Since Ablowitz and Musslimani[1] introduced an integrable nonlocal nonlinear Schr$\ddot{o}$dinger equation
$A_t+A_{xx}\pm A^2 B=0,\quad B=\hat{f}A=\hat{P}\hat{C} A=A^*(-x,t),$
where the operators $\hat{P}$ and $\hat{C}$ are the usual parity and charge conjugation, the study of the nonlocal system becomes one of the hot topics in nonlinear science.[2-12]
It is well known that there are various correlated and/or entangled events that may be happened in different times and places. To describe two-place physical problems, Alice-Bob (AB) systems[10] are proposed by using the shifted parity ($\hat{P}_{s}$), delayed time reversal ($\hat{T}_{d}$) and charge conjugate ($\hat{C}$) symmetries.
In addition to the nonlocal nonlinear Schr$\ddot{o}$dinger equation (1), there are many other types of two-place nonlocal models, such as the nonlocal KdV systems,[12] the nonlocal modified KdV systems,[4-5,13] the discrete nonlocal NLS systems,[6] the coupled nonlocal NLS systems[2] and the nonlocal Davey-Stewartson systems,[7-9] etc. Especially, in Ref. [10], one of us (Lou) proposed a series of integrable AB systems including the ABKdV systems, ABmKdV systems,[13] ABKP systems, AB-sine Gordon (ABsG) systems, ABNLS systems, AB-Toda (ABT) systems, and ABAKNS systems.
In addition, Lou established a most general ABKdV equation and presented its $\hat{P}^x_{s}$ and $ \hat{T}_{d}$ invariant Painlev$\acute{e}$ II reduction and soliton-cnoidal periodic wave interaction solutions for the ABKdV system.[11]
$\Bigl[A_t+A_{xxx}+\frac32A(3A_x+B_x)+\frac32B(A_x-B_x)\Bigr]_x \!+3\sigma^2A_{yy}=0, \\ B=\hat{P_s^x}\hat{ T_d}A\equiv A^{\hat{P_s^x}\hat{ T_d}}=A(-x+x_0, y, -t+t_0),$
with arbitrary constants $x_0, t_0$, and $y_0$.
The ABKP system Eq. (2) can be derived by applying the consistent correlated bang (CCB) approach to the usual KP equation
$(u_t+6uu_x+u_{xxx})_x+3\sigma^2u_{yy}=0,$
as follows in Ref. [11].
It is known that if $\hat{g}$ is a second order operator, then an arbitrary function $A$ can be separated to an invariant part and an antisymmetry part in the following way
$A=u+v,\quad u=\frac{A\!+\!A^{\hat{g}}}{2},\quad v=\frac{A\!-\!A^{\hat{g}}}{2},\quad A^{\hat{g}}\equiv \hat{g}A.$
To solve the ABKP system (2), we can take $\hat{g}=\hat{P_s^x}\hat{ T_d}$ in Eq. (3). Thus, the ABKP equation (2) becomes
$(u_t+6uu_x+u_{xxx})_x+3\sigma^2u_{yy}=0,$
$(v_t+6vu_x+6uv_x+v_{xxx})_x+3\sigma^2v_{yy}=0,$
with
$u=\frac{A+A^{\hat{P_s^x}\hat{ T_d}}}{2},\quad v=\frac{A-A^{\hat{P_s^x}\hat{ T_d}}}{2},$
and the trivial properties
$A=u+v,\quad \hat{P_s^x}\hat{ T_d}u=u,\quad \hat{P_s^x}\hat{T_d}v=-v.$
In other words, to solve the ABKP Eq. (2) is equivalent to solve the integrable coupling (4) and (5) with the parity conditions (6).
For the KP Eq. (4), it is well known that the multiple soliton solution possesses the form[14]
$u=2(\ln F)_{xx},$
$F=\sum_{\nu}\exp\Bigl(\sum_{j=1}^N\nu_j\xi_j+\sum_{i\leq j\leq}^N\nu_j\nu_i\theta_{ij}\Bigr),$
where the summation of $\nu$ should be done for all permutations of $\nu_i=0, 1$, $i=1, 2, \ldots, N$ and
$\xi_j = k_jx+l_jy-k_j^{-1}(k_j^4 +3\sigma^2l_j^2)t+\xi_{0j}, \\ exp(\theta_{ij}) =\frac{k_i^2k_j^2(k_j-k_i)^2 -\sigma^2(l_jk_i-l_ik_j)^2} {k_i^2k_j^2(k_j+k_l)^2-\sigma^2(l_jk_i-l_ik_j)^2},$
with arbitrary constants $k_i, l_i$, and $\xi_{i0}$ for all $i$.
From the expression (7) with (8) it is quite difficult to find its $\hat{P_s^x}\hat{T_d}$ invariant form. Fortunately, from the results of Ref. [10], we know that if we rewrite Eq. (9) as
$ \xi_j=\eta_j-\frac{1}{2}\sum_{i=1}^{j-1}\theta_{ij} -\frac12\sum_{i=j+1}^N\theta_{ij}, $
where
$ \eta_j = k_j\Bigl(x-\frac{1}{2}x_0\Bigr) +l_j\Bigl(y-\frac{1}{2}y_{0j}\Bigr) -k_j^{-1}(k_j^4+3\sigma^2l_j^2) \\ \times\Bigl(t-\frac{1}{2}t_0\Bigr)+\eta_{0j}, $
with arbitrary constants $y_{0j}$ and $\eta_{0j}$, then the N-soliton solution of the KP Eq. (4) can be rewritten as[10]
$u=2\Bigl[\ln\sum_{\nu} K_\nu\cosh\Bigl(\frac12\sum_{j=1}^N\nu_j\eta_j\Bigr)\Bigr]_{xx},$
where the summation of $\nu=\{\nu_1, \nu_2, \ldots, \nu_N\}$ should be done for all non-dual permutations of $\nu_i=1, -1, i=1, 2, \ldots, N$ ($\nu$ and $\nu'$ are dual if $\nu=-\nu'$), and
$K_\nu=\prod_{i>j}\sqrt{k_i^2k_j^2(k_i-\nu_i\nu_jk_j)^2-\sigma^2(l_ik_j-k_il_j)^2}.$
Similar to the AB-Boussinesq case,[15] the odd numbers of the multi-soliton solution are prohibited by the parity and time reversal condition (6).
For the even numbers of multi-soliton solutions, the paired conditions
$N=2n,\quad k_{n+i}=-k_i,\quad l_{n+i}=l_i,\quad y_{0(n+i)}=y_{0i}$
must be satisfied such that $\hat{P}^x_s\hat{T}_d \eta_{n+i}=-\eta_i$ and $\hat{P}^x_s\hat{T}_d F_{2n}=F_{2n}$ are satisfied.
For $\{n=1, N=2\}$ and $\{n=2, N=4\}$, we have (after rule out common constants for simplicity)
$F_2 = \cosh[k_1x-(k_1^3 +3\sigma^2l_1^2k_1^{-1})t] +\sqrt{1-\sigma^2k_1^4l_1^{-2}} \\ \times\cosh[l_1(y-y_{01})], \\ F_4 =\alpha_1\alpha_2[\delta_-\cosh(Y_+) +\delta_+\cosh(Y_-)] +\sigma^2l_1l_2 \\ \times[\beta_+\cosh(X_-) +\beta_-\cosh(X_+)] -\sqrt{-\sigma^2\beta_+\beta_-} \\ \times[\alpha_1l_2(\gamma_1 +\gamma_1^{\hat{P}^x_s\hat{T}_d})+\alpha_2l_1(\gamma_2 +\gamma_2^{\hat{P}^x_s\hat{T}_d}) ],$
where the variables $X_{\pm}, Y_{\pm}, \gamma_1$, and $\gamma_2$ are defined by
$ Y_{\pm}=l_1\Bigl(y-\frac{y_{01}}2\Bigr)\pm l_2\Bigl(y-\frac{y_{02}}2\Bigr), \\ X_{\pm}=(k_2\pm k_1)(x-\frac{x_0}2)-[k_2^3\pm k_1^3+3\sigma^2 \\ \times(l_2^2k_2^{-1}\pm l_1^2k_1^{-1})]\Bigl(t-\frac{t_0}2\Bigr), \\ \gamma_1=\cosh\Bigl[l_1\Bigl(y-\frac{y_{01}}2\Bigr) -k_2\Bigl(x-\frac{x_{0}}2\Bigr) +(k_2^3 \\ +3\sigma^2l_2^2k_2^{-1}) \Bigl(t-\frac{t_{0}}2\Bigr)\Bigr], \\ \gamma_2=\cosh\Bigl[l_2\Bigl(y-\frac{y_{02}}2\Bigr) -k_1\Bigl(x-\frac{x_{0}}2\Bigr) +(k_1^3 \\ +3\sigma^2l_1^2k_1^{-1}) \Bigl(t-\frac{t_{0}}2\Bigr)\Bigr], $
and the constants $\alpha_1, \alpha_2, \beta_{\pm}$, and $\delta_{\pm}$ are related to arbitrary constants $k_1$, $k_2$, $l_1$, and $l_2$ by
$ \alpha_i=\sqrt{k_i^4-3\sigma^2l_i^2},\quad i=1, 2, \\ \beta_{\pm}=k_1^4k_2^4(k_1\pm k_2)^4+(k_1^2l_2^2-k_2^2l_1^2)^2-2\sigma^2k_1^2k_2^2 \\ \times(k_1\pm k_2)^2(k_1^2l_2^2+k_2^2l_1^2), \\ \delta_{\pm}=2\sigma^2k_1^2k_2^2[k_1^2k_2^2(l_1\pm l_2)^2+(k_1^2l_2\pm k_2^2l_1)^2]-k_1^4k_2^4 \\ \times(k_1^2- k_2^2)^2-(k_1^2l_2^2-k_2^2l_1^2)^2. $
Whence the $u$ field Eq. (4) is solved. The $v$ field equation can be solved via known symmetries of the KP equation after considering the antisymmetric condition (6). It is clear that for any given solution $u$, say, the multi-soliton solution (10), there exist infinitely many solution $v$. Here we write down a special one,
$v=\Bigl[h\partial_x+g\partial_y+f\partial_t -\frac{\sigma^2}6y\dot{g}\partial_x +\Bigl(\frac{x}3\dot{f} -\frac{\sigma^2}{18}y^2\ddot{f}\Bigr)\partial_x \\ +\frac23\dot{f}\partial_yy\Bigr]u -\frac16\dot{h} +\frac{\sigma^2y}{36}\ddot{g} -\frac{x}{18}\ddot{f} +\frac{\sigma^2y^2}{108}\dddot{f},$
with $f, g$, and $h$ being arbitrary functions of $t$. To satisfy the $\hat{P}_s^x\hat{T}_d$ antisymmetric condition of $v$, The functions $f, g$, and $h$ in Eq. (14) should satisfy
$\hat{T}_d\{f,g,h\}=\{f, -g, h\}.$
In other words, the functions $f$ and $h$ are arbitrary even functions and $g$ is an arbitrary odd function with respect to time $t$.
Finally, we get a multi-soliton solution of the ABKP equation (2),
$A=2\Bigl[1+h\partial_x+g\partial_y+f\partial_t -\frac{\sigma^2}6y\dot{g}\partial_x \\ +\Bigl(\frac{x}3\dot{f} -\frac{\sigma^2}{18} \times y^2\ddot{f}\Bigr)\partial_x +\frac23\dot{f}\partial_yy\Bigr](\ln F_{2n})_{xx} \\ -\frac16\dot{h} +\frac{\sigma^2y}{36}\ddot{g}-\frac{x}{18}\ddot{f} +\frac{\sigma^2y^2}{108}\dddot{f}, \\ F_{2n}=\sum_{\nu} K_\nu\cosh\Bigl(\frac12\sum_{j=1}^{2n}\nu_j\eta_j\Bigr),$
with the relations (11) and (12). When the arbitrary functions $f,\ g$ and $h$ are all taken as zeros, the solution (15) becomes $\hat{P}_s^x\hat{T}_d$ invariant. For any nonzero $f$, $g$ and $h$, the solution (15) is a symmetry $\hat{P}_s^x\hat{T}_d$ breaking one.
Figure 1 displays the paired two soliton solution (15) with the parameters $\{n=1,\ k_2=-k_1=1,\ l_2=l_1=2,\ x_0=t_0=y_{01}=0\}$ and function selections $\{f=g=0, \ h=2\}$ at time $t=0$ for the ABKPI ($\sigma^2=-1$) equation.
Fig.1 (Color online) Two soliton interaction solution for the ABKPI equation. |
Fig.2 (Color online) The $y$-breather plot for the ABKPI equation expressed by Eq. (15) with the parameters $\{n=1, k_2=-k_1=1, l_2=l_1=2\sqrt{-1}, x_0=t_0=y_{01}=0\}$ and function selections $\{f=g=0, h=2\}$ at time $t=0$. |
From the expression (13), we know that there exists a further prohibition for the ABKPII ($\sigma^2=1$), the paired soliton is valid only for $k_1^4<l_1^2$.
For the ABKPI system, from (13) we know also that both $k_1$ and $l_1$ can be pure imaginary. If $k_1$ is real and $l_1$ is imaginary and $k_1^4<|l_1|^2$, then the expression (13) related solution $A$ becomes an analytic $y$-breather (periodic solution in the $y$ direction) for the ABKPI equation. Figure~2 displays the $y$-breather structure expressed by Eq. (15) with the same parameter and function selections as those in Fig.1 except for $l_1=2\sqrt{-1}$.
If $k_1$ is imaginary and $l_1$ is real, then the expression (13) related solution $A$ becomes an analytic $x$-breather (periodic solution in the $x$ direction) for the ABKPI equation. Figure 3 shows the $x$-breather structure expressed by Eq. (15) with the same parameter and function selections as those in Fig. 1 except for $k_1=\sqrt{-1}$.
Fig.3 (Color online) The density plot of the $x$-breather for the ABKPI equation expressed by Eq. (15) with the parameters $\{n=1, k_2=-k_1=\sqrt{-1}, l_2=l_1=2, x_0=t_0=y_{01}=0\}$ and function selections $\{f=g=0, h=2\}$ at time $t=0$. |
From Figs. 2 and 3, we can conclude that whence the period of the $x$-breather and/or $y$-breather tends to infinity, the breathers will become a special lump solution of the ABKP equation (2). In fact it is quit trivial to check the ABKP system (2) possesses a lump-type solution (15) with
$F_{2n}\sim 1+c\Bigl[\Bigl(x-\frac{x_0}2\Bigr)-3c\Bigl(t-\frac{t_0}2\Bigr)\Bigr]^2-\sigma^2(y-y_0)^2,$
for arbitrary constants $c$ and $y_0$.
Figure 4 displays three different structures for the ABKPI equation under the parameter and function selections
$f=g=x_0=t_0=y_0=0,\ c=-\sigma^2=1,$
Fig.4 (Color online) The lump structures (15) for the ABKPI system (2).The parameter and function selections are given by Eq. (16) at time $t=0$ while the function $h$ is fixed as (a) $h=0$ for four leaf lump, (b) $h=2.2$ for five leaf lump and (c) $h=300$ for six leaf. |
while $h$ is taken as $h=0$, $h=2.2$, and $h=300$ for Figs. 4(a)-4(c) respectively.
Figure 5 exhibits the interaction behavior for the ABKPI equation expressed by Eq. (15) with the parameter selections $\{n=2, l_1=-l_3=1, l_2=-l_4=2, k_3=k_1=k_4=k_2=2, x_0=t_0=y_{01}=0, y_{02}=1\}$ and the function selections $\{f=g=0, h=1\}$ at time $t=0$.
Fig.5 The density plot of the four-soliton interaction solution of the ABKP equation (2) with $\sigma^2=-1$. |
In summary, in this paper, a special ABKP system with $P_s^xT_d$ nonlocality is investigated. Some types of multi-soliton solutions including paired solitons, $x$- and $y$-breathers and lumps are obtained. The soliton structures of the ABKP systems are quite different from those of the usual KP system. The more about the AB systems and the multi-place systems[16] should be further studied.