1. Introduction
In 2013 Ablowitz and Musslimani introduced a new nonlocal nonlinear Schrödinger (nNLS) equation [1]1 ). Since the work of [1], nonlocal versions of many famous nonlinear systems, such as the Korteweg de-Vries (KdV) and modified KdV equation, the sine-Gordon equation, the Kadomtsev–Petviashivili (KP) equation, Sasa-Satsuma equation, etc are introduced and studied by applying various methods including inverse scattering transform [2, 3], Riemann–Hilbert method [4, 5], the Hirota's bilinear method [6–9], the Darboux transformations [10–12], Wronskian technique [13], symmetry analysis [14], deep learning neural network framework [15] and so on.
$\begin{eqnarray}{\rm{i}}{q}_{t}(x,t)={q}_{{xx}}(x,t)\pm q(x,t){q}^{* }(-x,t)q(x,t),\end{eqnarray}$
with q*(−x, t) being complex conjugate of q(−x, t), which is proved to be integrable under the meaning that it has a Lax pair and an infinite number of conservation laws. Contrary to local equations where dependent variables have the same independent variables, dependent variables of a nonlocal equation have two or more independent variables which are usually linked by space and/or time reversion, such as the variables of (−x, t) and (x, t) in equation (In recent years, Lou and Huang proposed the concept of the Alice–Bob (AB) system to describe two correlated events which can be assumed to be related by an operator $\hat{f}$, e.g. $A=\hat{f}B$, where $\hat{f}$ can be taken as shifted parity and delayed time reversal and so forth [16]. In other words, there exist at least two spacetime coordinates in one AB system. In this context, many AB-type nonlocal systems are constructed including the AB-KdV equation [17], AB-mKdV equation [18], AB-AKNS system [17], etc. In [19], a consistent correlated bang (CCB) method is proposed from which one can generate nonlocal systems from known local ones [20, 21].
The coupled KP (cKP) system [22–25] takes the form2 ) can be categorized as cKPI by taking σ2 = −1 and cKPII by taking σ2 = 1. In [23, 28], a host of solitonic interactions of the cKP are obtained, among which peculiar spider-web solutions are obtained and analyzed. In this paper, inspired by the CCB method, we introduce a nonlocal coupled KP (ncKP) system as
$\begin{eqnarray}{\left({u}_{t}+\displaystyle \frac{1}{4}{u}_{{xxx}}-\displaystyle \frac{3}{2}{{uu}}_{x}-\displaystyle \frac{3}{4}{\left({vw}\right)}_{x}\right)}_{x}+{\sigma }^{2}{u}_{{yy}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left({v}_{t}+\displaystyle \frac{1}{4}{v}_{{xxx}}-\displaystyle \frac{3}{2}{\left({uv}\right)}_{x}\right)}_{x}+{\sigma }^{2}{v}_{{yy}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left({w}_{t}+\displaystyle \frac{1}{4}{w}_{{xxx}}-\displaystyle \frac{3}{2}{\left({uw}\right)}_{x}\right)}_{x}+{\sigma }^{2}{w}_{{yy}}=0,\end{eqnarray}$
which was first appeared in a paper of Jimbo and Miwa in Hirota bilinear form [26], it has N-soliton solutions expressed in terms of Pfaffians [27]. The cKP system ( $\begin{eqnarray}\begin{array}{l}{A}_{{xt}}-\displaystyle \frac{9}{8}{A}_{x}^{2}-\displaystyle \frac{3}{4}{A}_{x}{B}_{x}+\displaystyle \frac{3}{8}{B}_{x}^{2}-\displaystyle \frac{3}{8}(3{F}_{x}+{G}_{x}){C}_{x}\\ \quad +\displaystyle \frac{3}{8}({G}_{x}-{F}_{x}){E}_{x}-\displaystyle \frac{3}{8}(3A+B){A}_{{xx}}+{\sigma }^{2}{A}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(B-A){B}_{{xx}}-\displaystyle \frac{3}{16}(3F+G){C}_{{xx}}+\displaystyle \frac{3}{16}(G-F){E}_{{xx}}\\ \quad -\displaystyle \frac{3}{16}(E+3C){F}_{{xx}}-\displaystyle \frac{3}{16}(C-E){G}_{{xx}}+\displaystyle \frac{1}{4}{A}_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{B}_{{xt}}-\displaystyle \frac{9}{8}{B}_{x}^{2}-\displaystyle \frac{3}{4}{A}_{x}{B}_{x}+\displaystyle \frac{3}{8}{A}_{x}^{2}-\displaystyle \frac{3}{8}(3{G}_{x}+{F}_{x}){E}_{x}\\ \quad +\displaystyle \frac{3}{8}({F}_{x}-{G}_{x}){C}_{x}-\displaystyle \frac{3}{8}(3B+A){B}_{{xx}}+{\sigma }^{2}{B}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(A-B){A}_{{xx}}-\displaystyle \frac{3}{16}(3G+F){E}_{{xx}}+\displaystyle \frac{3}{16}(F-G){C}_{{xx}}\\ \quad -\displaystyle \frac{3}{16}(C+3E){G}_{{xx}}-\displaystyle \frac{3}{16}(E-C){F}_{{xx}}+\displaystyle \frac{1}{4}{B}_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{C}_{{xt}}-\displaystyle \frac{3}{4}({C}_{x}-{E}_{x}){A}_{x}-\displaystyle \frac{3}{4}({E}_{x}+3{C}_{x}){B}_{x}-\displaystyle \frac{3}{8}(3C+E){A}_{{xx}}\\ \quad +\displaystyle \frac{3}{8}(E-C){B}_{{xx}}-\displaystyle \frac{3}{8}(3A+B){C}_{{xx}}+{\sigma }^{2}{C}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(B-A){E}_{{xx}}+\displaystyle \frac{1}{4}{C}_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{E}_{{xt}}-\displaystyle \frac{3}{4}({E}_{x}-{C}_{x}){B}_{x}-\displaystyle \frac{3}{4}({C}_{x}+3{E}_{x}){A}_{x}-\displaystyle \frac{3}{8}(3E+C){B}_{{xx}}\\ \quad +\displaystyle \frac{3}{8}(C-E){A}_{{xx}}-\displaystyle \frac{3}{8}(3B+A){E}_{{xx}}+{\sigma }^{2}{E}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(A-B){C}_{{xx}}+\displaystyle \frac{1}{4}{E}_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{F}_{{xt}}-\displaystyle \frac{3}{4}({F}_{x}-{G}_{x}){A}_{x}-\displaystyle \frac{3}{4}({F}_{x}+3{G}_{x}){B}_{x}-\displaystyle \frac{3}{8}(3F+G){A}_{{xx}}\\ \quad +\displaystyle \frac{3}{8}(G-F){B}_{{xx}}-\displaystyle \frac{3}{8}(3A+B){F}_{{xx}}+{\sigma }^{2}{F}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(B-A){G}_{{xx}}+\displaystyle \frac{1}{4}{F}_{{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{G}_{{xt}}-\displaystyle \frac{3}{4}({G}_{x}-{F}_{x}){B}_{x}-\displaystyle \frac{3}{4}({G}_{x}+3{F}_{x}){A}_{x}-\displaystyle \frac{3}{8}(3G+F){B}_{{xx}}\\ \quad +\displaystyle \frac{3}{8}(F-G){A}_{{xx}}-\displaystyle \frac{3}{8}(3B+A){G}_{{xx}}+{\sigma }^{2}{G}_{{yy}}\\ \quad +\displaystyle \frac{3}{8}(A-B){F}_{{xx}}+\displaystyle \frac{1}{4}{G}_{{xxxx}}=0,\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{rcl}B & = & {\hat{P}}_{s}^{x}{\hat{T}}_{d}A(x,y,t)=A(-x+{x}_{0},y,-t+{t}_{0}),\\ E & = & {\hat{P}}_{s}^{x}{\hat{T}}_{d}C(x,y,t)=C(-x+{x}_{0},y,-t+{t}_{0}),\\ G & = & {\hat{P}}_{s}^{x}{\hat{T}}_{d}F(x,y,t)=F(-x+{x}_{0},y,-t+{t}_{0}),\end{array}\end{eqnarray}$
and probe its exact solutions and symmetry properties.The paper is organized as follows. In section 2 , we convert the ncKPI (σ2 = −1) system into a local system by introducing some new variables with definite parity properties to replace the variables of the ncKPI system. Then we use N-soliton solutions of the cKP system to generate an even number of N-soliton solutions of the ncKPI system. In section 3 , we apply the standard Lie symmetry method on the ncKPII (σ2 = 1) system to give its Lie symmetry group and similarity reduction solutions. The last section is devoted to a summary.
2. Multiple soliton solutions of the ncKPI system
To convert the ncKPI system into a local system, considering the relation (9 ), we take10 ) into the ncKP system (3 )–(9 ) we split it into the following equations12 )–(14 ) are just the cKPI system (2 ) with σ2 = −1 while equations (15 )–(17 ) are linearized equations of the cKPI system.
$\begin{eqnarray}\begin{array}{rcl}A & = & u+{u}_{1},B=u-{u}_{1},C=v+{v}_{1},\\ E & = & v-{v}_{1},F=w+{w}_{1},G=w-{w}_{1},\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{\hat{P}}_{s}^{x}{\hat{T}}_{d}u=u,{\hat{P}}_{s}^{x}{\hat{T}}_{d}{u}_{1}=-{u}_{1},\\ {\hat{P}}_{s}^{x}{\hat{T}}_{d}v=v,{\hat{P}}_{s}^{x}{\hat{T}}_{d}{v}_{1}=-{v}_{1}\\ {\hat{P}}_{s}^{x}{\hat{T}}_{d}w=w,\\ {\hat{P}}_{s}^{x}{\hat{T}}_{d}{w}_{1}=-{w}_{1}.\end{array}\end{eqnarray}$
Substituting equation ( $\begin{eqnarray}{\left({u}_{t}+\displaystyle \frac{1}{4}{u}_{{xxx}}-\displaystyle \frac{3}{2}{{uu}}_{x}-\displaystyle \frac{3}{4}{\left({vw}\right)}_{x}\right)}_{x}-{u}_{{yy}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left({v}_{t}+\displaystyle \frac{1}{4}{v}_{{xxx}}-\displaystyle \frac{3}{2}{\left({uv}\right)}_{x}\right)}_{x}-{v}_{{yy}}=0,\end{eqnarray}$
$\begin{eqnarray}{\left({w}_{t}+\displaystyle \frac{1}{4}{w}_{{xxx}}-\displaystyle \frac{3}{2}{\left({uw}\right)}_{x}\right)}_{x}-{w}_{{yy}}=0,\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-3{u}_{1,x}{u}_{x}-\displaystyle \frac{3}{2}{u}_{{xx}}{u}_{1}-\displaystyle \frac{3}{2}{{uu}}_{1,{xx}}-\displaystyle \frac{3}{4}{v}_{1}{w}_{{xx}}\\ \quad -\displaystyle \frac{3}{4}{{vw}}_{1,{xx}}-\displaystyle \frac{3}{2}{w}_{x}{v}_{1,x}-\displaystyle \frac{3}{4}{v}_{{xx}}{w}_{1}\\ \quad -\displaystyle \frac{3}{4}{{wv}}_{1,{xx}}-\displaystyle \frac{3}{2}{v}_{x}{w}_{1,x}-{u}_{1,{yy}}+{u}_{1,{xt}}+\displaystyle \frac{1}{4}{u}_{1,{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{v}_{1,{yy}}-3{u}_{x}{v}_{1,x}+3{v}_{x}{u}_{1,x}-\displaystyle \frac{3}{2}{{vu}}_{1,{xx}}-\displaystyle \frac{3}{2}{u}_{{xx}}{v}_{1}+{v}_{1,{xt}}\\ \quad -\displaystyle \frac{3}{2}{{uv}}_{1,{xx}}-\displaystyle \frac{3}{2}{v}_{{xx}}{u}_{1}\\ \quad +\displaystyle \frac{1}{4}{v}_{1,{xxxx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-{w}_{1,{yy}}-3{u}_{x}{w}_{1,x}+3{w}_{x}{u}_{1,x}-\displaystyle \frac{3}{2}{{wu}}_{1,{xx}}\\ \quad -\displaystyle \frac{3}{2}{u}_{{xx}}{w}_{1}+{w}_{1,{xt}}-\displaystyle \frac{3}{2}{{uw}}_{1,{xx}}-\displaystyle \frac{3}{2}{w}_{{xx}}{u}_{1}\\ \quad +\displaystyle \frac{1}{4}{w}_{1,{xxxx}}=0.\end{array}\end{eqnarray}$
It can be seen that equations (The cKP system (2 ) has the following N-soliton solution [22]
$\begin{eqnarray}\begin{array}{rcl}u & = & -{\left(\mathrm{ln}F\right)}_{{xx}},v=\alpha {\left(\mathrm{ln}F\right)}_{{xx}},w=\displaystyle \frac{1}{\alpha }{\left(\mathrm{ln}F\right)}_{{xx}},\\ F & = & \displaystyle \sum _{\nu }\exp (\displaystyle \sum _{j=1}^{N}{\nu }_{j}{\xi }_{j}+\displaystyle \sum _{1\leqslant i^{\prime} \lt j}^{N}{\nu }_{i^{\prime} }{\nu }_{j}{\theta }_{i^{\prime} j}),\end{array}\end{eqnarray}$
with α being a nonzero real number, which is an extension of N-soliton solutions of the KP equation, the summations should be done for all permutations of ${\nu }_{i^{\prime} }=0,1(i^{\prime} =1,2,3,\cdots N)$, and $\begin{eqnarray*}\begin{array}{l}{\xi }_{j}={k}_{j}x+{r}_{j}y-\displaystyle \frac{{k}_{j}^{4}+4{\sigma }^{2}{r}_{j}^{2}}{4{k}_{j}}t+{\xi }_{j0},\exp ({\theta }_{i^{\prime} j})\\ \quad =\,\displaystyle \frac{3{k}_{i^{\prime} }^{2}{k}_{j}^{2}{\left({k}_{i^{\prime} }-{k}_{j}\right)}^{2}-4{\sigma }^{2}{\left({k}_{i^{\prime} }{r}_{j}-{k}_{j}{r}_{i^{\prime} }\right)}^{2}}{3{k}_{i^{\prime} }^{2}{k}_{j}^{2}{\left({k}_{i^{\prime} }+{k}_{j}\right)}^{2}-4{\sigma }^{2}{\left({k}_{i^{\prime} }{r}_{j}-{k}_{j}{r}_{i^{\prime} }\right)}^{2}},\end{array}\end{eqnarray*}$
with arbitrary constants kj, rj, ξj0, (j = 1, 2, 3, ⋯ ,N).Considering that u , v and w in equation (18 ) are invariant under the following transformation
$\begin{eqnarray*}F\to \beta \exp ({Kx}+{\rm{\Omega }}t+{X}_{0})F,\end{eqnarray*}$
we rewrite $\begin{eqnarray*}{\xi }_{j}={\eta }_{j}-\displaystyle \frac{1}{2}\displaystyle \sum _{i^{\prime} =1}^{j-1}{\theta }_{i^{\prime} j}-\displaystyle \frac{1}{2}\displaystyle \sum _{i^{\prime} =j+1}^{N}{\theta }_{{ji}^{\prime} },\end{eqnarray*}$
where $\begin{eqnarray*}\begin{array}{l}{\eta }_{j}={k}_{j}(x-\displaystyle \frac{{x}_{0}}{2})+{r}_{j}(y-\displaystyle \frac{{y}_{0j}}{2})\\ \quad -\displaystyle \frac{{k}_{j}^{4}+4{r}_{j}^{2}}{4{k}_{j}}(t-\displaystyle \frac{{t}_{0}}{2})+{\eta }_{0j},\end{array}\end{eqnarray*}$
with arbitrary constants y0j and η0j. So the N-soliton solutions of the cKPI system can be rewritten as [17] $\begin{eqnarray}\begin{array}{l}u=-\displaystyle \frac{1}{\alpha }v=-\alpha w=-\left[\mathrm{ln}\displaystyle \sum _{\nu }{K}_{\nu }\right.\\ \quad \,\times {\left.\cosh \left(\displaystyle \frac{1}{2}\displaystyle \sum _{j=1}^{N}{\nu }_{j}{\eta }_{j}\right)\right]}_{{xx}}\equiv {\left(\mathrm{ln}{F}_{N}\right)}_{{xx}},\end{array}\end{eqnarray}$
where the summation of ν being done for all non-dual permutations of ${\nu }_{i^{\prime} }=1,-1,(i^{\prime} =1,2,3,\cdots N)$ and $\begin{eqnarray*}{K}_{\nu }=\displaystyle \prod _{i^{\prime} \gt j}\sqrt{3{k}_{i^{\prime} }^{2}{k}_{j}^{2}{\left({k}_{i^{\prime} }-{\nu }_{i^{\prime} }{\nu }_{j}{k}_{j}\right)}^{2}+4{\left({k}_{i^{\prime} }{r}_{j}-{k}_{j}{r}_{i^{\prime} }\right)}^{2}}.\end{eqnarray*}$
In order to give ${\hat{P}}_{s}^{x}{\hat{T}}_{d}$ invariant part of the N-soliton solutions in equation (18 ), additional restrictions should be given as
$\begin{eqnarray}N=2n,{k}_{n+i^{\prime} }=-{k}_{i^{\prime} },{r}_{n+i^{\prime} }={r}_{i^{\prime} },{y}_{0(n+i^{\prime} )}={y}_{0i^{\prime} }.\end{eqnarray}$
It is clear that solutions of u1, v1, and w1 in equations (15 )–(17 ) are symmetries of the cKP system, which can be taken as21 ) with equation (22 ) satisfies the condition of equation (11 ). So, N-soliton solutions of the ncKPI system (12 )–(14 ) can be expressed by equation (10 ) with equations (19 ) and (21 ).
$\begin{eqnarray}\begin{array}{l}{u}_{1}=\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x+\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}+\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){u}_{x}+\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){u}_{y}\\ \quad +{f}_{1}{u}_{t}+\displaystyle \frac{2}{3}\dot{{f}_{1}}u+\displaystyle \frac{2}{9}\ddot{{f}_{1}}x+\displaystyle \frac{1}{9}\dddot{{f}_{1}}{y}^{2}+\displaystyle \frac{1}{3}\ddot{{f}_{2}}y+\displaystyle \frac{2}{3}\dot{{f}_{3}},\\ {v}_{1}=\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x+\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}+\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){v}_{x}\\ \quad +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){v}_{y}+{f}_{1}{v}_{t}+\displaystyle \frac{2}{3}\dot{{f}_{1}}v,\\ {w}_{1}=\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x+\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}+\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){w}_{x}\\ \quad +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){w}_{y}+{f}_{1}{w}_{t}+\displaystyle \frac{2}{3}\dot{{f}_{1}}w,\end{array}\end{eqnarray}$
where f1, f2 and f3 are arbitrary functions of t satisfying $\begin{eqnarray}\hat{{T}_{d}}\{{f}_{1},{f}_{2},{f}_{3}\}=\{{f}_{1},-{f}_{2},{f}_{3}\}.\end{eqnarray}$
It can be verified that equation (By the condition of equation (20 ), odd number solitons of the ncKPI system are prohibited. As for N = 2, 4, the explicit expressions of FN in equation (19 ) are
$\begin{eqnarray}\begin{array}{l}{F}_{2}=\sqrt{3{k}_{1}^{4}+4{r}_{1}^{2}}\cosh \left[{r}_{1}\left(y-\displaystyle \frac{{y}_{01}}{2}\right)\right]\\ \quad +2\cosh \left[{k}_{1}\left(x-\displaystyle \frac{{x}_{0}}{2}\right)-\displaystyle \frac{({k}_{1}^{4}-4{r}_{1}^{2})}{4{k}_{1}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)\right],\end{array}\end{eqnarray}$
and $\begin{eqnarray*}\begin{array}{l}{F}_{4}={\delta }_{14,-}{\delta }_{24,-}{\delta }_{34,-}{\delta }_{13,-}{\delta }_{23,-}{\delta }_{12,-}\\ \times \cosh \left[{r}_{1}\left(y-\displaystyle \frac{{y}_{01}}{2}\right)+{r}_{2}\left(y-\displaystyle \frac{{y}_{02}}{2}\right)\right]\\ +{\delta }_{14,+}{\delta }_{24,-}{\delta }_{34,-}{\delta }_{13,+}{\delta }_{23,-}{\delta }_{12,+}\\ \times \cosh \left[{k}_{1}\Space{0ex}{2.5ex}{0ex}(x-\displaystyle \frac{{x}_{0}}{2}\Space{0ex}{2.5ex}{0ex})-\displaystyle \frac{{k}_{1}^{4}-4{r}_{1}^{2}}{4{k}_{1}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)-{r}_{2}\left(y-\displaystyle \frac{{y}_{02}}{2}\right)\right]\\ +{\delta }_{14,-}{\delta }_{24,+}{\delta }_{34,-}{\delta }_{13,-}{\delta }_{23,+}{\delta }_{12,+}\\ \times \cosh \left[{r}_{1}\left(y-\displaystyle \frac{{y}_{01}}{2}\right)-{k}_{2}\left(x-\displaystyle \frac{{x}_{0}}{2}\right)+\displaystyle \frac{\left({k}_{2}^{4}-4{r}_{2}^{2}\right)}{4{k}_{2}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)\right]\end{array}\end{eqnarray*}$
$\begin{eqnarray}\begin{array}{l}+{\delta }_{14,-}{\delta }_{24,-}{\delta }_{34,+}{\delta }_{13,+}{\delta }_{23,+}{\delta }_{12,-}\\ \times \cosh \left[{k}_{1}\Space{0ex}{1ex}{0ex}(x-\displaystyle \frac{{x}_{0}}{2}\Space{0ex}{1ex}{0ex})-\displaystyle \frac{({k}_{1}^{4}-4{r}_{1}^{2})}{4{k}_{1}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)+{r}_{2}\left(y-\displaystyle \frac{{y}_{02}}{2}\right)\right]\\ +{\delta }_{14,+}{\delta }_{24,+}{\delta }_{34,+}{\delta }_{13,-}{\delta }_{23,-}{\delta }_{12,-}\\ \times \cosh \left[{r}_{1}\Space{0ex}{1ex}{0ex}(y-\displaystyle \frac{{y}_{01}}{2}\Space{0ex}{1ex}{0ex})+{k}_{2}\left(x-\displaystyle \frac{{x}_{0}}{2}\right)-\displaystyle \frac{{k}_{2}^{4}-4{r}_{2}^{2}}{4{k}_{2}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)\right]\\ +{\delta }_{14,+}{\delta }_{24,+}{\delta }_{34,-}{\delta }_{13,+}{\delta }_{23,+}{\delta }_{12,-}\\ \times \cosh \left[{k}_{1}\Space{0ex}{1ex}{0ex}(x-\displaystyle \frac{{x}_{0}}{2}\Space{0ex}{1ex}{0ex})-\displaystyle \frac{({k}_{1}^{4}-4{r}_{1}^{2})}{4{k}_{1}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)+{k}_{2}\left(x-\displaystyle \frac{{x}_{0}}{2}\right)\right.\\ \left.-\displaystyle \frac{({k}_{2}^{4}-4{r}_{2}^{2})}{4{k}_{2}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)\right]\\ +{\delta }_{14,+}{\delta }_{24,-}{\delta }_{34,+}{\delta }_{13,-}{\delta }_{23,+}{\delta }_{12,+}\\ \times \cosh \left[{r}_{1}\Space{0ex}{2.5ex}{0ex}(y-\displaystyle \frac{{y}_{01}}{2}\Space{0ex}{2.5ex}{0ex})-{r}_{2}\left(y-\displaystyle \frac{{y}_{02}}{2}\right)\right]\\ +{\delta }_{14,-}{\delta }_{24,+}{\delta }_{34,+}{\delta }_{13,+}{\delta }_{23,-}{\delta }_{12,+}\\ \times \cosh \left[{k}_{1}\Space{0ex}{2.5ex}{0ex}(x-\displaystyle \frac{{x}_{0}}{2}\Space{0ex}{2.5ex}{0ex})-\displaystyle \frac{({k}_{1}^{4}-4{r}_{1}^{2})}{4{k}_{1}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)-{k}_{2}\left(x-\displaystyle \frac{{x}_{0}}{2}\right)\right.\\ \left.+\displaystyle \frac{({k}_{2}^{4}-4{r}_{2}^{2})}{4{k}_{2}}\left(t-\displaystyle \frac{{t}_{0}}{2}\right)\right],\end{array}\end{eqnarray}$
where $\begin{eqnarray*}{\delta }_{i^{\prime} j,\pm }=\sqrt{3{k}_{i^{\prime} }^{2}{k}_{j}^{2}{\left({k}_{i^{\prime} }\pm {k}_{j}\right)}^{2}+4{\left({k}_{i^{\prime} }{r}_{j}-{k}_{j}{r}_{i^{\prime} }\right)}^{2}},\end{eqnarray*}$
and $\begin{eqnarray*}\begin{array}{rcl}{k}_{4} & = & -{k}_{2},{k}_{3}=-{k}_{1},{r}_{4}={r}_{2},\\ {r}_{3} & = & {r}_{1},{y}_{04}={y}_{02},{y}_{03}={y}_{01}.\end{array}\end{eqnarray*}$
At time t = 0, for N = 2 case, figures 1(a) and 2(a) give density plot and three-dimensional plot of A of the ncKP system expressed by equations (10 ) with equations (19 ) and (21 ) where the parameters are fixed by23 ) (or (24 )) depends similarly on the coordinates of x, y and t, the multiple soliton interaction behaviors of A depending on other variable pairs (x, t) and (y, t) are similar to those in figures 1 and 2.
$\begin{eqnarray}\begin{array}{rcl}{f}_{1} & = & {f}_{2}={x}_{0}={t}_{0}={\eta }_{01}={\eta }_{02}={y}_{01}={y}_{02}=0,\\ {f}_{3} & = & 2,\alpha =2,{k}_{1}=-{k}_{2}=1,{r}_{1}={r}_{2}=1;\end{array}\end{eqnarray}$
as for N = 4 case, figures 1(b) and 2(b) give density plot and three-dimensional plot of A of the ncKPI system where the parameters are fixed by $\begin{eqnarray}\begin{array}{rcl}{f}_{1} & = & {f}_{2}={x}_{0}={t}_{0}={\eta }_{0j}={y}_{0j}=0,(j=1,2,\cdots ,4),\\ {f}_{3} & = & 2,\alpha =1,{k}_{1}=-{k}_{3}=1,\\ {k}_{2} & = & -{k}_{4}=2,{r}_{1}={r}_{2}={r}_{3}={r}_{4}=1.\end{array}\end{eqnarray}$
Because F2 (or F4) in equations (
Figure 1. (a) The density plot of the solution A of the ncKPI system at time t = 0 for N = 2 case with parameters being fixed by equation ( |
Figure 2. (a) The three-dimensional plot of the solution A of the ncKPI system at time t = 0 for N = 2 case with parameters being fixed by equation ( |
When rj(j = 1, 2, 3, ⋯ ,N) in equation (19 ) are taken to be a pure imaginary number, these N-soliton solutions become breather solutions. To illustrate this point, for the N = 2 case, when we take the parameters as
$\begin{eqnarray}\begin{array}{rcl}{f}_{1} & = & {f}_{2}={x}_{0}={y}_{01}={y}_{02}={t}_{0}={\eta }_{01}={\eta }_{02}=0,\\ {f}_{3} & = & 2,{k}_{1}=-{k}_{2}=1,\\ {r}_{1} & = & {r}_{2}=\sqrt{-1},\alpha =2,\end{array}\end{eqnarray}$
we get breathers at time t = 0 in figure 3.
Figure 3. Breather solutions of the ncKPI system at time t = 0 for: (a) the density plots of the variable A; (b) the three-dimensional plots of the variable A. The parameters are fixed by equation ( |
It is well known that the KP equation has lump solutions, we can verify that the cKPI system has the following solution28 ) into equations (10 ) with equation (21 ). Figure 4 demonstrates a lump-type solution of the ncKPI system for the variable A, where the parameters are fixed by
$\begin{eqnarray}\begin{array}{l}u=-\displaystyle \frac{1}{\alpha }v=-\alpha w=\displaystyle \frac{{L}_{1}}{{L}_{2}},\\ {L}_{1}=32{d}^{2}{x}^{2}+32[{d}^{2}({{dt}}_{0}-{x}_{0})-2{d}^{3}t]x-32{d}^{3}{y}^{2}\\ +64{d}^{3}{{yy}}_{0}+32{d}^{4}{t}^{2}-32{d}^{3}({{dt}}_{0}-{x}_{0})t\\ +8d({d}^{3}{t}_{0}^{2}-2{d}^{2}{t}_{0}{x}_{0}-4{d}^{2}{y}_{0}^{2}+{{dx}}_{0}^{2}-3),\\ {L}_{2}=\left[4{{dx}}^{2}+4d[({{dt}}_{0}-{x}_{0})-2{dt}]x+4{d}^{2}{y}^{2}-8{d}^{2}{{yy}}_{0}\right.\\ {\left.+4{d}^{3}{t}^{2}-4{d}^{2}({{dt}}_{0}-{x}_{0})t+d{\left({{dt}}_{0}-{x}_{0}\right)}^{2}+4{d}^{2}{y}_{0}^{2}+3\right]}^{2},\end{array}\end{eqnarray}$
with arbitrary constants d, x0, y0, t0, which leads to lump-type solutions of the ncKP system by substituting equation ( $\begin{eqnarray}{f}_{1}={f}_{2}={f}_{3}={x}_{0}={y}_{0}={t}_{0}=0,d=1.\end{eqnarray}$
Figure 4. Plots of lump solution of A of the ncKPI system at time t = 0, while the parameters being fixed by equation ( |
3. Symmetry reduction solutions of the the ncKPII system
Symmetry analysis plays an important role in solving nonlinear systems [29, 30], in this section we apply the standard Lie symmetry method on the ncKPII system. To this end, we first give the Lie point symmetry of this system in the form30 ) can be written in function form as
$\begin{eqnarray}\begin{array}{rcl}V & = & X\displaystyle \frac{\partial }{\partial x}+Y\displaystyle \frac{\partial }{\partial y}+T\displaystyle \frac{\partial }{\partial t}+{{\rm{\Gamma }}}_{1}\displaystyle \frac{\partial }{\partial A}+{{\rm{\Lambda }}}_{1}\displaystyle \frac{\partial }{\partial B}\\ & & +{{\rm{\Gamma }}}_{2}\displaystyle \frac{\partial }{\partial C}+{{\rm{\Lambda }}}_{2}\displaystyle \frac{\partial }{\partial E}+{{\rm{\Gamma }}}_{3}\displaystyle \frac{\partial }{\partial F}+{{\rm{\Lambda }}}_{2}\displaystyle \frac{\partial }{\partial G},\end{array}\end{eqnarray}$
where X, Y, T, Γ1, Γ2, Γ3, Λ1, Λ2, Λ3 are functions of x, y, t, A, B, C, E, F, G that needs to be determined. In other words, the ncKPII system is invariant under the following transformation $\begin{eqnarray*}\begin{array}{l}\{x,y,t,A,B,C,E,F,G\}\\ \quad \to \{x+\epsilon X,y+\epsilon Y,t+\epsilon T,A+\epsilon {{\rm{\Gamma }}}_{1},B+\epsilon {{\rm{\Lambda }}}_{1},\\ \quad C+\epsilon {{\rm{\Gamma }}}_{2},E+\epsilon {{\rm{\Lambda }}}_{2},\\ \quad F+\epsilon {{\rm{\Gamma }}}_{3},G+\epsilon {{\rm{\Lambda }}}_{3}\},\end{array}\end{eqnarray*}$
with infinitesimal parameter ε. The symmetry of equation ( $\begin{eqnarray}{\sigma }_{A}={{XA}}_{x}+{{YA}}_{y}+{{TA}}_{t}-{{\rm{\Gamma }}}_{1},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{B}={{XB}}_{x}+{{YB}}_{y}+{{TB}}_{t}-{{\rm{\Lambda }}}_{1},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{C}={{XC}}_{x}+{{YC}}_{y}+{{TC}}_{t}-{{\rm{\Gamma }}}_{2},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{E}={{XB}}_{x}+{{YB}}_{y}+{{TB}}_{t}-{{\rm{\Lambda }}}_{2},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{F}={{XF}}_{x}+{{YF}}_{y}+{{TF}}_{t}-{{\rm{\Gamma }}}_{3},\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{G}={{XG}}_{x}+{{YG}}_{y}+{{TG}}_{t}-{{\rm{\Lambda }}}_{3},\end{eqnarray}$
which satisfy the linearized equations of the ncKPII system $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3}{4}({\sigma }_{E,x}-{\sigma }_{C,x}){A}_{x}-\displaystyle \frac{3}{4}({\sigma }_{E,x}+3{\sigma }_{C,x}){B}_{x}\\ \quad -\displaystyle \frac{3}{4}(3{\sigma }_{B,x}+{\sigma }_{A,x}){C}_{x}+\displaystyle \frac{3}{4}({\sigma }_{A,x}-{\sigma }_{B,x}){E}_{x}\\ -\displaystyle \frac{3}{8}(3{\sigma }_{C}+{\sigma }_{E}){A}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{E}-{\sigma }_{C}){B}_{{xx}}\\ \quad -\displaystyle \frac{3}{8}(3{\sigma }_{A}+{\sigma }_{B}){C}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{B}-{\sigma }_{A}){E}_{{xx}}\\ \quad +\displaystyle \frac{1}{4}{\sigma }_{C,{xxxx}}-\displaystyle \frac{3}{8}(3C+E){\sigma }_{A,{xx}}\\ \quad +\displaystyle \frac{3}{8}(E-C){\sigma }_{B,{xx}}+{\sigma }_{C,{xt}}-\displaystyle \frac{3}{8}(3A+B){\sigma }_{C,{xx}}\\ \quad +{\sigma }_{C,{yy}}+\displaystyle \frac{3}{8}(B-A){\sigma }_{E,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3}{4}({\sigma }_{C,x}-{\sigma }_{E,x}){B}_{x}-\displaystyle \frac{3}{4}({\sigma }_{C,x}+3{\sigma }_{E,x}){A}_{x}\\ \quad -\displaystyle \frac{3}{4}(3{\sigma }_{A,x}+{\sigma }_{B,x}){E}_{x}+\displaystyle \frac{3}{4}({\sigma }_{B,x}-{\sigma }_{A,x}){C}_{x}\\ \quad -\displaystyle \frac{3}{8}(3{\sigma }_{E}+{\sigma }_{C}){B}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{C}-{\sigma }_{E}){A}_{{xx}}\\ \quad -\displaystyle \frac{3}{8}(3{\sigma }_{B}+{\sigma }_{A}){E}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{A}-{\sigma }_{B}){C}_{{xx}}\\ \quad +\displaystyle \frac{1}{4}{\sigma }_{E,{xxxx}}-\displaystyle \frac{3}{8}(3E+C){\sigma }_{B,{xx}}\\ \quad +\displaystyle \frac{3}{8}(C-E){\sigma }_{A,{xx}}+{\sigma }_{E,{xt}}-\displaystyle \frac{3}{8}(3B+A){\sigma }_{E,{xx}}\\ \quad +{\sigma }_{E,{yy}}+\displaystyle \frac{3}{8}(A-B){\sigma }_{C,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3}{4}(3{\sigma }_{F,x}+{\sigma }_{G,x}){A}_{x}+\displaystyle \frac{3}{4}({\sigma }_{F,x}-{\sigma }_{G,x}){B}_{x}\\ \quad +\displaystyle \frac{3}{4}({\sigma }_{B,x}+3{\sigma }_{A,x}){F}_{x}\\ \quad -\displaystyle \frac{3}{4}({\sigma }_{B,x}-{\sigma }_{A,x}){G}_{x}+\displaystyle \frac{3}{8}(3{\sigma }_{F}+{\sigma }_{G}){A}_{{xx}}\\ +\displaystyle \frac{3}{8}({\sigma }_{F}-{\sigma }_{G}){B}_{{xx}}+\displaystyle \frac{3}{8}(3{\sigma }_{A}+{\sigma }_{B}){F}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{A}-{\sigma }_{B}){G}_{{xx}}\\ \quad -\displaystyle \frac{1}{4}{\sigma }_{F,{xxxx}}+\displaystyle \frac{3}{8}(3F+G){\sigma }_{A,{xx}}+\displaystyle \frac{3}{8}(F-G){\sigma }_{B,{xx}}\\ -{\sigma }_{F,{xt}}+\displaystyle \frac{3}{8}(3A+B){\sigma }_{F,{xx}}-{\sigma }_{F,{yy}}+\displaystyle \frac{3}{8}(A-B){\sigma }_{G,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{3}{4}(3{\sigma }_{G,x}+{\sigma }_{F,x}){B}_{x}+\displaystyle \frac{3}{4}({\sigma }_{G,x}-{\sigma }_{F,x}){A}_{x}\\ \quad +\displaystyle \frac{3}{4}({\sigma }_{A,x}+3{\sigma }_{B,x}){G}_{x}\\ \quad -\displaystyle \frac{3}{4}({\sigma }_{A,x}-{\sigma }_{B,x}){F}_{x}+\displaystyle \frac{3}{8}(3{\sigma }_{G}+{\sigma }_{F}){B}_{{xx}}\\ +\displaystyle \frac{3}{8}({\sigma }_{G}-{\sigma }_{F}){A}_{{xx}}+\displaystyle \frac{3}{8}(3{\sigma }_{B}+{\sigma }_{A}){G}_{{xx}}\\ \quad +\displaystyle \frac{3}{8}({\sigma }_{B}-{\sigma }_{A}){F}_{{xx}}-\displaystyle \frac{1}{4}{\sigma }_{G,{xxxx}}\\ \quad +\displaystyle \frac{3}{8}(3G+F){\sigma }_{B,{xx}}+\displaystyle \frac{3}{8}(G-F){\sigma }_{A,{xx}}\\ -{\sigma }_{G,{xt}}+\displaystyle \frac{3}{8}(3B+A){\sigma }_{G,{xx}}-{\sigma }_{G,{yy}}+\displaystyle \frac{3}{8}(B-A){\sigma }_{F,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\sigma }_{A,{xt}}-\displaystyle \frac{3}{4}({\sigma }_{B,x}+3{\sigma }_{A,x}){A}_{x}\\ \quad +\displaystyle \frac{3}{4}({\sigma }_{B,x}-{\sigma }_{A,x}){B}_{x}\\ \quad -\displaystyle \frac{3}{8}(3{\sigma }_{F,x}+{\sigma }_{G,x}){C}_{x}+\displaystyle \frac{3}{8}({\sigma }_{G,x}-{\sigma }_{F,x}){E}_{x}\\ \quad -\displaystyle \frac{3}{8}({\sigma }_{E,x}+3{\sigma }_{C,x}){F}_{x}+\displaystyle \frac{3}{8}({\sigma }_{E,x}-{\sigma }_{C,x}){G}_{x}-\displaystyle \frac{3}{8}(3{\sigma }_{A}\\ \quad +{\sigma }_{B}){A}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{B}-{\sigma }_{A}){B}_{{xx}}-\displaystyle \frac{3}{16}(3{\sigma }_{F}+{\sigma }_{G}){C}_{{xx}}\\ \quad +\displaystyle \frac{3}{16}({\sigma }_{G}-{\sigma }_{F}){E}_{{xx}}-\displaystyle \frac{3}{16}({\sigma }_{E}+3{\sigma }_{C}){F}_{{xx}}\\ \quad -\displaystyle \frac{3}{16}({\sigma }_{C}-{\sigma }_{E}){G}_{{xx}}+\displaystyle \frac{1}{4}{\sigma }_{A,{xxxx}}-\displaystyle \frac{3}{8}(3A+B){\sigma }_{A,{xx}}+{\sigma }_{A,{yy}}\\ \quad +\displaystyle \frac{3}{8}(B-A){\sigma }_{B,{xx}}-\displaystyle \frac{3}{16}(3F+G){\sigma }_{C,{xx}}\\ \quad +\displaystyle \frac{3}{16}(G-F){\sigma }_{E,{xx}}\\ \quad -\displaystyle \frac{3}{16}(E+3C){\sigma }_{F,{xx}}-\displaystyle \frac{3}{16}(C-E){\sigma }_{G,{xx}}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\sigma }_{B,{xt}}-\displaystyle \frac{3}{4}({\sigma }_{A,x}+3{\sigma }_{B,x}){B}_{x}+\displaystyle \frac{3}{4}({\sigma }_{A,x}-{\sigma }_{B,x}){A}_{x}\\ \quad -\displaystyle \frac{3}{8}(3{\sigma }_{G,x}+{\sigma }_{F,x}){E}_{x}+\displaystyle \frac{3}{8}({\sigma }_{F,x}-{\sigma }_{G,x}){C}_{x}\\ -\displaystyle \frac{3}{8}({\sigma }_{C,x}+3{\sigma }_{E,x}){G}_{x}+\displaystyle \frac{3}{8}({\sigma }_{C,x}-{\sigma }_{E,x}){F}_{x}-\displaystyle \frac{3}{8}(3{\sigma }_{B}\\ \quad +{\sigma }_{A}){B}_{{xx}}+\displaystyle \frac{3}{8}({\sigma }_{A}-{\sigma }_{B}){A}_{{xx}}-\displaystyle \frac{3}{16}(3{\sigma }_{G}+{\sigma }_{F}){E}_{{xx}}\\ \quad +\displaystyle \frac{3}{16}({\sigma }_{F}-{\sigma }_{G}){C}_{{xx}}-\displaystyle \frac{3}{16}({\sigma }_{C}+3{\sigma }_{E}){G}_{{xx}}\\ \quad -\displaystyle \frac{3}{16}({\sigma }_{E}-{\sigma }_{C}){F}_{{xx}}+\displaystyle \frac{1}{4}{\sigma }_{B,{xxxx}}-\displaystyle \frac{3}{8}(3B+A){\sigma }_{B,{xx}}+{\sigma }_{B,{yy}}\\ \quad +\displaystyle \frac{3}{8}(A-B){\sigma }_{A,{xx}}-\displaystyle \frac{3}{16}(3G+F){\sigma }_{E,{xx}}\\ \quad +\displaystyle \frac{3}{16}(F-G){\sigma }_{C,{xx}}\\ \quad -\displaystyle \frac{3}{16}(C+3E){\sigma }_{G,{xx}}-\displaystyle \frac{3}{16}(E-C){\sigma }_{F,{xx}}=0,\end{array}\end{eqnarray}$
and also the nonlocal condition $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{A} & = & {\sigma }_{B}(-x+{x}_{0},y,-t+{t}_{0}),\\ {\sigma }_{C} & = & {\sigma }_{E}(-x+{x}_{0},y,-t+{t}_{0}),\\ {\sigma }_{F} & = & {\sigma }_{G}(-x+{x}_{0},y,-t+{t}_{0}).\end{array}\end{eqnarray}$
By substituting equations (31 ) into equation (32 ) and eliminating Axt, Bxt, Cxt, Ext, Fxt, Gxt by the ncKPII system, we obtain a system of the functions X, Y, T, Γ1, Γ2, Γ3, Λ1, Λ2, Λ3. By vanishing all independent partial derivatives of variables A, B, C, E, F, G we obtain a system of over determined linear equations, which can be solved by software like maple. After considering the nonlocal relation of equation (32g ), we have22 ). So the explicit expressions of equation (31 ) are33 ) under the condition σA = σB = σC = σE = σF = σG = 0, which is equivalent to solving the characteristic equation34 ) we get symmetry reduction solutions of the ncKPII system41 ), m1, m2, m3 are arbitrary functions of t which related to f1, f2, f3 by35 )–(40 ) into the ncKPII system, we get corresponding symmetry reduction equations
$\begin{eqnarray*}\begin{array}{rcl}X & = & \displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3},\\ Y & = & \displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2},T={f}_{1},\\ {{\rm{\Gamma }}}_{1} & = & \displaystyle \frac{1}{9}\dddot{{f}_{1}}{y}^{2}-\displaystyle \frac{2}{3}\dot{{f}_{1}}A-\displaystyle \frac{2}{9}\ddot{{f}_{1}}x+\displaystyle \frac{1}{3}\ddot{{f}_{2}}y-\displaystyle \frac{2}{3}\dot{{f}_{3}},\\ {{\rm{\Lambda }}}_{1} & = & \displaystyle \frac{1}{9}\dddot{{f}_{1}}{y}^{2}-\displaystyle \frac{2}{3}\dot{{f}_{1}}B-\displaystyle \frac{2}{9}\ddot{{f}_{1}}x\\ & & +\displaystyle \frac{1}{3}\ddot{{f}_{2}}y-\displaystyle \frac{2}{3}\dot{{f}_{3}},\\ {{\rm{\Gamma }}}_{2} & = & -\displaystyle \frac{2}{3}C\dot{{f}_{1}}-{e}_{1}E-{e}_{1}C,{{\rm{\Lambda }}}_{2}=-\displaystyle \frac{2}{3}\dot{{f}_{1}}E+{e}_{1}C+{e}_{1}E,\\ {{\rm{\Gamma }}}_{3} & = & -\displaystyle \frac{2}{3}\dot{{f}_{1}}F+{e}_{1}G+{e}_{1}F,\\ {{\rm{\Lambda }}}_{3} & = & -\displaystyle \frac{2}{3}\dot{{f}_{1}}G-{e}_{1}G-{e}_{1}F,\end{array}\end{eqnarray*}$
where f1, f2, f3 are arbitrary functions of t satisfying the condition of ( $\begin{eqnarray}\begin{array}{rcl}{\sigma }_{A} & = & {f}_{1}{A}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){A}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){A}_{y}-\displaystyle \frac{1}{9}\dddot{{f}_{1}}{y}^{2}+\displaystyle \frac{2}{3}\dot{{f}_{1}}A\\ & & +\displaystyle \frac{2}{9}\ddot{{f}_{1}}x-\displaystyle \frac{1}{3}\ddot{{f}_{2}}y+\displaystyle \frac{2}{3}\dot{{f}_{3}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{B} & = & {f}_{1}{B}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){B}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){B}_{y}-\displaystyle \frac{1}{9}\dddot{{f}_{1}}{y}^{2}+\displaystyle \frac{2}{3}\dot{{f}_{1}}B\\ & & +\displaystyle \frac{2}{9}\ddot{{f}_{1}}x-\displaystyle \frac{1}{3}\ddot{{f}_{2}}y+\displaystyle \frac{2}{3}\dot{{f}_{3}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{C} & = & {f}_{1}{C}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){C}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){C}_{y}+\displaystyle \frac{2}{3}C\dot{{f}_{1}}+{e}_{1}(C+E),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{E} & = & {f}_{1}{E}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){E}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){E}_{y}+\displaystyle \frac{2}{3}E\dot{{f}_{1}}\\ & & -{e}_{1}(C+E),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{F} & = & {f}_{1}{F}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){F}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){F}_{y}+\displaystyle \frac{2}{3}F\dot{{f}_{1}}-{e}_{1}(F+G),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\sigma }_{G} & = & {f}_{1}{G}_{t}+\left(\displaystyle \frac{1}{3}\dot{{f}_{1}}x-\displaystyle \frac{1}{6}\ddot{{f}_{1}}{y}^{2}-\displaystyle \frac{1}{2}\dot{{f}_{2}}y+{f}_{3}\right){G}_{x}\\ & & +\left(\displaystyle \frac{2}{3}\dot{{f}_{1}}y+{f}_{2}\right){G}_{y}+\displaystyle \frac{2}{3}G\dot{{f}_{1}}\\ & & +{e}_{1}(F+G).\end{array}\end{eqnarray}$
Group invariant solutions of the ncKPII system can be obtained by solving equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{dx}}{\tfrac{1}{3}\dot{{f}_{1}}x-\tfrac{1}{6}\ddot{{f}_{1}}{y}^{2}-\tfrac{1}{2}\dot{{f}_{2}}y+{f}_{3}}=\displaystyle \frac{{dt}}{{f}_{1}}\\ \quad =\displaystyle \frac{{dA}}{\tfrac{1}{9}\dddot{{f}_{1}}{y}^{2}-\tfrac{2}{3}\dot{{f}_{1}}A-\tfrac{2}{9}\ddot{{f}_{1}}x+\tfrac{1}{3}\ddot{{f}_{2}}y-\tfrac{2}{3}\dot{{f}_{3}}}\\ =\displaystyle \frac{{dB}}{\tfrac{1}{9}\dddot{{f}_{1}}{y}^{2}-\tfrac{2}{3}\dot{{f}_{1}}B-\tfrac{2}{9}\ddot{{f}_{1}}x+\tfrac{1}{3}\ddot{{f}_{2}}y-\tfrac{2}{3}\dot{{f}_{3}}}\\ \quad =\displaystyle \frac{{dC}}{-\tfrac{2}{3}C\dot{{f}_{1}}-{e}_{1}E-{e}_{1}C}\\ \quad =\displaystyle \frac{{dE}}{-\tfrac{2}{3}\dot{{f}_{1}}E+{e}_{1}C+{e}_{1}E}\\ \quad =\displaystyle \frac{{dF}}{-\tfrac{2}{3}\dot{{f}_{1}}F+{e}_{1}G+{e}_{1}F}\\ \quad =\displaystyle \frac{{dG}}{-\tfrac{2}{3}\dot{{f}_{1}}G-{e}_{1}G-{e}_{1}F}.\end{array}\end{eqnarray}$
After solving equation ( $\begin{eqnarray}\begin{array}{rcl}A & = & -\displaystyle \frac{{\left({m}_{1}+\eta \right)}^{2}\dddot{{m}_{3}}}{9{\dot{{m}_{3}}}^{\tfrac{7}{3}}}+\displaystyle \frac{({m}_{1}+\eta )\ddot{{m}_{1}}}{3{\dot{{m}_{3}}}^{\tfrac{4}{3}}}\\ & & +\left[-\displaystyle \frac{4({m}_{1}+\eta )\dot{{m}_{1}}}{9{\dot{{m}_{3}}}^{\tfrac{7}{3}}}+\displaystyle \frac{{m}_{2}+2\xi }{9{\dot{{m}_{3}}}^{\tfrac{4}{3}}}\right]\ddot{{m}_{3}}\\ & & +\displaystyle \frac{5{\left({m}_{1}+\eta \right)}^{2}{\ddot{{m}_{3}}}^{2}}{27{\dot{{m}_{3}}}^{\tfrac{10}{3}}}+\displaystyle \frac{{\dot{{m}_{1}}}^{2}}{6{\dot{{m}_{3}}}^{\tfrac{4}{3}}}-\displaystyle \frac{\dot{{m}_{2}}}{3{\dot{{m}_{3}}}^{\tfrac{1}{3}}}+{\dot{{m}_{3}}}^{\tfrac{2}{3}}{A}_{1},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}B & = & -\displaystyle \frac{{\left({m}_{1}+\eta \right)}^{2}\dddot{{m}_{3}}}{9{\dot{{m}_{3}}}^{\tfrac{7}{3}}}+\displaystyle \frac{({m}_{1}+\eta )\ddot{{m}_{1}}}{3{\dot{{m}_{3}}}^{\tfrac{4}{3}}}\\ & & +\left[-\displaystyle \frac{4({m}_{1}+\eta )\dot{{m}_{1}}}{9{\dot{{m}_{3}}}^{\tfrac{7}{3}}}+\displaystyle \frac{{m}_{2}+2\xi }{9{\dot{{m}_{3}}}^{\tfrac{4}{3}}}\right]\ddot{{m}_{3}}\\ & & +\displaystyle \frac{5{\left({m}_{1}+\eta \right)}^{2}{\ddot{{m}_{3}}}^{2}}{27{\dot{{m}_{3}}}^{\tfrac{10}{3}}}+\displaystyle \frac{{\dot{{m}_{1}}}^{2}}{6{\dot{{m}_{3}}}^{\tfrac{4}{3}}}\\ & & -\displaystyle \frac{\dot{{m}_{2}}}{3{\dot{{m}_{3}}}^{\tfrac{1}{3}}}+{\dot{{m}_{3}}}^{\tfrac{2}{3}}{B}_{1},\end{array}\end{eqnarray}$
$\begin{eqnarray}C=-{\dot{{m}_{3}}}^{\displaystyle \frac{2}{3}}[({e}_{1}{m}_{3}-1){C}_{1}+{E}_{1}],\end{eqnarray}$
$\begin{eqnarray}E={\dot{{m}_{3}}}^{\displaystyle \frac{2}{3}}({e}_{1}{m}_{3}{C}_{1}+{E}_{1}),\end{eqnarray}$
$\begin{eqnarray}F={\dot{{m}_{3}}}^{\displaystyle \frac{2}{3}}({e}_{1}{m}_{3}{G}_{1}+{F}_{1}),\end{eqnarray}$
$\begin{eqnarray}G=-{\dot{{m}_{3}}}^{\displaystyle \frac{2}{3}}[({e}_{1}{m}_{3}-1){G}_{1}+{F}_{1}],\end{eqnarray}$
where A1, B1, C1, E1, F1, G1 are invariant functions of two new invariant variables $\begin{eqnarray}\begin{array}{l}\xi =\displaystyle \frac{\dot{{m}_{1}}}{2{\dot{{m}_{3}}}^{\tfrac{1}{3}}}y+{\dot{{m}_{3}}}^{\tfrac{1}{3}}x-\displaystyle \frac{\ddot{{m}_{3}}}{6{\left(\dot{{m}_{3}}\right)}^{\tfrac{2}{3}}}{y}^{2}\\ \quad -\displaystyle \frac{1}{2}{m}_{2},\eta =-{m}_{1}+{\dot{{m}_{3}}}^{-\tfrac{2}{3}}y.\end{array}\end{eqnarray}$
In equation ( $\begin{eqnarray}{f}_{1}=\displaystyle \frac{1}{\dot{{m}_{3}}},{f}_{2}=\displaystyle \frac{\dot{{m}_{1}}}{{\dot{{m}_{3}}}^{\tfrac{5}{3}}},{f}_{3}=-\displaystyle \frac{{\dot{{m}_{1}}}^{2}}{2{\dot{{m}_{3}}}^{\tfrac{7}{3}}}+\displaystyle \frac{\dot{{m}_{2}}}{2{\dot{{m}_{3}}}^{\tfrac{4}{3}}},\end{eqnarray}$
and satisfy $\begin{eqnarray}\hat{{T}_{d}}\{{m}_{1},{m}_{2},{m}_{3}\}=\{{m}_{1},-{m}_{2},-{m}_{3}\}.\end{eqnarray}$
By substituting equations ( $\begin{eqnarray}\begin{array}{l}-18{A}_{1,\xi }^{2}-12{A}_{1,\xi }{B}_{1,\xi }+6{B}_{1,\xi }^{2}-6({G}_{1,\xi }+2{F}_{1,\xi }){C}_{1,\xi }\\ \quad +12{G}_{1,\xi }{E}_{1,\xi }+16{A}_{1,\eta \eta }-6({B}_{1}+3{A}_{1}){A}_{1,\xi \xi }\\ \quad +6({B}_{1}-{A}_{1}){B}_{1,\xi \xi }-3({G}_{1}+2{F}_{1}){C}_{1,\xi \xi }+6{E}_{1,\xi \xi }{G}_{1}\\ \quad -6{F}_{1,\xi \xi }{C}_{1}+3(2{E}_{1}-{C}_{1}){G}_{1,\xi \xi }+4{A}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}-18{B}_{1,\xi }^{2}-12{A}_{1,\xi }{B}_{1,\xi }+6{A}_{1,\xi }^{2}-6({G}_{1,\xi }-2{F}_{1,\xi }){C}_{1,\xi }\\ \quad -12{G}_{1,\xi }{E}_{1,\xi }+16{B}_{1,\eta \eta }-6({B}_{1}-{A}_{1}){A}_{1,\xi \xi }\\ \quad -6(3{B}_{1}+{A}_{1}){B}_{1,\xi \xi }-3({G}_{1}-2{F}_{1}){C}_{1,\xi \xi }-6{E}_{1,\xi \xi }{G}_{1}\\ \quad +6{F}_{1,\xi \xi }{C}_{1}-3(2{E}_{1}+{C}_{1}){G}_{1,\xi \xi }+4{B}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left[6(2{e}_{1}{m}_{3}-1){C}_{1,\xi }+12{E}_{1,\xi }\right]{A}_{1,\xi }+[6(2{m}_{3}{e}_{1}-3){C}_{1,\xi }\\ \quad +12{E}_{1,\xi }]{B}_{1,\xi }-8{C}_{1,\xi }{e}_{1}\\ \quad +3(2{C}_{1}{e}_{1}{m}_{3}+2{E}_{1}-3{C}_{1}){A}_{1,\xi \xi }+3(2{C}_{1}{e}_{1}{m}_{3}-{C}_{1}\\ \quad +2{E}_{1}){B}_{1,\xi \xi }-8({m}_{3}{e}_{1}-1){C}_{1,\eta \eta }\\ \quad +3(2{B}_{1}{m}_{3}{e}_{1}+2{A}_{1}{m}_{3}{e}_{1}-{B}_{1}-3{A}_{1}){C}_{1,\xi \xi }-8{E}_{1,\eta \eta }\\ \quad +6({B}_{1}+{A}_{1}){E}_{1,\xi \xi }-2({m}_{3}{e}_{1}-1){C}_{1,\xi \xi \xi \xi }\\ \quad -2{E}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\left[6(2{e}_{1}{m}_{3}+1){C}_{1,\xi }+12{E}_{1,\xi }\right]{A}_{1,\xi }\\ \quad +[6(2{m}_{3}{e}_{1}-1){C}_{1,\xi }+12{E}_{1,\xi }]{B}_{1,\xi }-8{C}_{1,\xi }{e}_{1}\\ \quad +3(2{C}_{1}{e}_{1}{m}_{3}+2{E}_{1}-{C}_{1}){A}_{1,\xi \xi }+3(2{C}_{1}{e}_{1}{m}_{3}\\ \quad +{C}_{1}+2{E}_{1}){B}_{1,\xi \xi }-8{m}_{3}{e}_{1}{C}_{1,\eta \eta }\\ \quad +3(2{B}_{1}{m}_{3}{e}_{1}+2{A}_{1}{m}_{3}{e}_{1}+{B}_{1}-{A}_{1}){C}_{1,\xi \xi }-8{E}_{1,\eta \eta }\\ \quad +6({B}_{1}+{A}_{1}){E}_{1,\xi \xi }-2{m}_{3}{e}_{1}{C}_{1,\xi \xi \xi \xi }-2{E}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}6[2{F}_{1,\xi }+(2{m}_{3}{e}_{1}+1){G}_{1,\xi }]{A}_{1,\xi }\\ \quad +6[2{F}_{1,\xi }+(2{m}_{3}{e}_{1}-1){G}_{1,\xi }]{B}_{1,\xi }-8{G}_{1,\xi }{e}_{1}\\ \quad +3(2{G}_{1}{e}_{1}{m}_{3}+{G}_{1}+2{F}_{1}){A}_{1,\xi \xi }\\ \quad +3(2{G}_{1}{e}_{1}{m}_{3}-{G}_{1}+2{F}_{1}){B}_{1,\xi \xi }-8{F}_{1,\eta \eta }\\ \quad +6({B}_{1}+{A}_{1}){F}_{1,\xi \xi }-8{G}_{1,\eta \eta }{m}_{3}{e}_{1}\\ \quad +3(2{A}_{1}{m}_{3}{e}_{1}+2{B}_{1}{m}_{3}{e}_{1}+{A}_{1}-{B}_{1}){G}_{1,\xi \xi }\\ \quad -2{G}_{1,\xi \xi \xi \xi }{e}_{1}{m}_{3}-2{F}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}6[2{F}_{1,\xi }+(2{m}_{3}{e}_{1}-1){G}_{1,\xi }]{A}_{1,\xi }+6[2{F}_{1,\xi }\\ \quad +(2{m}_{3}{e}_{1}-3){G}_{1,\xi }]{B}_{1,\xi }-8{G}_{1,\xi }{e}_{1}\\ +3(2{G}_{1}{e}_{1}{m}_{3}-{G}_{1}+2{F}_{1}){A}_{1,\xi \xi }\\ \quad +3(2{G}_{1}{e}_{1}{m}_{3}-3{G}_{1}+2{F}_{1}){B}_{1,\xi \xi }-8{F}_{1,\eta \eta }\\ +6({B}_{1}+{A}_{1}){F}_{1,\xi \xi }-8({m}_{3}{e}_{1}-1){G}_{1,\eta \eta }\\ \quad +3(2{A}_{1}{m}_{3}{e}_{1}+2{B}_{1}{m}_{3}{e}_{1}-{A}_{1}-3{B}_{1}){G}_{1,\xi \xi }\\ -2({e}_{1}{m}_{3}-1){G}_{1,\xi \xi \xi \xi }-2{F}_{1,\xi \xi \xi \xi }=0,\end{array}\end{eqnarray}$
along with $\begin{eqnarray*}\begin{array}{l}{A}_{1}(-\xi ,\eta )={B}_{1}(\xi ,\eta ),{C}_{1}(-\xi ,\\ \eta )={C}_{1}(\xi ,\eta ),{E}_{1}(-\xi ,\eta )={C}_{1}(\xi ,\eta )-{E}_{1}(\xi ,\eta ),\\ {F}_{1}(-\xi ,\eta )={G}_{1}(\xi ,\eta )-{F}_{1}(\xi ,\eta ),\\ {G}_{1}(\xi ,\eta )={G}_{1}(-\xi ,\eta ).\end{array}\end{eqnarray*}$
4. Summary
In summary, a nonlocal coupled KP system is introduced and studied by converting it into a localized system. Via this method, new solutions of the ncKP system are generated from known ones of the cKP system. An even number of singular soliton solutions are obtained in a general form, among which N = 2 and N = 4 soliton solutions are plotted and analyzed. By fixing appropriate parameters, soliton solutions of the ncKPI system become breathers and we also attained lump-type solutions. The standard Lie symmetry method is carried on the ncKPII system to obtain symmetry reduction solutions.