1. Introduction
Since the integrable nonlocal nonlinear Schrödinger (NLS) [1] equation
$\begin{eqnarray}\begin{array}{l}{{\rm{i}}{A}}_{t}+{A}_{{xx}}\pm {A}^{2}B=0,\\ \quad B=\hat{f}A=\hat{{ \mathcal P }}\hat{{ \mathcal C }}A={A}^{{\prime} }\left(-x,t\right),\end{array}\end{eqnarray}$
has been introduced by Ablowitz and Musslimani, where the operators $\hat{{ \mathcal P }}$ and $\hat{C}$ are the usual parity and charge conjugation, nonlocal systems have become one of the most popular fields in nonlinear science. There are a number of nonlocal integrable systems that have been established in different ways, such as the nonlocal Toda systems [2, 3], the nonlocal Korteweg-De-Vries (KdV) systems [2–5], the nonlocal modified KdV (MKdV) systems [2, 6, 7], the nonlocal Davey-Stewartson systems, the nonlocal Kadomtsev-Petviashvili (KP) systems [8–11], the nonlocal Sawada Kortera equations [12], the nonlocal sine-Gordon/sinh-Gordon equation with nonzero boundary conditions [13], the nonlinear Klein-Gordon systems [14] and so on.Symmetry is an important method to construct nonlocal integrable systems. In physics, there are some important symmetries, such as charge conjugation $\left({ \mathcal C }\right)$ symmetry, parity $\left({ \mathcal P }\right)$ symmetry, time reversal $\left({ \mathcal T }\right)$ symmetry, and their combinations. Some nonlocal integrable systems have been constructed by using the symmetries described above, such as the NNLS equation [1] with ${ \mathcal P }{ \mathcal C }$ symmetry, the coupled KdV system [2] with ${ \mathcal P }{ \mathcal T }$ symmetry, and so on.
In section 2 of this paper, the nonlocal sine-Gordon (SG) systems are constructed by using ${ \mathcal P }{ \mathcal T }$ symmetry, space-time exchange (Ex,t) symmetry, and their combination ${ \mathcal P }{ \mathcal T }{E}_{x,t}$. The nonlocal SG system is rewritten as a special integrable SG coupling consisting of the usual SG equation and its symmetry equation by using the symmetric-antisymmetric separation approach in section 2 . For the common local SG equation, two types of N-soliton solutions and three types of periodic solutions are presented in section 3 . Some exact solutions of the ABSG systems are proposed in sections 3 , 4 and 5 , which are the linear superposition of the solutions to the usual SG equation and its symmetry equation. The last section is a summary and some discussions.
2. Two-place nonlocal integrable sine-Gordon systems
In light cone coordinates, the sine-Gordon (SG) equation can be generally written as
$\begin{eqnarray}{u}_{{xt}}=\sin u,\end{eqnarray}$
where ${u}_{{xt}}\equiv \tfrac{{\partial }^{2}u}{\partial x\partial t}$.It is clear that the usual SG equation (2 ) possesses a discrete symmetry group that can be expressed by
$\begin{eqnarray}{ \mathcal G }\equiv \{I,{ \mathcal P }{ \mathcal T },{E}_{x,t},{ \mathcal P }{ \mathcal T }{E}_{x,t}\},\end{eqnarray}$
where the operators I, ${ \mathcal P }$, ${ \mathcal T }$ and Ex,t are the identity operator, parity, time reversal and space-time exchange symmetry defined by $\begin{eqnarray}\begin{array}{rcl}I\left\{x,t\right\} & = & \left\{x,t\right\},\quad { \mathcal P }x=-x,\\ { \mathcal T }t & = & -t,\quad {E}_{x,t}\{x,t\}=\{t,x\}.\end{array}\end{eqnarray}$
It is clear that the SG equation is ${ \mathcal G }$ invariant. Then, the consistent correlated bang method [15] can be used to construct the nonlocal SG systems. Introducing2 ), the usual SG equation can be written as
$\begin{eqnarray}\begin{array}{rcl}u & = & {u}_{1}-{u}_{2},{u}_{1}={{gu}}_{2},\\ {u}_{2} & = & {{gu}}_{1},{g}^{2}=I,g\ \in { \mathcal G }\end{array}\end{eqnarray}$
into the usual SG equation ( $\begin{eqnarray}\begin{array}{l}{u}_{1,{xt}}-{u}_{2,{xt}}-\sin ({u}_{1}-{u}_{2})\\ \quad =\,(I-g)\left[{u}_{1,{xt}}-\displaystyle \frac{1}{2}\sin ({u}_{1}-{u}_{2})\right]\\ \quad =\,{\left.(I-g)\left[{u}_{1,{xt}}-\displaystyle \frac{1}{2}\sin ({u}_{1}-{u}_{2})-G\right]\right|}_{(I-g)G=0}\\ \quad =\,0,\end{array}\end{eqnarray}$
where ${u}_{i,{xt}}\equiv \tfrac{{\partial }^{2}{u}_{i}}{\partial x\partial t},i=1,2$ and G is an arbitrary function with the condition $\begin{eqnarray}\left(I-g\right)G=G-{gG}=0.\end{eqnarray}$
It is clear that the nonlocal SG equation (6 ) can be solved by a two-place SG system
$\begin{eqnarray}{u}_{1,{xt}}=\displaystyle \frac{1}{2}\sin \left({u}_{1}-{u}_{2}\right)+G,\quad {u}_{2}={{gu}}_{1}.\end{eqnarray}$
Interestingly, there are many different integrable cases by properly choosing the function G, such as, $G=0,G=c\sin ({u}_{1}+{u}_{2}),G\,=\,c({u}_{1}+{u}_{2})\cos ({u}_{1}-{u}_{2})$ and so on, where c is an arbitrary real number. We choose a special case, $G=\tfrac{1}{2}\left({u}_{1}+{u}_{2}\right)\cos ({u}_{1}-{u}_{2})$ in this paper.When G is chosen as $G=\tfrac{1}{2}\left({u}_{1}+{u}_{2}\right)\cos ({u}_{1}-{u}_{2})$, the two-place SG system (8 ) can be rewritten as9 ) becomes a special equation u1,xt = u1, which has some simple, but interesting solutions, such as9 ) possesses the ${ \mathcal P }{ \mathcal T }$ nonlocality. When g = Ex,t, the nonlocal SG equation possesses the space-time exchange (Ex,t) nonlocality. When $g={ \mathcal P }{ \mathcal T }{E}_{x,t}$, the nonlocal SG equation possesses the combination nonlocality of ${ \mathcal P }{ \mathcal T }$ and Ex,t.
$\begin{eqnarray}\begin{array}{rcl}{u}_{1,{xt}} & = & \displaystyle \frac{1}{2}\sin ({u}_{1}-{{gu}}_{1})+\displaystyle \frac{1}{2}\left({u}_{1}+{{gu}}_{1}\right)\cos \left({u}_{1}-{{gu}}_{1}\right),\\ g\in { \mathcal G } & \equiv & \left\{I,{ \mathcal P }{ \mathcal T },{E}_{x,t},{ \mathcal P }{ \mathcal T }{E}_{x,t}\right\}.\end{array}\end{eqnarray}$
When g = I, the two-place nonlocal SG equation ( $\begin{eqnarray}{u}_{1}\left(x,t\right)=\left\{\begin{array}{l}{c}_{1}\cos \left(\displaystyle \frac{x}{\omega }-\omega t+\alpha \right)\\ {c}_{2}\sin \left(\displaystyle \frac{x}{\omega }-\omega t+\alpha \right)\\ {d}_{1}\cosh \left(\displaystyle \frac{x}{\omega }+\omega t+\alpha \right)\\ {d}_{2}\sinh \left(\displaystyle \frac{x}{\omega }+\omega t+\alpha \right)\end{array}\right.,\end{eqnarray}$
where c1, c2, d1, d2, α, ω are all arbitrary real numbers. When $g={ \mathcal P }{ \mathcal T }$, the nonlocal SG equation (In order to be more concise, the SG equation (9 ) can be rewritten by using the notation transformation (u1 → v),
$\begin{eqnarray}\begin{array}{rcl}{v}_{{xt}} & = & \displaystyle \frac{1}{2}\sin (v-{gv})+\displaystyle \frac{1}{2}\left(v+{gv}\right)\cos \left(v-{gv}\right),\\ g\in { \mathcal G } & \equiv & \left\{I,{ \mathcal P }{ \mathcal T },{E}_{x,t},{ \mathcal P }{ \mathcal T }{E}_{x,t}\right\}.\end{array}\end{eqnarray}$
In order to solve the nonlocal equation (11 ), the symmetric-antisymmetric separation approach [2] is the most useful and simplest method. Because ${g}^{2}=I,g\in { \mathcal G }$, the solution of the nonlocal SG equation can be separated into the combination of the g-symmetric part and the g-antisymmetric part.
Thus, introducing11 ) and separating the resulting equation into the g-symmetric part and the g-antisymmetric part, we have
$\begin{eqnarray}v=\displaystyle \frac{1}{2}\left(A+B\right),\quad {gA}=-A,\quad {gB}=B,\end{eqnarray}$
to the nonlocal SG equation ( $\begin{eqnarray}{A}_{{xt}}=\sin A,\quad {gA}=-A,\end{eqnarray}$
$\begin{eqnarray}{B}_{{xt}}=B\cos A,\quad {gB}=B.\end{eqnarray}$
It is clear that equations (13 ) and (14 ) are simply the usual SG equation and its symmetry equation. In other words, the approach used in (13 ) and (14 ) is simply a special integrable SG coupling. Obviously, A is a g-antisymmetric solution of the usual SG equation.
There are different solutions of the nonlocal SG equation by properly choosing the function B, such as B = 0, B = cAx, B = cAt and so on. If we choose B = 0, the nonlocal SG equation becomes a usual SG equation with the condition gA = −A. In this paper, we choose a special trivial symmetry B = cAx, which is related to the space translation invariance. If $A\left(x,t\right)$ and $B\left(x,t\right)={{cA}}_{x}$ are the solutions of equations (13 ) and (14 ), the condition gA = − A, gB = gAx = Ax must be satisfied, where $g\in \left\{{ \mathcal P }{ \mathcal T },{E}_{x,t},{ \mathcal P }{ \mathcal T }{E}_{x,t}\right\}$. Different choices of g reflect different physical properties of the solution of the SG equation. For example, ${ \mathcal P }{ \mathcal T }A=-A,{ \mathcal P }{ \mathcal T }B={ \mathcal P }{ \mathcal T }{{cA}}_{x}={A}_{x}$ means A is a ${ \mathcal P }{ \mathcal T }$--antisymmetric solution of SG equation, but the derivative of A with respect to x is ${ \mathcal P }{ \mathcal T }$-symmetric. So the selection of g should satisfy the physical properties of the solution of the SG equation.
So, the solutions of the usual SG equation (2 ) and the nonlocal SG equation (11 ) have the following relationship that
$\begin{eqnarray}v=\displaystyle \frac{A}{2}+\displaystyle \frac{{{cA}}_{x}}{2},\quad {gA}=-A,\quad {{gA}}_{x}={A}_{x},\end{eqnarray}$
where A is a g-antisymmetric solution of the usual SG equation.3. The exact solutions of nonlocal SG equation with parity and time reversal nonlocalities
When $g={ \mathcal P }{ \mathcal T }$, the SG equation (11 ) becomes a nonlocal SG equation with ${ \mathcal P }{ \mathcal T }$ nonlocality,
$\begin{eqnarray}\begin{array}{l}{v}_{\,,{xt}}=\displaystyle \frac{1}{2}\sin (v-{ \mathcal P }{ \mathcal T }v)\\ \quad +\,\displaystyle \frac{1}{2}\left(v+{ \mathcal P }{ \mathcal T }v\right)\cos \left(v-{ \mathcal P }{ \mathcal T }v\right).\end{array}\end{eqnarray}$
It is clear that there are many periodic solutions for the usual SG equation (2 ). Here, we list three types of periodic solutions of the usual SG equation (2 ),
$\begin{eqnarray}u=\pm 4\arctan \left[\sqrt{m}{\rm{sn}}\left({kx}-\displaystyle \frac{t}{k{\left(m+1\right)}^{2}}+\xi ,m\right)\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}u & = & \pm 4\arctan \left[\displaystyle \frac{1}{\sqrt[4]{1-{m}_{0}^{2}}}{\rm{dn}}\left(\Space{0ex}{4.2ex}{0ex}{k}_{0}x\right.\right.\\ & & \left.\left.+\displaystyle \frac{{\left(1-\sqrt{1-{m}_{0}^{2}}\right)}^{2}t}{{k}_{0}{m}_{0}^{4}}+{\xi }_{0},{m}_{0}\right)\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}u=\pm 4\arctan \left[\displaystyle \frac{{m}_{1}{\rm{sn}}\left(\tfrac{{k}_{1}x}{{A}^{* }+{m}_{1}}-\tfrac{{A}^{* }t}{{k}_{1}(1+{m}_{1}{A}^{* })}+{\xi }_{10},{m}_{1}\right)}{b\ {\rm{dn}}\left(\tfrac{{{bk}}_{1}x}{{A}^{* }+{m}_{1}}+\tfrac{{{bA}}^{* }t}{{k}_{1}(1+{m}_{1}{A}^{* })}+{\xi }_{20},{m}_{2}\right)}\right],\end{eqnarray}$
where ${A}^{* }=\sqrt{1-{m}_{2}^{2}}$, $b=\sqrt[4]{{m}_{1}^{2}/\left(1-{m}_{2}^{2}\right)}$, ${\rm{sn}}(\xi ,m)$ and ${\rm{dn}}(\xi ,m)$ are two types of Jacobi elliptic functions with the modulus m, {k, m, m1, m2, ξ0, ξ10, ξ20} are all arbitrary constants.It is clear that (17 ) and (19 ) are two types of ${ \mathcal P }{ \mathcal T }$-antisymmetric periodic solutions of the local SG equation, when $k={\left(m+1\right)}^{-1}$, ξ = 0, ${k}_{1}=\sqrt{{A}^{* }({A}^{* }+{m}_{1})/(1+{A}^{* }{m}_{1})}$, ξ10 = 0 and ξ20 = 0. So, two types of periodic solutions of the nonlocal SG equation (16 ) with ${ \mathcal P }{ \mathcal T }$ nonlocality can be written as
$\begin{eqnarray}v=\pm \left(2+2c{\partial }_{x}\right)\arctan \left[\sqrt{m}{\rm{sn}}\left(\displaystyle \frac{x-t}{m+1},m\right)\right],\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}v & = & \pm \left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \frac{{m}_{1}{\rm{sn}}\left({B}^{* }(x-t),{m}_{1}\right)}{b\ {\rm{dn}}\left({{bB}}^{* }(x+t),{m}_{2}\right)}\right],\\ {B}^{* } & = & \sqrt{\displaystyle \frac{{A}^{* }}{({A}^{* }+{m}_{1})({A}^{* }{m}_{1}+1)}}.\end{array}\end{eqnarray}$
In addition to the periodic solutions, there are many other types of exact solutions for the usual SG equation (2 ). The bilinear method [16–20] is an important method in finding exact solutions to the soliton equation. When it is difficult to directly study the nonlinear evolution equation, we can apply the bilinear method. We can transform the nonlinear equation into a bilinear form by introducing appropriate new variables. From Hirota's work [20], we can know that the solution of the local SG (2 ) equation has the following structure,
$\begin{eqnarray}\begin{array}{rcl}u & = & m\pi \pm 2{\rm{i}}\mathrm{ln}\displaystyle \frac{\bar{F}}{F},\\ & = & m\pi \pm 4\arctan \displaystyle \frac{b}{a},\qquad F=a+{\rm{i}}b,\end{array}\end{eqnarray}$
where i defined by ${\rm{i}}=\sqrt{-1}$, $\bar{F}$ is the complex conjugate of F, m is an arbitrary integer, a is the real part of F and b is the imaginary part of F. F can be found by solving the following bilinear [20] equation $\begin{eqnarray}{D}_{x}{D}_{t}F\cdot F+{\left(-1\right)}^{m}({\bar{F}}^{2}-{F}^{2})=0,\quad {D}_{x}^{2}\bar{F}\cdot F=0,\end{eqnarray}$
where the bilinear operators [20] Dx and Dt are defined as $\begin{eqnarray*}\begin{array}{l}{D}_{x}^{n}{D}_{t}^{p}F\cdot G\equiv \mathop{\mathrm{lim}}\limits_{c=0,d=0}\displaystyle \frac{{{\rm{d}}}^{n}}{{\rm{d}}{a}^{n}}\\ \quad \times \displaystyle \frac{{{\rm{d}}}^{p}}{{\rm{d}}{b}^{p}}F(x+c,t+d)G(x-c,t-d).\end{array}\end{eqnarray*}$
To get the multi-soliton solutions of the usual SG equation, the auxiliary functions $a\left(x,t\right)$ and $b\left(x,t\right)$ can be defined as
$\begin{eqnarray}\begin{array}{rcl}b & = & \displaystyle \sum _{{v}_{e}}{K}_{v}\sinh \left(\displaystyle \frac{1}{2}\displaystyle \sum _{j=1}^{N}{v}_{j}{\eta }_{j}\right),\\ a & = & \displaystyle \sum _{{v}_{o}}{K}_{v}\cosh \left(\displaystyle \frac{1}{2}\displaystyle \sum _{j=1}^{N}{v}_{j}{\eta }_{j}\right),\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\eta }_{j}={k}_{j}x+\omega t+{\eta }_{0j}={k}_{j}x+{\left(-1\right)}^{m}\displaystyle \frac{t}{{k}_{j}}+{\eta }_{0j},\end{eqnarray}$
the summation of v should be done to all the permutations of $\left\{{v}_{1},{v}_{2},\ldots {v}_{N}\right\}$ for vi = 1, −1, the summation of ve means all nondual even permutations of $\left\{{v}_{1},{v}_{2},\ldots {v}_{N}\right\}$ for vi = 1, −1 and the summation of vo means all nondual odd permutations of $\left\{{v}_{1},{v}_{2},\ldots {v}_{N}\right\}$ for vi = 1, −1, with $\begin{eqnarray}{K}_{v}\equiv \displaystyle \prod _{i\gt j}\left({k}_{i}-{v}_{i}{v}_{j}{k}_{j}\right),\end{eqnarray}$
η0j is an arbitrary real constant indicating the initial positions of each soliton.The N-soliton solutions to the SG equation have two different forms because the SG equation (2 ) has two different dispersion relations, $\omega ={k}_{j}^{-1}$ and $\omega =-{k}_{j}^{-1}$. These two types of N-soliton solutions can be written as
$\begin{eqnarray}\begin{array}{l}u=2n\pi +\displaystyle \frac{1+{\left(-1\right)}^{N}}{2}\pi \pm 4\,\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}u=2n\pi +\displaystyle \frac{1-{\left(-1\right)}^{N}}{2}\pi \pm 4\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}\right],\end{array}\end{eqnarray}$
where $\begin{eqnarray}{\eta }_{j}^{\pm }\equiv {k}_{j}x\pm {k}_{j}^{-1}t+{\eta }_{0j},\end{eqnarray}$
and n is an arbitrary integer that can be chosen to be zero for simplicity.For soliton solution (27 ), the ${ \mathcal P }{ \mathcal T }$-antisymmetric conditions can take N as an odd number and η0j = 0. Similarly, for soliton solution (28 ), the ${ \mathcal P }{ \mathcal T }$-antisymmetric conditions can be taken as N is an even number and η0j = 0.
So the ${ \mathcal P }{ \mathcal T }$-antisymmetric N-soliton solutions to the usual SG equation can be written as
$\begin{eqnarray}u=\left\{\begin{array}{lll}\pm 4\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}\right] & {\rm{for}} & N\quad {\rm{is}}\,{\rm{an}}\,{\rm{even}}\,{\rm{number}}\\ \pm 4\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}\right] & {\rm{for}} & N\quad {\rm{is}}\,{\rm{an}}\,{\rm{odd}}\,{\rm{number}}\end{array}\right.,\end{eqnarray}$
where η0j = 0.It is clear that the N-soliton solution u (30 ) is ${ \mathcal P }{ \mathcal T }$-antisymmetric and ux is ${ \mathcal P }{ \mathcal T }$-symmetric. Thus, the multi-soliton solutions of the nonlocal SG equation (16 ) can be expressed
$\begin{eqnarray}v=\left\{\begin{array}{lll}\pm \left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{+}\right)}\right] & {\rm{for}} & N\quad {\rm{is}}\,{\rm{an}}\,{\rm{even}}\,{\rm{number}}\\ \pm \left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \frac{{\sum }_{{v}_{e}}{K}_{v}\sinh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}{{\sum }_{{v}_{o}}{K}_{v}\cosh \left(\tfrac{1}{2}{\sum }_{j=1}^{N}{v}_{j}{\eta }_{j}^{-}\right)}\right] & {\rm{for}} & N\quad {\rm{is}}\,{\rm{an}}\,{\rm{odd}}\,{\rm{number}}\end{array}\right.,\end{eqnarray}$
where η0j = 0.For N = 1, the single soliton solution of the nonlocal SG equation can be written as
$\begin{eqnarray}\begin{array}{rcl}v & = & 2\arctan \left[\tanh \left(\displaystyle \frac{{k}_{1}x}{2}-\displaystyle \frac{t}{2{k}_{1}}\right)\right]+\displaystyle \frac{{{ck}}_{1}\left(1-\tanh {\left(\tfrac{{k}_{1}x}{2}-\tfrac{t}{2{k}_{1}}\right)}^{2}\right)}{\tanh {\left(\tfrac{{k}_{1}x}{2}-\tfrac{t}{2{k}_{1}}\right)}^{2}+1}.\end{array}\end{eqnarray}$
For N = 2, the two-soliton solution of the nonlocal SG equation can be written as
$\begin{eqnarray}\begin{array}{l}v=2\arctan \left[\displaystyle \frac{\left({k}_{2}-{k}_{1}\right)\sinh \tfrac{{\eta }_{1}^{+}+{\eta }_{2}^{+}}{2}}{\left({k}_{2}+{k}_{1}\right)\cosh \tfrac{{\eta }_{1}^{+}-{\eta }_{2}^{+}}{2}}\right]+\displaystyle \frac{2c\left({k}_{2}^{2}-{k}_{1}^{2}\right)\left({k}_{2}\cosh {\eta }_{1}^{+}+{k}_{1}\cosh {\eta }_{2}^{+}\right)}{\left({k}_{2}+{k}_{1}^{2}\right)\cosh \left({\eta }_{1}^{+}-{\eta }_{2}^{+}\right)+\left({k}_{1}-{k}_{2}^{2}\right)\cosh \left({\eta }_{1}^{+}+{\eta }_{2}^{+}\right)+4{k}_{1}{k}_{2}}.\end{array}\end{eqnarray}$
Selecting ${k}_{1}=\bar{{k}_{2}}=a+{\rm{i}}b$ in (33 ), the soliton solution becomes a breather solution adding its derivative with respect to x, where $\bar{{k}_{2}}$ is the complex conjugate of k2. So the solution of the nonlocal SG equation can be written as
$\begin{eqnarray}v=\left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \frac{a\sin \left({bx}-\tfrac{{bt}}{{a}^{2}+{b}^{2}}\right)}{b\cosh \left({ax}+\tfrac{{at}}{{a}^{2}+{b}^{2}}\right)}\right],\end{eqnarray}$
which is the linear combination of the breather solution of the usual SG equation and its derivative with respect to x.Selecting k1 = kl, k2 = − kr in (33 ), the soliton solution becomes a double kink solution adding its derivative with respect to x, which can be written as
$\begin{eqnarray}\begin{array}{l}v=\left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \Space{0ex}{4.2ex}{0ex}\frac{{k}_{r}+{k}_{l}}{{k}_{r}-{k}_{l}}\right.\\ \quad \left.\times \displaystyle \frac{\sinh \tfrac{1}{2}\left(\left({k}_{l}-{k}_{r}\right)x+\left({k}_{l}^{-1}-{k}_{r}^{-1}\right)t\right)}{\cosh \tfrac{1}{2}\left(\left({k}_{l}+{k}_{r}\right)x+\left({k}_{l}^{-1}+{k}_{r}^{-1}\right)t\right)}\right].\end{array}\end{eqnarray}$
Figure 1 displays an interaction periodic solution consisting of two periodic solitons for the nonlocal SG equation with space-time exchange nonlocality, expressed by equation (21 ) with the parameters $\left\{{m}_{1}=2,{m}_{2}=0.5\right\}$. Figure 2 displays an interaction breather solution consisting of two overlapping breather solitons for the nonlocal SG equation with space-time exchange nonlocality, expressed by equation (34 ) with the parameters $\left\{a=1,b=1\right\}$.
Figure 1. The periodic solution of the nonlocal SG equation with ${ \mathcal P }{ \mathcal T }$ nonlocality. |
Figure 2. The breather solution of the nonlocal SG equation with ${ \mathcal P }{ \mathcal T }$ nonlocality. |
4. The exact solutions of the nonlocal SG equation with space-time exchange nonlocality
When g = Ex,t, the SG equation (11 ) becomes a nonlocal SG equation with space-time exchange nonlocality,
$\begin{eqnarray}\begin{array}{l}{v}_{{xt}}=\displaystyle \frac{1}{2}\sin (v-{E}_{x,t}v)\\ \quad +\displaystyle \frac{1}{2}\left(v+{E}_{x,t}v\right)\cos \left(v-{E}_{x,t}v\right).\end{array}\end{eqnarray}$
The Ex,t-antisymmetric solution can be obtained from the two types (27 ) and (28 ) of the N-soliton solutions to the SG equation.
For N = 1 soliton solution (27 ), the conditions Ex,tu = − u and Ex,tux = ux can be taken as k1 = 1. So, the solution of the nonlocal SG equation with space-time exchange nonlocality can be written as
$\begin{eqnarray}\begin{array}{l}v=2\arctan \left[\tanh \left(\displaystyle \frac{x-t}{2}\right)\right]\\ \quad +\displaystyle \frac{c\left(1-\tanh {\left(\tfrac{x-t}{2}\right)}^{2}\right)}{\tanh {\left(\tfrac{x-t}{2}\right)}^{2}+1}.\end{array}\end{eqnarray}$
It is well-known that the double kink solution of the usual SG equation can be written as38 ) is equivalent to (28 ) for N = 2 which k1 = kl, k2 = − kr and η02 = η02 = 0. To get the solution of the nonlocal SG equation (36 ), the parameter k2 can be selected as ${k}_{r}=\tfrac{1}{{k}_{l}}$. Thus, the solution of the nonlocal SG equation with space-time exchange nonlocality is obtained that
$\begin{eqnarray}\begin{array}{l}u=4\arctan \left[\displaystyle \Space{0ex}{1.95em}{0ex}\frac{{k}_{r}+{k}_{l}}{{k}_{r}-{k}_{l}}\right.\\ \quad \left.\times \displaystyle \frac{\sinh \tfrac{1}{2}\left(\left({k}_{l}-{k}_{r}\right)x+\left({k}_{l}^{-1}-{k}_{r}^{-1}\right)t\right)}{\cosh \tfrac{1}{2}\left(\left({k}_{l}+{k}_{r}\right)x+\left({k}_{l}^{-1}+{k}_{r}^{-1}\right)t\right)}\right],\end{array}\end{eqnarray}$
Obviously, the result ( $\begin{eqnarray}\begin{array}{l}v=\left(2+2c{\partial }_{x}\right)\arctan \left[\displaystyle \Space{0ex}{4.2ex}{0ex}\frac{1+{k}_{l}^{2}}{1-{k}_{l}^{2}}\right.\\ \quad \left.\times \displaystyle \frac{\sinh \tfrac{1}{2}\left(\left({k}_{l}-{k}_{l}^{-1}\right)\left(x-t\right)\right)}{\cosh \tfrac{1}{2}\left(\left({k}_{l}+{k}_{l}^{-1}\right)\left(x+t\right)\right)}\right].\end{array}\end{eqnarray}$
From the periodic solutions (17 )–(19 ) of the local SG equation (2 ), we can find two types of Ext-antisymmetric periodic solutions of the nonlocal SG equation.
$\begin{eqnarray}v=\pm \left(2+2{\partial }_{x}\right)\arctan \left[\sqrt{m}{\rm{sn}}\left(\displaystyle \frac{x-t}{m+1},m\right)\right],\end{eqnarray}$
$\begin{eqnarray}v=\pm \left(2+2{\partial }_{x}\right)\arctan \left[\displaystyle \frac{{m}_{1}{\rm{sn}}\left({B}^{* }(x-t),{m}_{1}\right)}{b\ {\rm{dn}}\left({{bB}}^{* }(x+t)+{\xi }_{20},{m}_{2}\right)}\right].\end{eqnarray}$
Figure 3 displays an interaction solution consisting of a single soliton and a periodic soliton for the nonlocal SG equation with space-time exchange nonlocality, expressed by equation (37 ). Figure 4 displays an interaction solution consisting of two periodic solitons for the nonlocal SG equation with space-time exchange nonlocality, expressed by equation (40 ) with the parameter m = 2.
Figure 3. The interaction solution consists of a single soliton and a periodic soliton for the nonlocal SG equation with space-time exchange nonlocality. |
Figure 4. The interaction solution consists of two periodic solitons for the nonlocal SG equation with space-time exchange nonlocality. |
5. The exact solutions of the nonlocal sine-Gordon equation with space-time exchange and ${ \mathcal P }{ \mathcal T }$ nonlocalities
When $g={ \mathcal P }{ \mathcal T }{E}_{x,t}$, the SG equation (11 ) becomes a nonlocal SG equation with space-time exchange and ${ \mathcal P }{ \mathcal T }$ nonlocalities,
$\begin{eqnarray}\begin{array}{l}{v}_{{xt}}=\displaystyle \frac{1}{2}\sin (v-{ \mathcal P }{ \mathcal T }{E}_{x,t}v)\\ \quad +\displaystyle \frac{1}{2}\left(v+{ \mathcal P }{ \mathcal T }{E}_{x,t}v\right)\cos \left(v-{ \mathcal P }{ \mathcal T }{E}_{x,t}v\right).\end{array}\end{eqnarray}$
The ${ \mathcal P }{ \mathcal T }$-antisymmetric two-soliton solution to the usual SG equation (2 ) can be written as43 ). So, the parameter k2 can be chosen as ${k}_{2}={k}_{1}^{-1}$, then the ${ \mathcal P }{ \mathcal T }{E}_{x,t}$-antisymmetric two-soliton solution of the usual SG equation can be written as
$\begin{eqnarray}u=4\arctan \left[\displaystyle \frac{\left({k}_{2}-{k}_{1}\right)\sinh \left(\tfrac{{\eta }_{1}+{\eta }_{2}}{2}\right)}{\left({k}_{2}+{k}_{1}\right)\cosh \left(\tfrac{{\eta }_{1}-{\eta }_{2}}{2}\right)}\right],\end{eqnarray}$
where η01 = η02 = 0. In order to get the ${ \mathcal P }{ \mathcal T }{E}_{x,t}$-antisymmetric solution of the usual SG equation, the condition Ex,tu = u must be satisfied for solution ( $\begin{eqnarray}\begin{array}{l}v=\left(2+2c{\partial }_{x}\right)\arctan \\ \quad \times \left[\displaystyle \frac{\left(1-{k}_{1}^{2}\right)\sinh \tfrac{1}{2}\left(\left({k}_{1}+\tfrac{1}{{k}_{1}}\right)\left(x+t\right)\right)}{\left(1+{k}_{1}^{2}\right)\cosh \tfrac{1}{2}\left(\left({k}_{1}-\tfrac{1}{{k}_{1}}\right)\left(x-t\right)\right)}\right].\end{array}\end{eqnarray}$
Figure 5 displays an interaction solution consisting of a double kink soliton and a two periodic soliton for the nonlocal SG equation with ${ \mathcal P }{ \mathcal T }$ and space-time exchange nonlocalities, expressed by equation (44 ) with the parameter k1 = 2.
Figure 5. The interaction solution consists of a double kink soliton and two periodic soliton for the nonlocal SG equation with ${ \mathcal P }{ \mathcal T }$ and space-time exchange nonlocalities. |
6. Summary and discussions
In summary, three nonlocal integrable SG systems with ${ \mathcal P }{ \mathcal T }$ and/or space-time exchange nonlocalities are presented in this paper and it is interesting that there are various integrable cases by appropriately selecting the function G. Charge conjugation $\left({ \mathcal C }\right)$ symmetry, parity $\left({ \mathcal P }\right)$ symmetry, time reversal $\left({ \mathcal T }\right)$ symmetry, and their combinations have been used to construct nonlocal physical systems. But, the Ex,t symmetry was first introduced to nonlocal integrable systems theoretically by Jia and Lou [14] in 2022. More attention needs to be paid to the Ex,t symmetry.
The nonlocal equations can be transformed into an integrable coupled SG equation, by using the symmetric-antisymmetric separation approach. So, the solutions of the nonlocal SG equation are linear superpositions of the solutions to the usual SG equation and its symmetry equation. There are various solutions of the symmetry equation but in this paper, we just use the trivial symmetry B = cAx. More kinds of symmetric equations can be discussed. In conclusion, there are many interesting phenomena in a multi-place system that deserve further exploration.