楼森岳)1,2** Affiliations 1School of Physical Science and Technology, Ningbo University, Ningbo 315211 2Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062 Received 11 February 2019, online 17 April 2019 *Supported by the National Natural Science Foundation of China under Grant No 11435005, and the K. C. Wong Magna Fund in Ningbo University.
**Corresponding author. Email: lousenyue@nbu.edu.cn Citation Text: Li H and Lou S Y 2019 Chin. Phys. Lett. 36 050501    Abstract Three Alice–Bob Boussinesq (ABB) nonlocal systems with shifted parity ($\hat{P}_{\rm s}$), delayed time reversal ($\hat{T}_{\rm d}$) and $\hat{P}_{\rm s}\hat{T}_{\rm d}$ nonlocalities are investigated. The multi-soliton solutions of these models are systematically found from the $\hat{P}_{\rm s}$, $\hat{T}_{\rm d}$ and $\hat{P}_{\rm s}\hat{T}_{\rm d}$ symmetry reductions of a coupled local Boussinesq system. The result shows that for ABB equations with $\hat{P}_{\rm s}$ and/or $\hat{T}_{\rm d}$ nonlocality, an odd number of solitons is prohibited. The solitons of the $\hat{P}_{\rm s}$ nonlocal ABB and $\hat{T}_{\rm d}$ nonlocal ABB equations must be paired, while any number of solitons is allowed for the $\hat{P}_{\rm s}\hat{T}_{\rm d}$ nonlocal ABB system. $t$-breathers, $x$-breathers and rogue waves exist for all three types of nonlocal ABB system. In particular, different from classical local cases, the first-order rogue wave can have not only four leaves but also five and six leaves. DOI:10.1088/0256-307X/36/5/050501 PACS:05.45.Yv, 11.30.Er, 42.65.Tg © 2019 Chinese Physics Society Article Text In quantum mechanics, physical quantities are described by Hermitian operators such that the physical observable quantities possess real eigenvalues. In 1998, Bender and Boettcher found that a non-Hermitian Hamiltonian with parity and charge conjugate (PC) symmetry or equivalently parity and time reversal (PT) symmetry may also possess real eigenvalues if the potential of the Hamiltonian is time-independent.[1] This phenomenon has attracted a great deal of attention from scientists because it can be realized in optical systems with the help of gain or lost effects.[2] In 2013, a PT symmetric (or more exactly PC symmetric because nonlinear potentials are time-dependent) nonlinear nonlocal Schrd̈inger system was proposed by Ablowitz and Musslimani.[3] After that, the study of nonlocal systems (named Alice–Bob (AB) systems) became one of the hot topics in nonlinear science.[4–19] In this Letter, using the obtained idea[4,5] we investigate a special Alice–Bob Boussinesq (ABB) system $$\begin{align} &A_{\rm tt}-\Big[A_{xx}+\delta A+\frac94A^2+\frac32AB-\frac34B^2\Big]_{xx}=0,\\ &B=\hat{f}A\equiv A^{\hat{f}},~\hat{f}\in {\cal G}\equiv\{1,\hat{P_{\rm s}},\hat{ T_{\rm d}},\hat{P_{\rm s}}\hat{T_{\rm d}}\},\\ &A^{\hat{P_{\rm s}}}=A(-x+x_0,t),~ A^{\hat{T_{\rm d}}}=A(x,-t+t_0),~~ \tag {1} \end{align} $$ with arbitrary constants $x_0$ and $t_0$. For $\hat{f}=1$, Eq. (1) is just the usual local Boussinesq equation which is widely used in physics, especially in fluid dynamics. For $\hat{f}\neq 1$, Eq. (1) expresses three types of nonlocal ABB systems with three different nonlocalities, the parity nonlocal ABB (PNABB) system ($B=A^{\hat{P}_{\rm s}}$), the time-reversal nonlocal ABB (TNABB) system ($B=A^{\hat{T}_{\rm d}}$) and the parity and time-reversal nonlocal ABB (PTNABB) ($B=A^{\hat{P}_{\rm s}\hat{T}_{\rm d}}$) system. The ABB system in Eq. (1) can be derived by applying the consistent correlated bang approach to the usual Bussinesq equation[4,6] $$ u_{\rm tt}-(3u^2+\delta u+u_{xx})_{xx}=0.~~ \tag {2} $$ It is known that if $\hat{g}$ is a second-order operator with $\hat{g}^2=1$, then an arbitrary function $A$ can be separated into an invariant part and an antisymmetric part in the following way $$ A=u+v,~u=\frac{A+A^{\hat{g}}}{2},~v=\frac{A-A^{\hat{g}}}{2}.~~ \tag {3} $$ Substituting Eq. (3) into Eq. (1) and separating the symmetric and antisymmetric parts, we have $$\begin{align} u_{\rm tt}-(u_{xx}+\delta u+3u^2)_{xx}=\,&0,~~ \tag {4} \end{align} $$ $$\begin{align} v_{\rm tt}-(v_{xx}+\delta v+6uv)_{xx}=\,&0,~~ \tag {5} \end{align} $$ under the symmetric and antisymmetric conditions $$ A=u+v,~\hat{f}u=u,~\hat{f}v=-v,~\hat{f}\in \{\hat{P_{\rm s}}, \hat{T_{\rm d}}, \hat{P_{\rm s}}\hat{T_{\rm d}}\}.~~ \tag {6} $$ In other words, the nonlocal system (1) is just a (discrete) symmetry reduction of the enlarged system (4) and (5). The integrable system (4) and (5) is just one of the simplest integrable couplings, named as the $Z_2$-Boussinesq equation. Thus solving the ABB in Eq. (1) is equivalent to solving the integrable coupling (4) and (5) with the parity and time reversal conditions (6). Because of the integrability of the $Z_n$-Boussinesq system for arbitrary $n$,[20] it is straightforward to write the Lax pair of the nonlocal ABB in Eq. (1) as $$\begin{alignat}{1} \!\!\!\!\!\!\psi_{xxx}=\,&\left(\begin{matrix} M & 0\\ Z & M \end{matrix}\right) \psi,~~ \tag {7} \end{alignat} $$ $$\begin{alignat}{1} \!\!\!\!\!\!\psi_t=\,&\sqrt{-3}\psi_{xx}+\frac{\sqrt{-3}}2\left(\begin{matrix} A+A^{\hat{f}} & 0\\ A-A^{\hat{f}} & A+A^{\hat{f}} \end{matrix}\right)\psi,~~ \tag {8} \end{alignat} $$ with $M\equiv -\frac14[3(A+A^{\hat{f}})+\delta]\partial_x+\frac12\int (A+A^{\hat{f}})_t\rm dx+\lambda_1-\frac38(A+A^{\hat{f}})_x$, $Z\equiv -\frac{3(A-A^{\hat{f}})}4\partial_x+\frac12\int (A-A^{\hat{f}})_t\rm dx+\lambda_2-\frac38(A-A^{\hat{f}})_x$ and $\lambda_1$ and $\lambda_2$ being arbitrary constants. For the Boussinesq Eq. (4), it is well known that the multiple soliton solution possesses the form[4,21] $$\begin{align} u=\,&2(\ln F)_{xx},~~ \tag {9} \end{align} $$ $$\begin{align} F=\,&\sum_{\nu}\exp\Big(\sum_{j=1}^N\nu_j\xi_j+\sum_{i\leq j}^N\nu_j\nu_i\theta_{ij}\Big),~~ \tag {10} \end{align} $$ where the summation of $\nu$ should be carried out for all permutations of $\nu_i=0, 1$, $i=1, 2, \ldots, N$ and $$\begin{alignat}{1} \!\!\!\!\!\!&\xi_j=k_jx-\omega_jt+\xi_{0j}, ~~~\omega_i^2=k_i^4+\delta k_i^2,\\ \!\!\!\!\!\!&\exp(\theta_{ij})=\frac{k_ik_j(2k_i^2+2k_j^2-3k_ik_j+\delta)-\omega_i\omega_j} {k_ik_j(2k_i^2+2k_j^2+3k_ik_j+\delta)-\omega_i\omega_j},~~ \tag {11} \end{alignat} $$ with arbitrary constants $k_i$ and $\xi_{i0}$ for all $i$. From the expressions (9) and (10) it is quite difficult to find its $\hat{f}(\hat{f}\in \{\hat{P}_{\rm s},\hat{T}_{\rm d},\hat{P}_{\rm s}\hat{T}_{\rm d}\})$ invariant form. Fortunately, from the obtained results[4] we know that if we rewrite Eq. (11) as $$\begin{align} \xi_j=\,&\eta_j-\frac{1}{2}\sum_{i=1}^{j-1}\theta_{ij} -\frac12\sum_{i=j+1}^N\theta_{ij},~~ \tag {12} \end{align} $$ $$\begin{align} \eta_j=\,&k_j\Big(x-\frac{1}{2}x_{0j}\Big)-\omega_j\Big(t-\frac{1}{2}t_{0j}\Big),~~ \tag {13} \end{align} $$ with arbitrary constants $x_{0j}$ and $t_{0j}$, then the $N$-soliton solution of the Boussinesq Eq. (2) can be rewritten as[4] $$ u=2(\ln F_{N})_{xx}=2\Big[\ln\sum_{\nu} K_\nu\cosh\Big(\frac12\sum_{j=1}^N\nu_j\eta_j\Big)\Big]_{xx},~~ \tag {14} $$ where the summation of $\nu=\{\nu_1,\nu_2,\ldots,\nu_N\}$ should be carried out for all non-dual permutations of $\nu_i=1,-1,i=1,2,\ldots,N$ ($\nu$ and $\nu'$ are dual if $\nu=-\nu'$), and $$\begin{align} K_\nu=\,&\prod_{i>j}\sqrt{k_ik_j(2k_i^2+2k_j^2-3\nu_i\nu_jk_ik_j+\delta) -\omega_i\omega_j}\\ &\equiv \prod_{i>j}a_{ij}^{\nu_i\nu_j},a_{ij}^{\nu_i\nu_j}\equiv a_{ij}^{\pm} \rm if\ \nu_i\nu_j=\pm1.~~ \tag {15} \end{align} $$ From the expressions (13) and (14), it is straightforward to see that for $\hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d}$, the $\hat{f}$-invariant condition $\hat{f}u=u$ requires $$ x_{0j}=x_0,~~t_{0j}=t_{0},~~j=1,2,\ldots,N,~~ \tag {16} $$ with arbitrary positive integer $N$. From the constraint condition (16), we know that for the PTNABB system (Eq. (1) with $\hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d}$), arbitrary numbers of solitons are allowed. However, all the soliton interaction positions are fixed at $\{x=\frac{x_0}2,t=\frac{t_0}2\}$ because of the loss of the $x$ and $t$ translation invariants. Similar to the analyses for another ABB system[8] and the ABKP system,[9] there may be some constraints on the multi-soliton solution (14) for the other two nonlocal ABB systems, the PNABB system (Eq. (1) with $\hat{f}=\hat{P}_{\rm s}$) and the TNABB system (Eq. (1) with $\hat{f}=\hat{T} _{\rm d}$). For $N=1$, Eq. (14) possesses the form $$\begin{alignat}{1} u=\,&2(\ln F_1)_{xx},~F_1=\cosh\Big(\frac{\eta_1}2\Big),\\ \eta_1=\,&k_1\Big(x-\frac{x_{01}}2\Big)\pm \sqrt{k_1^4+\delta k_1^2}\Big(t-\frac{t_{01}}2\Big).~~ \tag {17} \end{alignat} $$ It is clear that it is impossible for the single soliton solution (17) to select the arbitrary constants $k_1$, $x_{01}$ and $t_{01}$ such that the $\hat{f}$ invariant condition $\hat{f}u=u$ is satisfied for $\hat{f}=\hat{P}_{\rm s}$ or $\hat{f}=\hat{T}_{\rm d}$. In fact, for any nontrivial traveling wave solution $u=u(kx+\omega t+\theta_0)$ is not $\hat{P}_{\rm s}$ or $\hat{T}_{\rm d}$-invariant. Furthermore, one can obtain that for any odd $N=2n-1$, the solution (14) is not $\hat{f}$-invariant for $\hat{f}\in \{\hat{P}_{\rm s},\hat{T}_{\rm d}\}$. In other words, an odd number of solitons for the PNABB and TNABB systems is prohibited. For $N=2$, solution (14) becomes $$\begin{alignat}{1} u=\,&2(\ln F_{2})_{xx},\\ F_2=\,&a_{12}^-\cosh\Big(\frac{\eta_1+\eta_2}2\Big)+a_{12}^+ \cosh\Big(\frac{\eta_1-\eta_2}2\Big),\\ a_{12}^{\pm}=\,&\sqrt{k_1k_2(2k_1^2+2k_2^2\pm 3k_1k_2+\delta)-\omega_1\omega_2}.~~ \tag {18} \end{alignat} $$ From expression (18), it is obvious that if we select $$ \eta_2=\pm \eta_1^{\hat{f}},~~ \tag {19} $$ then the field $u$ will be naturally $\hat{f}$-invariant. Fortunately, the condition (19) can be readily satisfied by fixing the parameters $k_2$, $\omega_2$, $x_{0i}$ and $t_{0i}$, that is, for $\hat{f}=\hat{P}_{\rm s}$, $$ x_{0i}=x_0,~~t_{02}=t_{01},~~k_2=\mp k_1,~~\omega_2=\pm \omega_1;~~ \tag {20} $$ and for $\hat{f}=\hat{T}_{\rm d}$, $$ t_{0i}=t_0,~~x_{02}=x_{01},~~k_2=\pm k_1,~~\omega_2=\mp \omega_1.~~ \tag {21} $$ Thus we have two soliton solutions $$\begin{alignat}{1} F_2=\,&\sqrt{4k_1^2+\delta}\cosh\Big[\omega_1 \Big(t-\frac{t_{01}}2\Big)\Big]\\ &+\sqrt{k_1^2+\delta}\cosh\Big[k_1\Big(x-\frac{x_{0}}2\Big)\Big], ~\hat{f}=\hat{P}_{\rm s},~~ \tag {22} \end{alignat} $$ with arbitrary constant $t_{01}$ for the PNABB system and $$\begin{alignat}{1} \!\!\!\!\!\!F_2=\,&\sqrt{4k_1^2+\delta}\cosh\Big[\omega_1 \Big(t-\frac{t_{0}}2\Big)\Big]\\ \!\!\!\!\!\!&+\sqrt{k_1^2+\delta}\cosh\Big[k_1 \Big(x-\frac{x_{01}}2\Big)\Big],~~\hat{f}=\hat{P}_{\rm s},~~ \tag {23} \end{alignat} $$ with arbitrary constant $x_{01}$ for the TNABB system. For $N=4$, $F_4$ in Eq. (14) becomes $$\begin{align} F_4=\,&K_{\{2,4\}}\cosh(\xi_1^+-\xi_2^+)+K_{\{2,3\}}\cosh(\xi_1^--\xi_2^-)\\ &+K_{\{3\}}\cosh(\xi_1^-+\xi_2^+)+K_{\{2\}}\cosh(\xi_1^+-\xi_2^-)\\ &+K_{\{1\}}\cosh(-\xi_1^-+\xi_2^+)+K_{\{3,4\}}\cosh(\xi_1^-+\xi_2^-)\\ &+K_{\{\}}\cosh(\xi_1^++\xi_2^+)+K_{\{4\}}\cosh(\xi_1^++\xi_2^-),~~ \tag {24} \end{align} $$ $$\begin{align} \xi_1^{\pm}=\,&\frac12(\eta_1\pm \eta_3),~~~\xi_2^{\pm}=\frac12(\eta_2\pm \eta_4),~~ \tag {25} \end{align} $$ where the notation $K_{\{i_1,\ldots,i_j\}}$ is a simplified one for $K_{\{\nu\}}$ with the permutation $\{\nu\}\equiv \{\nu_i=-1,i=i_1,\ldots,i_j; \nu_i=1,i\neq i_1,\ldots,i_j\}$. For instance, for $n=4$, $K_{\{\nu\}}$ in Eq. (24) possesses the forms $K_{\{\}}=K_{\{1,1,1,1\}}$, $K_{\{4\}}=K_{\{1,1,1,-1\}}$, $K_{\{2,4\}}=K_{\{1,-1,1,-1\}}$ and so on. From Eq. (25) we know that if we select $$ \eta_{3}=\eta_1^{\hat{f}},~~\eta_{4}=\eta_2^{\hat{f}},~~ \tag {26} $$ then $\xi_i^{\pm}$ will be all symmetric or antisymmetric with respect to the operator $\hat{f}$. To realize that the pair condition (26) is quite trivial as in the $N=2$ case, we fix the constants as $$\begin{align} k_3=\,&-k_1,~~k_4=-k_2,~~\omega_3=\omega_1,~~\omega_4=\omega_2, \\ x_{01}=\,&x_{02}=x_{03}=x_{04}=x_0,~~t_{03}=t_{01},~~t_{04}=t_{02},~~ \tag {27} \end{align} $$ for the PNABB system and $$\begin{align} k_3=\,&k_1,~~k_4=k_2,~~\omega_3=-\omega_1,~~\omega_4=-\omega_2,\\ t_{01}=\,&t_{02}=t_{03}=t_{04}=t_0,~~x_{03}=x_{01},~~x_{04}=x_{02},~~ \tag {28} \end{align} $$ for the TNABB system. For the parameter selections (27) and/or (28), we have $$ a_{14}^{\pm}=a_{23}^{\pm},~~a_{34}^{\pm}=a_{12}^{\pm}.~~ \tag {29} $$ After considering the relations (29), one can directly check $$ K_{\{4\}}=K_{\{2\}},~~K_{\{3\}}=K_{\{1\}}.~~ \tag {30} $$ Using the relation (30) or equivalently (29), Eq. (24) becomes $$\begin{alignat}{1} F_4=\,&K_{\{\}}\cosh(\xi_1^++\xi_2^+) +K_{\{34\}}\cosh(\xi_1^-+\xi_2^-)\\ &+K_{\{24\}}\cosh(\xi_1^+-\xi_2^+)+K_{\{23\}}\cosh(\xi_1^--\xi_2^-)\\ &+K_{\{4\}}[\cosh(\xi_1^++\xi_2^-)+\cosh(\xi_1^+-\xi_2^-)]\\ &+K_{\{3\}}[\cosh(\xi_2^++\xi_1^-)+\cosh(\xi_2^+-\xi_1^-)].~~ \tag {31} \end{alignat} $$ Now we are ready to write down the invariant conditions for the $N=2n$ soliton solutions (14) for the shifted parity nonlocal and delayed time reversal nonlocal ABB Eq. (1). Taking the constant pair conditions $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!\!\!\!k_{n+i}=-k_{i},\omega_{n+i}=\omega_{i},~x_{0i}=x_0,~i=1,2,\ldots,n,~~ \tag {32} \end{alignat} $$ with arbitrary constants $k_i$ and $t_{0i}=t_{0(n+i)}$ for the PNABB system and $$ k_{n+i}=k_i,~~\omega_{n+i}=-\omega_i,~~t_{0i}=t_0,~~i=1,2,\ldots,n,~~ \tag {33} $$ with arbitrary constants $k_i$ and $x_{0i}=x_{0(n+i)}$ for the TNABB equation, one can directly verify that the following relations $$\begin{alignat}{1} \eta_{n+i}=\,&\eta_i^{\hat{f}},~~ \tag {34} \end{alignat} $$ $$\begin{alignat}{1} a_{i(n+j)}^{\nu_i\nu_{n+j}}=\,&a_{j(n+i)}^{\nu_j\nu_{n+i}},~~ a_{(n+i)(n+j)}^{\nu_{n+i}\nu_{n+j}}=a_{ij}^{\nu_{i}\nu_{j}}~~ \tag {35} \end{alignat} $$ are correct for both $\hat{f}=\hat{P}_{\rm s}$ and $\hat{f}=\hat{T}_{\rm d}$. Thus thanks to the paired conditions (32) and/or (33), the $N=2n$ solution (14) becomes an $\hat{f}$-invariant solution. To solve $\hat{f}$-antisymmetric $v$ Eq. (5), we can apply the symmetry knowledge on the usual Boussinesq Eq. (4). It is known that there are infinitely many symmetries for the integrable Boussinesq Eq. (4). Two of the simplest symmetry solutions are $u_t$ and $u_x$ related to the space-time translation invariance. Thus we have a special $\hat{f}$-antisymmetric $v$-solution $$ v=\left\{\begin{matrix} a u_x+bu_t,& \hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d},\\ a u_x,& \hat{f}=\hat{P}_{\rm s},\\ bu_t,& \hat{f}=\hat{T}_{\rm d}, \end{matrix}\right.~~ \tag {36} $$ with arbitrary constants $a$ and $b$. Finally, the multiple soliton solutions for ABB can be written as $$ A=\left\{\begin{matrix} (1+a \partial_x+b\partial_t)u,& \hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d},\\ (1+a \partial_x)u,& \hat{f}=\hat{P}_{\rm s},\\ (1+b\partial_t)u,& \hat{f}=\hat{T}_{\rm d}, \end{matrix}\right.~~ \tag {37} $$ with $u$ given by (14) with (13) and (15) under the conditions (16), (32) and (33) for $\hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d}$, $\hat{f}=\hat{P}_{\rm s}$ and $\hat{f}=\hat{T}_{\rm d}$, respectively.
cpl-36-5-050501-fig1.png
Fig. 1. Two-soliton interaction of the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$.
Figure 1 is a density plot of the two-soliton interaction solution described by Eq. (37) for the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$ under the parameter selections ($t_{0j}=x_{0j}=x_0=t_0=\delta=0$ for Figs. 1–4) $$ N=2,~~ k_1=-k_2=a=1.~~ \tag {38} $$ Figure 2 is a density plot of the $t$-breather described by Eq. (37) for the PNABB (1) with $\hat{f}=\hat{P}_{\rm s}$ under the parameter selections $$ N=2,~~k_1=-k_2=0.5,~~d=-2,~~a=1.~~ \tag {39} $$
cpl-36-5-050501-fig2.png
Fig. 2. The $t$-breather of the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$.
cpl-36-5-050501-fig3.png
Fig. 3. The $x$-breather of the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$.
Figure 3 is a density plot of the $x$-breather described by Eq. (37) for the PNABB (1) with $\hat{f}=\hat{P}_{\rm s}$ under the parameter selections $$ N=2,~~k_1=-k_2=\sqrt{-1},~~d=-2,~~a=1.~~ \tag {40} $$
cpl-36-5-050501-fig4.png
Fig. 4. Four-soliton interaction solution of the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$.
Figure 4 is a density plot of the four-soliton interaction solution described by Eq. (37) for the PNABB Eq. (1) with $\hat{f}=\hat{P}_{\rm s}$ under the parameter selections $$ N=4,~~k_1=-k_3=3,~~k_2=-k_4=-d=2,~~a=0.1.~~ \tag {41} $$ From the $t$-breather (Fig. 2) and $x$-breather (Fig. 3), we know that there are rogue wave solutions which correspond to the limit cases of the breathers with infinite periods. It is quite simple to verify that the single rogue wave solution of the nonlocal ABB Eq. (1) possesses the form (37) with $$ u=2\partial_x^2\ln\Big[-\frac3{\delta}+x'^2-\delta t'^2\Big],~~x'=x-x_1,~~t'=t-t_1~~ \tag {42} $$ where the constants $x_1$ and $t_1$ should be fixed as $$\begin{align} x_1=\,&\frac{x_0}{2},~t_1=\frac{t_0}{2},~{\rm for}~\hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d},\\ x_1=\,&\frac{x_0}{2},~~{\rm for}~\hat{f}=\hat{P}_{\rm s},\\ t_1=\,&\frac{t_0}{2},~~{\rm for}~\hat{f}=\hat{T}_{\rm d}. \end{align} $$ From Eq. (42) we know that the algebraic solution (37) with (42) exists only for $\delta\neq0$ and is analytical for $\delta < 0$. Figure 5 displays the structures of the rogue wave for the PNABB equation. Figure 5(a) shows the structure of the four-leaf rogue wave described by (37) with (42) and the parameter selections $\delta=-3$ and $a=x_0=t_1=0$. Figure 5(b) shows the structure of the six-leaf rogue wave under the parameter selections $x_0=t_1=0$, $\delta=-3$ and $a=30$. In fact, the first-order rogue waves of the nonlocal ABB systems can be changed from four leaves to five leaves and six leaves as the parameter $a$ increases. However, in local systems, first-order rogue waves can only have four leaves.
cpl-36-5-050501-fig5.png
Fig. 5. Rogue wave solution of the PNABB system (1) with $\hat{f}=\hat{P}_{\rm s}$. (a) Four-leaf rogue wave and (b) six-leaf rogue wave.
In fact, multiple breather solutions can be read off from the multiple soliton solution expression (37) by appropriately selecting parameters, and the multiple higher order rogue waves can be obtained from the limit cases of the multiple breathers. The final form of the multiple rogue waves of the nonlocal ABB also possesses the form (37) with $$ u=2(\ln G_n)_{xx},~~G_n=\sum_{m=0}^{n(n+1)/2}\sum_{j=0}^ma_{jm}x'^{2j}t_1^{2m-2j}, $$ with suitable constants $a_{jm}$. For instance, for the second rogue wave (for $\delta < 0$), we have $$\begin{align} G_2=\,&\delta^3(\delta t'^2-x'^2)^3+\delta^2 (17\delta t'^2-5 x'^2)(\delta t'^2-5 x'^2)\\ &+25\delta (5 x'^2 +19\delta t'^2)+2\alpha\delta x'(3\delta^2 t'^2+\delta x'^2+1)\\ &+2 t'\delta\beta (\delta^2 t'^2+3\delta x'^2-5)-\alpha^2\delta+\beta^2+1875, \end{align} $$ where $x'$ and $t'$ are defined the same as in Eq. (42), and $\alpha$ and $\beta$ are arbitrary constants. In summary, three special ABB systems with different nonlocalities $\hat{f}=\hat{P}_{\rm s}\hat{T}_{\rm d}$, $\hat{P}_{\rm s}$ and $\hat{T}_{\rm d}$ are investigated. Some types of multi-soliton solutions including paired solitons, $x$- and $t$-breathers, and rogue waves are obtained. The soliton structures of the ABB systems are quite different from those of the usual Boussinesq equation. In particular, the odd numbers of solitons are prohibited for the PNABB and TNABB system. Quite different from the local cases, first-order rogue waves of the ABB systems may have four leaves, five leaves or six leaves depending on the different selections of the antisymmetric parameter $a$. References Real Spectra in Non-Hermitian Hamiltonians Having P T SymmetryNonlinear waves in PT -symmetric systemsIntegrable Nonlocal Nonlinear Schrödinger EquationAlice-Bob systems, P^-T^-Ĉ symmetry invariant and symmetry breaking soliton solutionsAlice-Bob Physics: Coherent Solutions of Nonlocal KdV SystemsFrom Nothing to Something II: Nonlinear Systems via Consistent Correlated BangExact PT invariant and PT symmetric breaking solutions, symmetry reductions and Bäcklund transformations for an AB–KdV systemProhibitions caused by nonlocality for nonlocal Boussinesq‐KdV type systemsMulti-place nonlocal systemsContinuation of periodic orbits in the Sun-Mercury elliptic restricted three-body problemComplex solitary waves and soliton trains in KdV and mKdV equationsInverse scattering transform for the integrable nonlocal nonlinear Schrödinger equationSoliton solutions of an integrable nonlocal modified Korteweg–de Vries equation through inverse scattering transformIntegrable discrete P T symmetric modelDavey-Stewartson type equations in 4+2 and 3+1 possessing soliton solutionsIntegrable multidimensional versions of the nonlocal nonlinear Schrödinger equationReal spectra in non-Hermitian Hamiltonians having PT symmetryRogue waves of the nonlocal Davey–Stewartson I equationDynamics of high-order solitons in the nonlocal nonlinear Schrödinger equationsIntegrable KP Coupling and Its Exact SolutionExact Solution of the Korteweg—de Vries Equation for Multiple Collisions of Solitons
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