1. Introduction
Recently, ${ \mathcal P }{ \mathcal T }$-symmetric integrable nonlocal nonlinear Schrödinger equation
$\begin{eqnarray}{\rm{i}}{q}_{t}(x,t)+{q}_{{xx}}(x,t)\pm {q}^{2}(x,t){q}^{* }(-x,t)=0,\end{eqnarray}$
was proposed by Ablowitz and Musslimani [1], which is reduced from the second-order Ablowitz–Kaup–Newell–Segur (AKNS) system with reduction $r(x,t)=\mp {q}^{* }(-x,t)$. It is ${ \mathcal P }{ \mathcal T }$-symmetric because of the potential $V(x,t)\,=\pm q(x,t){q}^{* }(-x,t)={V}^{* }(-x,t)$ (see [2, 3]). Since then, more and more integrable nonlocal systems, such as nonlocal Davey–Stewartson model, nonlocal Korteweg–de Vries model and so forth, were proposed (e.g. [4–9]). Classical solving methods, such as the inverse scattering transform and Darboux transformation (e.g. [1, 5, 10–14]), have been successfully used to find solutions of nonlocal integrable systems. However, the bilinear method cannot be directly applied to the nonlocal case as it is difficult to define a ‘nonlocal’ Hirota’s bilinear operator. Recently, Chen et al [15–18] developed a reduction technique of double Wronskians on unreduced bilinear equations to obtain solutions to those reduced equations, including nonlocal ones. One advantage of this method is that one only needs to make use of the solutions (most of them are known) of the unreduced bilinear equations. The other advantage is one can obtain N-soliton solutions for the whole hierarchy, rather than implementing reductions solution by solution and equation by equation (see [19]).The (2+1)-dimensional breaking solitons equations ware first systematically investigated by Bogoyavlenskii [20–26]. Such models can be used to describe two-dimensional interaction of a Riemann wave with transverse long waves (see [20, 21]). Mathematically, these equations can be constructed based on (1+1)-D Lax pairs by imposing an evolution in y-direction in both spectral parameters and eigenfunctions (see [21, 24]). The (2+1)-D breaking AKNS(BAKNS in brief) equations were first proposed in [24], and has been investigated from many aspects (e.g. [27–31]). And recently, bilinear form and double Wronskian solutions of (2+1)-D BAKNS hierarchy were worked out [32]. In this paper, we will take into account of bilinearization-reduction technique and derive double Wronskian solutions for the classical and nonlocal (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy. Note that the negative order AKNS equations also exhibits interesting dynamical behaviors (see [33]).
This paper is organized as follows. In section 2 we recall the unreduced (2+1)-D BAKNS hierarchy, its bilinear form and double Wronskian solutions. In section 3 we present classical and nonlocal reductions of the (2+1)-D BAKNS hierarchy. In section 4 , solutions in double Wronskian form of the reduced (2+1)-D BAKNS hierarchies are derived by means of the reduction technique from those of the unreduced hierarchy. Section 5 is contributed to the negative order AKNS hierarchy. Finally, section 6 consists of concluding remarks.
2. Unreduced (2+1)-D BAKNS hierarchy and solutions
Let us recall the unreduced (2+1)-D BAKNS hierarchy, its bilinear form and double Wronskian solutions.
The unreduced (2+1)-D BAKNS hierarchy reads [27]
$\begin{eqnarray}{u}_{{t}_{n}}={K}_{n}=\left(\begin{array}{c}{K}_{1,n}\\ {K}_{2,n}\end{array}\right)={L}^{n}{u}_{y},\,\,n=1,2,3,\cdot \cdot \cdot ,\end{eqnarray}$
where $u={(q,r)}^{{\rm{T}}}$, L is a recursion operator $\begin{eqnarray}L=\left(\begin{array}{cc}-{\partial }_{x}+2q{\partial }_{x}^{-1}r & 2q{\partial }_{x}^{-1}q\\ -2r{\partial }_{x}^{-1}r & {\partial }_{x}-2r{\partial }_{x}^{-1}q\end{array}\right),\end{eqnarray}$
in which ${\partial }_{x}=\tfrac{\partial }{{\partial }_{x}}$ and ${\partial }_{x}{\partial }_{x}^{-1}={\partial }_{x}^{-1}{\partial }_{x}=1$. This hierarchy is related to the AKNS spectral problem [24] $\begin{eqnarray}{\phi }_{x}=\left(\begin{array}{cc}\lambda & q\\ r & -\lambda \end{array}\right)\phi ,\end{eqnarray}$
and $\begin{eqnarray*}{\phi }_{{t}_{n}}={\left(2\lambda \right)}^{n}{\phi }_{y}+\left(\begin{array}{cc}{A}_{n} & {B}_{n}\\ {C}_{n} & -{A}_{n}\end{array}\right)\phi ,\end{eqnarray*}$
where ${\lambda }_{{t}_{n}}={\left(2\lambda \right)}^{n}{\lambda }_{y}$ and $\begin{eqnarray*}\begin{array}{rcl}{A}_{n} & = & {\partial }^{-1}(r,q){\left(-{B}_{n},{C}_{n}\right)}^{{\rm{T}}},\,\,{\left(-{B}_{n},{C}_{n}\right)}^{{\rm{T}}}\\ & = & \sum _{j=1}^{n}{2}^{n-j}{L}^{j-1}{\left({q}_{y},{r}_{y}\right)}^{{\rm{T}}}{\lambda }^{n-j}.\end{array}\end{eqnarray*}$
The hierarchy (2 ) can be rewritten as
$\begin{eqnarray}{q}_{{t}_{n}}=-{q}_{x,{t}_{n-1}}+2q{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{n-1}},\end{eqnarray}$
$\begin{eqnarray}{r}_{{t}_{n}}={r}_{x,{t}_{n-1}}-2r{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{n-1}}\end{eqnarray}$
with n = 1, 2, 3, ⋯ and setting t0 = y. Introducing transformation $\begin{eqnarray}q=\displaystyle \frac{g}{f},\,\,r=-\displaystyle \frac{h}{f},\end{eqnarray}$
the hierarchy is written as the following bilinear form [32] $\begin{eqnarray}({D}_{{t}_{n}}+{D}_{x}{D}_{{t}_{n}-1})g\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{n}}-{D}_{x}{D}_{{t}_{n}-1})h\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}{D}_{x}^{2}f\cdot f=2{gh},\end{eqnarray}$
for n ≥ 1, where Hirota’s bilinear operator D is defined as [34] $\begin{eqnarray*}\begin{array}{l}{D}_{x}^{m}{D}_{y}^{n}f(x,y)\cdot g(x,y)\\ \quad =\,{\left({\partial }_{x}-{\partial }_{x^{\prime} }\right)}^{m}{\left({\partial }_{y}-{\partial }_{y^{\prime} }\right)}^{n}f(x,y)g(x^{\prime} ,y^{\prime} ){| }_{x^{\prime} =x,y^{\prime} =y}.\end{array}\end{eqnarray*}$
System (
$\begin{eqnarray}f={W}^{n+1,m+1}(\varphi ,\psi )=| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(m)}| ,\end{eqnarray}$
$\begin{eqnarray}g=2{W}^{n+2,m}(\varphi ,\psi )=2| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(m-1)}| ,\end{eqnarray}$
$\begin{eqnarray}h=2{W}^{n,m+2}(\varphi ,\psi )=2| {\widehat{\varphi }}^{(n-1)};{\widehat{\psi }}^{(m+1)}| .\end{eqnarray}$
Here, $\varphi $ and $\psi $ are respectively $(n+m+2)$th order column vectors $\begin{eqnarray}\varphi ={\left({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{n+m+2}\right)}^{{\rm{T}}},\,\,\psi ={\left({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n+m+2}\right)}^{{\rm{T}}}\end{eqnarray}$
defined by $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \exp \left(-{Ax}+{A}^{2}y+\sum _{j=1}^{\infty }{2}^{j}{A}^{j+2}{t}_{j}\right){C}^{+},\\ \psi & = & \exp \left({Ax}-{A}^{2}y-\sum _{j=1}^{\infty }{2}^{j}{A}^{j+2}{t}_{j}\right){C}^{-},\end{array}\end{eqnarray}$
where $A$ is a $(n+m+2)\times (n+m+2)$ constant matrix in ${{\mathbb{C}}}_{(m+n+2)\times (m+n+2)}$ and $\begin{eqnarray}{C}^{\pm }={\left({c}_{1}^{\pm },{c}_{2}^{\pm },\cdots ,{c}_{n+m+2}^{\pm }\right)}^{{\rm{T}}},\,\,{c}_{i}^{\pm }\in {\mathbb{C}},\end{eqnarray}$
${\widehat{\varphi }}^{(n)}$ and ${\widehat{\psi }}^{(m)}$ respectively denote $(n+m+2)\times (n+1)$ and $(n+m+2)\times (m+1)$ Wronski matrices $\begin{eqnarray}\begin{array}{rcl}{\widehat{\varphi }}^{(n)} & = & \left(\varphi ,{\partial }_{x}\varphi ,{\partial }_{x}^{2}\varphi ,\cdots ,{\partial }_{x}^{n}\varphi \right),\\ {\widehat{\psi }}^{(m)} & = & \left(\psi ,{\partial }_{x}\psi ,{\partial }_{x}^{2}\psi ,\cdots ,{\partial }_{x}^{m}\psi \right).\end{array}\end{eqnarray}$
It is easy to find that for the odd members in the unreduced (2+1)-D BAKNS hierarchy (2 )6 ) and (8 ) where6 ) and (8 ) where
$\begin{eqnarray}{u}_{{t}_{2l+1}}={K}_{2l+1},\,\,l=0,1,2,\cdots ,\end{eqnarray}$
their solutions are given through ( $\begin{eqnarray}\displaystyle \begin{array}{rcl}\varphi & = & \exp \left(-{Ax}+{A}^{2}y+\sum _{j=0}^{\infty }{2}^{2j+1}{A}^{2j+3}{t}_{2j+1}\right){C}^{+},\\ \psi & = & \exp \left({Ax}-{A}^{2}y-\sum _{j=0}^{\infty }{2}^{2j+1}{A}^{2j+3}{t}_{2j+1}\right){C}^{-},\end{array}\end{eqnarray}$
and for the even members $\begin{eqnarray}{u}_{{t}_{2l}}={K}_{2l},\,\,l=1,2,\cdots ,\end{eqnarray}$
their solutions are given through ( $\begin{eqnarray}\displaystyle \begin{array}{rcl}\varphi & = & \exp \left(-{Ax}+{A}^{2}y+\sum _{j=1}^{\infty }{2}^{2j}{A}^{2j+2}{t}_{2j}\right){C}^{+},\\ \psi & = & \exp \left({Ax}-{A}^{2}y-\sum _{j=1}^{\infty }{2}^{2j}{A}^{2j+2}{t}_{2j}\right){C}^{-}.\end{array}\end{eqnarray}$
3. Reductions of the (2+1)-D BAKNS hierarchy
3.1. Classical reductions
The odd hierarchy (13 ) allows a classical complex reduction
$\begin{eqnarray}r(x,y,t)=\delta {q}^{* }(x,y,t),\,\,\delta =\pm 1,\,y\to {\rm{i}}y,\end{eqnarray}$
where i is the imaginary unit and ∗ stands for complex conjugate. And the representative one is $\begin{eqnarray}{q}_{{t}_{1}}={\rm{i}}{q}_{{xy}}-2{\rm{i}}\delta q{\partial }_{x}^{-1}{\left({{qq}}^{* }\right)}_{y},\,\,\delta =\pm 1.\end{eqnarray}$
The even hierarchy (15 ) allows classical complex reduction
$\begin{eqnarray}r(x,y,t)=\delta {q}^{* }(x,y,t)\,\,\delta =\pm 1,\,\,y\to {\rm{i}}y,{t}_{2l}\to {\rm{i}}{t}_{2l},\end{eqnarray}$
when l = 1, the corresponding equation is $\begin{eqnarray}\begin{array}{rcl}{q}_{{t}_{2}} & = & {q}_{{xxy}}-4\delta {q}^{* }{{qq}}_{y}-2\delta {q}_{x}{\partial }_{x}^{-1}{\left({{qq}}^{* }\right)}_{y}\\ & & -2\delta q{\partial }_{x}^{-1}({q}_{x}{q}_{y}^{* }-{q}_{x}^{* }{q}_{y}),\,\,\delta =\pm 1,\end{array}\end{eqnarray}$
3.2. Nonlocal reductions
For the odd hierarchy (13 ), first, it allows a real nonlocal reduction13 ) also allows a complex nonlocal reduction
$\begin{eqnarray}r(x,y,t)=\delta q(\sigma x,-y,\sigma t),\,\,\delta ,\sigma =\pm 1,\,\,x,y,t\in {\mathbb{R}},\end{eqnarray}$
where the simplest one-component equation reads $\begin{eqnarray}{q}_{{t}_{1}}=-{q}_{{xy}}+2\delta q{\partial }_{x}^{-1}{\left({qq}(\sigma x,-y,\sigma t\right)}_{y},\,\,\delta ,\sigma =\pm 1.\end{eqnarray}$
Equation ( $\begin{eqnarray}r(x,y,t)=\delta {q}^{* }(\sigma x,-y,\sigma t),\,\,\delta ,\sigma =\pm 1,\,\,x,y,t\in {\mathbb{R}},\end{eqnarray}$
and the resulted simplest one-component equation is $\begin{eqnarray}{q}_{{t}_{1}}=-{q}_{{xy}}+2\delta q{\partial }_{x}^{-1}{\left({{qq}}^{* }(\sigma x,-y,\sigma t\right)}_{y},\,\,\delta ,\sigma =\pm 1.\end{eqnarray}$
The even hierarchy (15 ) also allows two types of reductions. One is real27 ).
$\begin{eqnarray}r(x,y,t)=\delta q(\sigma x,-y,-t),\,\,\delta ,\sigma =\pm 1,\end{eqnarray}$
and the resulted simplest one-component equation is $\begin{eqnarray}\begin{array}{rcl}{q}_{{t}_{2}} & = & {q}_{{xxy}}-4\delta \widetilde{q}{{qq}}_{y}-2\delta {q}_{x}{\partial }_{x}^{-1}{\left(q\widetilde{q}\right)}_{y}\\ & & -2\delta q{\partial }_{x}^{-1}({q}_{x}{\widetilde{q}}_{y}-{\widetilde{q}}_{x}{q}_{y}),\,\,\delta ,\sigma =\pm 1,\end{array}\end{eqnarray}$
where $\begin{eqnarray}\widetilde{q}=q(\sigma x,-y,-t).\end{eqnarray}$
The other reduction is complex $\begin{eqnarray}r(x,y,t)=\delta {q}^{* }(\sigma x,-y,-t),\,\,\delta ,\sigma =\pm 1,\end{eqnarray}$
and the resulted simplest one-component equation is $\begin{eqnarray}\begin{array}{rcl}{q}_{{t}_{2}} & = & {q}_{{xxy}}-4\delta {\widetilde{q}}^{* }{{qq}}_{y}-2\delta {q}_{x}{\partial }_{x}^{-1}{\left(q{\widetilde{q}}^{* }\right)}_{y}\\ & & -2\delta q{\partial }_{x}^{-1}({q}_{x}{\widetilde{q}}_{y}^{* }-{\widetilde{q}}_{x}^{* }{q}_{y}),\,\,\delta ,\sigma =\pm 1,\end{array}\end{eqnarray}$
where $\widetilde{q}$ is given as (It is remarkable that the integration operator ${\partial }_{x}^{-1}$ should specially take the form5 ) with n = 1, one can handle it as the following. First, introduce
$\begin{eqnarray}{\partial }_{x}^{-1}=\displaystyle \frac{1}{2}\left({\int }_{-\infty }^{x}-{\int }_{x}^{+\infty }\right)\cdot {\rm{d}}x,\end{eqnarray}$
which has played an important role in nonlocal reduction involved with x (see [16, 18]). For example, for the integration term ${\partial }_{x}^{-1}{\left({qr}\right)}_{y}$ in ( $\begin{eqnarray*}z(x,y,t)={\partial }_{x}^{-1}{\left({qr}\right)}_{y}.\end{eqnarray*}$
Then we have $\begin{eqnarray*}\begin{array}{l}z(\sigma x,-y,\sigma t)\\ =\,\displaystyle \frac{-1}{2}\left({\displaystyle \int }_{-\infty }^{\sigma x}-{\displaystyle \int }_{\sigma x}^{+\infty }\right){\left[q(x,-y,\sigma t)r(x,-y,\sigma t)\right]}_{y}{\rm{d}}x\\ =\,\displaystyle \frac{-\sigma }{2}\left({\displaystyle \int }_{-\infty }^{x}-{\displaystyle \int }_{x}^{+\infty }\right){\left[q(\sigma x^{\prime} ,-y,\sigma t)r(\sigma x^{\prime} ,-y,\sigma t)\right]}_{y}{\rm{d}}x^{\prime} \\ =\,\displaystyle \frac{-\sigma }{2}\left({\displaystyle \int }_{-\infty }^{x}-{\displaystyle \int }_{x}^{+\infty }\right){\left[r(x^{\prime} ,y,t)q(x^{\prime} ,y,t)\right]}_{y}{\rm{d}}x^{\prime} \\ =\,-\sigma z(x,y,t),\end{array}\end{eqnarray*}$
which provides a clear expression for the integration term with nonlocal reduction.4. Reduction of solutions
4.1. Nonlocal cases
In the following we implement the reduction procedure on the double Wronskian solutions, by which solutions for the reduced nonlocal hierarchies can be obtained from those of the unreduced (2+1)-D BAKNS hierarchy (2 ) presented in theorem 1 .
4.1.1. Reduction (21 )
The nonlocal hierarchy
$\begin{eqnarray}{q}_{{t}_{2l+1}}={K}_{\mathrm{1,2}l+1}{| }_{(21)},\,\,l=0,1,2,\cdots \end{eqnarray}$
allows the following solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$(i.e. m = n in ( $\begin{eqnarray}\psi (x,y,t)=T\varphi (\sigma x,-y,\sigma t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+},\end{eqnarray}$
in which $T$ is a constant matrix determined through $\begin{eqnarray}{AT}+\sigma {TA}=0,\,\,\end{eqnarray}$
$\begin{eqnarray}{T}^{2}=\sigma \delta I,\,\,\sigma ,\delta =\pm 1.\end{eqnarray}$
It follows from (
$\begin{eqnarray*}\begin{array}{rcl}\psi (x,y,t) & = & \exp \left({Ax}-{A}^{2}y-\sum _{j=0}^{\infty }{2}^{2j+1}{A}^{2j+3}{t}_{2j+1}\right){C}^{-}\\ & = & \exp \left(-\sigma ({{TAT}}^{-1})x-{A}^{2}y\right.\\ & & \left.-\sum _{j=0}^{\infty }{2}^{2j+1}(-\sigma {{TA}}^{2j+3}{T}^{-1}){t}_{2j+1}\right){{TC}}^{+}\\ & = & T\exp \left(-A(\sigma x)+{A}^{2}(-y)\right.\\ & & \left.+\sum _{j=0}^{\infty }{2}^{2j+1}{A}^{2j+3}(\sigma {t}_{2j+1}\right){C}^{+}\\ & = & T\varphi (\sigma x,-y,\sigma t)^{\prime} ,\end{array}\end{eqnarray*}$
which gives rise to the the constraint ( $\begin{eqnarray}\begin{array}{rcl}f(x,y,t) & = & | {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| \\ & = & | {\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[x]};T{\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}g(x,y,t) & = & 2| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| \\ & = & 2| {\widehat{\varphi }}^{(n+1)}{\left(x,y,t\right)}_{[x]};T{\widehat{\varphi }}^{(n-1)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| ,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}h(x,y,t) & = & 2| {\widehat{\varphi }}^{(n-1)};{\widehat{\psi }}^{(n+1)}| \\ & = & 2| {\widehat{\varphi }}^{(n-1)}{\left(x,y,t\right)}_{[x]};T{\widehat{\varphi }}^{(n+1)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| ,\end{array}\end{eqnarray}$
where we employed a notation (see [15]) $\begin{eqnarray*}{\widehat{\varphi }}^{(n)}{\left({ax}\right)}_{[{bx}]}=\left(\varphi ({ax}),{\partial }_{{bx}}\varphi ({ax}),{\partial }_{{bx}}^{2}\varphi ({ax}),\cdots ,{\partial }_{{bx}}^{n}\varphi ({ax}\right).\end{eqnarray*}$
Next, with assumption ( $\begin{eqnarray*}\begin{array}{l}f(\sigma x,-y,\sigma t)\\ =\,| {\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[\sigma x]};T{\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[\sigma x]}| \\ =\,| T| | {T}^{-1}{\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[\sigma x]};{\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[\sigma x]}| \\ =\,{\left(-1\right)}^{{\left(n+1\right)}^{2}}| T| | {\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[x]};{T}^{-1}{\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| \\ =\,{\left(-1\right)}^{{\left(n+1\right)}^{2}}| T| | {\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[x]};\sigma \delta T{\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| \\ =\,{\left(-1\right)}^{{\left(n+1\right)}^{2}}{\left(\sigma \delta \right)}^{n+1}| T| | {\widehat{\varphi }}^{(n)}{\left(x,y,t\right)}_{[x]};T{\widehat{\varphi }}^{(n)}{\left(\sigma x,-y,\sigma t\right)}_{[x]}| \\ =\,{\left(-1\right)}^{{\left(n+1\right)}^{2}}{\left(\sigma \delta \right)}^{n+1}| T| f(x,y,t),\end{array}\end{eqnarray*}$
and in a similar way $\begin{eqnarray*}g(\sigma x,-y,\sigma t)={\left(-1\right)}^{(n+2)n}{\sigma }^{n+1}{\delta }^{n}| T| h(x,y,t),\end{eqnarray*}$
which then give rise to $\begin{eqnarray*}\begin{array}{rcl}\delta q(\sigma x,-y,\sigma t) & = & \delta \displaystyle \frac{g(\sigma x,-y,\sigma t)}{f(\sigma x,-y,\sigma t)}\\ & = & -\displaystyle \frac{h(x,y,t)}{f(x,y,t)}=r(x,y,t).\end{array}\end{eqnarray*}$
This is nothing but the reduction (Solutions of (34 ) can be written out by assuming T and A to be block matrices
$\begin{eqnarray}T=\left(\begin{array}{cc}{T}_{1} & {T}_{2}\\ {T}_{3} & {T}_{4}\end{array}\right),\,\,A=\left(\begin{array}{cc}{K}_{1} & 0\\ 0 & {K}_{4}\end{array}\right),\end{eqnarray}$
where Ti and Ki are (n + 1) × (n + 1) matrices and Ki is a complex matrix. We list them in table 1.One can prove that A and any of its similar forms lead to same solution (32 ). We only need to consider canonical form of A. When
$\begin{eqnarray}{{\bf{K}}}_{n+1}=\mathrm{Diag}({k}_{1},{k}_{2},\cdots ,{k}_{n+1}),\end{eqnarray}$
we get $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\zeta ({k}_{1})},{c}_{2}{{\rm{e}}}^{\zeta ({k}_{2})},\cdots ,{c}_{n+1}{{\rm{e}}}^{\zeta ({k}_{n+1})},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\zeta (-{k}_{1})},{d}_{2}{{\rm{e}}}^{\zeta (-{k}_{2})},\cdots ,{d}_{n+1}{{\rm{e}}}^{\zeta (-{k}_{n+1})}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
When ${{\bf{K}}}_{n+1}$ is a $(n+1)\times (n+1)$ Jordan matrix ${J}_{n+1}(k)$ $\begin{eqnarray}{J}_{n+1}(k)={\left(\begin{array}{cccc}k & 0 & \cdots & 0\\ 1 & k & \cdots & 0\\ \vdots & \ddots & \ddots & \vdots \\ 0 & \cdots & 1 & k\end{array}\right)}_{(n+1)\times (n+1)},\end{eqnarray}$
we get $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\zeta (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\zeta (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\zeta (k)}),\right.\\ & & {\left.d{{\rm{e}}}^{\zeta (-k)},\displaystyle \frac{{\partial }_{k}}{1!}(d{{\rm{e}}}^{\zeta (-k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(d{{\rm{e}}}^{\zeta (-k)}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\zeta ({k}_{i})=-{k}_{i}x+{k}_{i}^{2}y+\sum _{j=0}^{\infty }{2}^{2j+1}{k}_{i}^{2j+3}{t}_{2j+1}.\end{eqnarray}$
As examples, for equation (22 ) with different (σ, δ), its one-soliton solution (1SS) are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{4{cdk}{{\rm{e}}}^{2{k}^{2}y}}{{c}^{2}{{\rm{e}}}^{-2{kx}+4{k}^{3}t}+{d}^{2}{{\rm{e}}}^{2{kx}-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{4{cdk}{{\rm{e}}}^{2{k}^{2}y}}{{c}^{2}{{\rm{e}}}^{-2{kx}+4{k}^{3}t}-{d}^{2}{{\rm{e}}}^{2{kx}-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{4k{{\rm{e}}}^{2{k}^{2}y}}{-{{\rm{e}}}^{-2{kx}+4{k}^{3}t}-{{\rm{e}}}^{2{kx}-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{4k{{\rm{e}}}^{2{k}^{2}y}}{-{\mathrm{ie}}^{-2{kx}+4{k}^{3}t}-{\mathrm{ie}}^{2{kx}-4{k}^{3}t}},\end{eqnarray}$
where we have taken $k={k}_{1},\,t={t}_{1},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.Next, let us quickly investigate dynamics of ${q}_{(\sigma ,\delta )}={q}_{(1,-1)}$ which is governed by equation42a ) is rewritten as
$\begin{eqnarray}{q}_{{t}_{1}}=-{q}_{{xy}}+2\delta q{\partial }_{x}^{-1}{\left({qq}(x,-y,t\right)}_{y}.\end{eqnarray}$
Its 1SS ( $\begin{eqnarray}q(x,y,t)=\displaystyle \frac{2{kcd}\,{{\rm{e}}}^{2{k}^{2}y}}{| {cd}| }{\rm{sech}} \left(-2{kx}+4{k}^{3}t+\mathrm{ln}\displaystyle \frac{| c| }{| d| }\right),\end{eqnarray}$
which is a moving wave with an initial phase $\mathrm{ln}\tfrac{| c| }{| d| }$ and a y-dependent amplitude $\tfrac{2{cdk}\,{{\rm{e}}}^{2{k}^{2}y}}{| {cd}| }$. The top trajectory is $x(t)=2{k}^{2}t+\tfrac{\mathrm{ln}\left|\tfrac{c}{d}\right|}{2k}$, and travel velocity is 2k2. Fixing y, the trajectory of one-solton in coordinate frame {x, t} is depicted as figure 1(a).
Figure 1. (a). Shape and motion of 1SS ( |
The 2SS of equation (43 ) can be written as45 ) in the following coordinate
$\begin{eqnarray}q(x,y,t)=\displaystyle \frac{A}{B},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}A & = & 4\left({k}_{1}^{2}-{k}_{2}^{2}\right){{\rm{e}}}^{2\left({k}_{1}^{2}+{k}_{2}^{2}\right)y}\left({c}_{2}{c}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}\left(2{k}_{2}^{2}t+{k}_{2}y+x\right)+8{k}_{1}^{3}t}\right.\\ & & \left.+{c}_{2}{d}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{4{k}_{2}^{3}t+2{k}_{2}x+4{k}_{1}x+2{k}_{2}^{2}y}\right)\\ & & -4{c}_{1}{d}_{1}{k}_{1}\left({k}_{1}^{2}-{k}_{2}^{2}\right)\left({c}_{2}^{2}{{\rm{e}}}^{8{k}_{2}^{3}t}\right.\\ & & \left.+{d}_{2}^{2}{{\rm{e}}}^{4{k}_{2}x}\right){{\rm{e}}}^{\left(2{k}_{1}\left(2{k}_{1}^{2}t+{k}_{1}y+x\right)+2\left({k}_{1}^{2}+{k}_{2}^{2}\right)y\right)},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}B & = & -{c}_{2}^{2}\left({{\rm{e}}}^{2{k}_{2}^{2}\left(4{k}_{2}t+y\right)+2{k}_{1}^{2}y}\right)\\ & & \times \left({c}_{1}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}{{\rm{e}}}^{8{k}_{1}^{3}t}+{d}_{1}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}{{\rm{e}}}^{4{k}_{1}x}\right)\\ & & +4{c}_{1}{c}_{2}{d}_{1}{d}_{2}{k}_{1}{k}_{2}\left({{\rm{e}}}^{4{k}_{1}^{2}y}+{{\rm{e}}}^{4{k}_{2}^{2}y}\right){{\rm{e}}}^{4{k}_{1}^{3}t+4{k}_{2}^{3}t+2{k}_{1}x+2{k}_{2}x}\\ & & -{d}_{2}^{2}{{\rm{e}}}^{4{k}_{2}x+2{k}_{1}^{2}y+2{k}_{2}^{2}y}\left({c}_{1}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}{{\rm{e}}}^{8{k}_{1}^{3}t}\right.\\ & & \left.+{d}_{1}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}{{\rm{e}}}^{4{k}_{1}x}\right),\end{array}\end{eqnarray*}$
and we note that 2SS is non-singular if ${c}_{1}{c}_{2}{d}_{1}{d}_{2}{k}_{1}{k}_{2}\lt 0$. As shown in figures 1(b)–(d), there are three types of two-soliton interactions, respectively are soliton-soliton interaction, soliton-anti-soliton interaction and breather soliton. To investigate asymptotic behavior of the first two types solution interactions, we make asymptotic analysis for t going to infinity in the case of ${k}_{1}\gt -{k}_{2}\gt 0$. To do that, we first rewrite the 2SS ( $\begin{eqnarray}({X}_{1}=x-2{k}_{1}^{2}t,t),\end{eqnarray}$
fix X1, let $t\to \pm \infty $, and we get $\begin{eqnarray}\begin{array}{l}q(x,y,t)\\ \sim \,\left\{\begin{array}{ll}\tfrac{2{c}_{1}{d}_{1}{k}_{1}{{\rm{e}}}^{2{k}_{1}^{2}y}}{| {c}_{1}| | {d}_{1}| }{\rm{sech}} (2{k}_{1}{X}_{1}+\mathrm{ln}\tfrac{| {d}_{1}| ({k}_{1}+{k}_{2})}{| {c}_{1}| ({k}_{1}-{k}_{2})}), & t\to +\infty ,\\ \tfrac{2{c}_{1}{d}_{1}{k}_{1}{{\rm{e}}}^{2{k}_{1}^{2}y}}{| {c}_{1}| | {d}_{1}| }{\rm{sech}} (2{k}_{1}{X}_{1}+\mathrm{ln}\tfrac{| {d}_{1}| ({k}_{1}-{k}_{2})}{| {c}_{1}| ({k}_{1}+{k}_{2})}), & t\to -\infty .\end{array}\right.\end{array}\end{eqnarray}$
It shows that along the line X1 = constant, there is a soliton; when $t\to \pm \infty $, the soliton asymptotically follows $\begin{eqnarray}{\rm{top}}\,{\rm{trajectory}}:\,\,\,x(t)=2{k}_{1}^{2}t-\displaystyle \frac{\mathrm{ln}| \tfrac{{d}_{1}}{{c}_{1}}| }{2{k}_{1}}\pm \displaystyle \frac{\mathrm{ln}\tfrac{{k}_{1}+{k}_{2}}{{k}_{1}-{k}_{2}}}{2{k}_{1}},\end{eqnarray}$
$\begin{eqnarray}{\rm{amplitude}}:\,\,\,\displaystyle \frac{2{c}_{1}{d}_{1}{k}_{1}{{\rm{e}}}^{2{k}_{1}^{2}y}}{| {c}_{1}| | {d}_{1}| },\end{eqnarray}$
and phase shift after interaction is $\tfrac{{\rm{ln}}\tfrac{{k}_{1}+{k}_{2}}{{k}_{1}-{k}_{2}}}{{k}_{1}}$.We can also rewrite the 2SS (45 ) in the coordinate $({X}_{2}=x-2{k}_{2}^{2}t,t)$. By fixing X2, we can obtain
$\begin{eqnarray}\begin{array}{l}q(x,y,t)\\ \sim \,\left\{\begin{array}{ll}\tfrac{-2{c}_{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}^{2}y}}{| {c}_{2}| | {d}_{2}| }{\rm{sech}} (2{k}_{2}{X}_{2}+\mathrm{ln}\tfrac{| {d}_{2}| ({k}_{1}+{k}_{2})}{| {c}_{2}| ({k}_{1}-{k}_{2})}), & t\to +\infty ,\\ \tfrac{-2{c}_{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}^{2}y}}{| {c}_{2}| | {d}_{2}| }{\rm{sech}} (2{k}_{2}{X}_{2}+\mathrm{ln}\tfrac{| {d}_{2}| ({k}_{1}-{k}_{2})}{| {c}_{2}| ({k}_{1}+{k}_{2})}), & t\to -\infty .\end{array}\right.\end{array}\end{eqnarray}$
It indicates that along the line X2= constant, there is a soliton; when $t\to \pm \infty $, the soliton asymptotically follows $\begin{eqnarray}{\rm{top}}\,{\rm{trajectory}}:\,\,\,x(t)=2{k}_{2}^{2}t-\displaystyle \frac{\mathrm{ln}| \tfrac{{d}_{2}}{{c}_{2}}| }{2{k}_{2}}\pm \displaystyle \frac{\mathrm{ln}\tfrac{{k}_{1}+{k}_{2}}{{k}_{1}-{k}_{2}}}{2{k}_{2}},\end{eqnarray}$
$\begin{eqnarray}{\rm{amplitude}}:\,\,\,\displaystyle \frac{-2{c}_{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}^{2}y}}{| {c}_{2}| | {d}_{2}| },\end{eqnarray}$
and the phase shift after interaction is $\tfrac{{\rm{ln}}\tfrac{{k}_{1}+{k}_{2}}{{k}_{1}-{k}_{2}}}{{k}_{2}}$.Supposing ${k}_{1}\gt -{k}_{2}\gt 0,{c}_{1}{c}_{2}{d}_{1}{d}_{2}{k}_{1}{k}_{2}\lt 0$, from the asymptotic analysis, we can obtain that if $\mathrm{sgn}[{c}_{1}{d}_{1}]\,=\mathrm{sgn}[{c}_{2}{d}_{2}]$, it will appear soliton-soliton interaction, such as figure 1(b); if $\mathrm{sgn}[{c}_{1}{d}_{1}]=-\mathrm{sgn}[{c}_{2}{d}_{2}]$, it will appear soliton-anti-soliton interaction, such as figure 1(c).
We list out solutions for other cases of nonlocal reductions without giving proof.
4.1.2. Reduction (23 )
For the reduction (
$\begin{eqnarray}{q}_{{t}_{2l+1}}={K}_{\mathrm{1,2}l+1}{| }_{(23)},\,\,l=0,1,2,\cdots \end{eqnarray}$
allow a solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi $ and $\psi $ are $(2n+2)$th order column vectors (i.e. $m=n$ in ( $\begin{eqnarray}\psi (x,y,t)=T{\varphi }^{* }(\sigma x,-y,\sigma t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+* },\end{eqnarray}$
in which $T$ is a constant matrix determined through $\begin{eqnarray}{AT}+\sigma {{TA}}^{* }=0,\,\,\end{eqnarray}$
$\begin{eqnarray}{{TT}}^{* }=\sigma \delta I,\,\,\sigma ,\delta =\pm 1.\end{eqnarray}$
Table 2. T and A for ( |
| σ,δ | T | A |
|---|---|---|
| (1, −1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}=-{T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}=-{K}_{4}^{* }={{\bf{K}}}_{n+1}$ |
| (1, 1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}={T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}=-{K}_{4}^{* }={{\bf{K}}}_{n+1}$ |
| (−1, −1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}={T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}={K}_{4}^{* }={{\bf{K}}}_{n+1}$ |
| (−1, 1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}=-{T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}={K}_{4}^{* }={{\bf{K}}}_{n+1}$ |
When ${{\bf{K}}}_{n+1}$ is a diagonal matrix (37 ), the vector φ in (52 ) can be given as39 ), we have
$\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\left.\theta ({k}_{1}\right)},{c}_{2}{{\rm{e}}}^{\left.\theta ({k}_{2}\right)},\cdots ,{c}_{n+1}{{\rm{e}}}^{\left.\theta ({k}_{n+1}\right)},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\theta (-\sigma {k}_{1}^{* })},{d}_{2}{{\rm{e}}}^{\theta (-\sigma {k}_{2}^{* })},\cdots ,{d}_{n+1}{{\rm{e}}}^{\theta (-\sigma {k}_{n+1}^{* })}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\theta ({k}_{i})=-{k}_{i}x+{k}_{i}^{2}y+\sum _{j=0}^{\infty }{2}^{2j+1}{k}_{i}^{2j+3}{t}_{2j+1}.\end{eqnarray}$
When ${{\bf{K}}}_{n+1}={J}_{n+1}(k)$ ( $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\theta (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\theta (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\theta (k)}),d{{\rm{e}}}^{\theta (-\sigma {k}^{* })},\right.\\ & & {\left.\displaystyle \frac{{\partial }_{{k}^{* }}}{1!}(d{{\rm{e}}}^{\theta (-\sigma {k}^{* })}),\cdots ,\displaystyle \frac{{\partial }_{{k}^{* }}^{n}}{n!}(d{{\rm{e}}}^{\theta (-\sigma {k}^{* })}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
For the equation (24 ) with different (σ, δ), its 1SS are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{k}^{* 2}y+4{k}^{* 3}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{k}^{* 2}y+4{k}^{* 3}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{2{cd}(k-{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x-2{k}^{* 2}y+4{k}^{* 3}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{2{cd}(k-{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x-2{k}^{* 2}y+4{k}^{* 3}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{1},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.4.1.3. Reduction (25 )
For the reduction (
$\begin{eqnarray}{q}_{{t}_{2l}}={K}_{\mathrm{1,2}l}{| }_{(25)},\,\,l=1,2,\cdots \end{eqnarray}$
allow a solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\hat{\varphi }}^{(n+1)};{\hat{\psi }}^{(n-1)}| }{| {\hat{\varphi }}^{(n)};{\hat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$, defined by ( $\begin{eqnarray}\psi (x,y,t)=T\varphi (\sigma x,-y,-t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+},\end{eqnarray}$
in which $T$ is a constant matrix satisfying (If T and A are block matrices (36 ), solutions to (34 ) haven been given by table 1. In the case that ${{\bf{K}}}_{n+1}$ is a diagonal matrix (37 ), the vector φ in (60 ) can be given as39 ), we have
$\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\xi ({k}_{1})},{c}_{2}{{\rm{e}}}^{\xi ({k}_{2})},\cdots ,{c}_{n+1}{{\rm{e}}}^{\xi ({k}_{n+1})},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\xi (-{k}_{1})},{d}_{2}{{\rm{e}}}^{\xi (-{k}_{2})},\cdots ,{d}_{n+1}{{\rm{e}}}^{\xi (-{k}_{n+1})}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
where $\begin{eqnarray}\xi ({k}_{i})=-{k}_{i}x+{k}_{i}^{2}y+\sum _{j=1}^{\infty }{2}^{2j}{k}_{i}^{2j+2}{t}_{2j}.\end{eqnarray}$
When ${{\bf{K}}}_{n+1}={J}_{n+1}(k)$ ( $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\xi (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\xi (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\xi (k)}),\right.\\ & & {\left.d{{\rm{e}}}^{\xi (-k)},\displaystyle \frac{{\partial }_{k}}{1!}(d{{\rm{e}}}^{\xi (-k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(d{{\rm{e}}}^{(-k)}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
Table 1. T and A for ( |
| (σ, δ) | T | A |
|---|---|---|
| (1, −1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}=-{T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}=-{K}_{4}={{\bf{K}}}_{n+1}$ |
| (1, 1) | ${T}_{1}={T}_{4}={{\bf{0}}}_{n+1},{T}_{3}={T}_{2}={{\bf{I}}}_{n+1}$ | ${K}_{1}=-{K}_{4}={{\bf{K}}}_{n+1}$ |
| (−1, −1) | ${T}_{1}=-{T}_{4}={{\bf{I}}}_{n+1},{T}_{3}={T}_{2}={{\bf{0}}}_{n+1}$ | ${K}_{1}=-{K}_{4}={{\bf{K}}}_{n+1}$ |
| (−1, 1) | ${T}_{1}=-{T}_{4}=i{{\bf{I}}}_{n+1},{T}_{3}={T}_{2}={{\bf{0}}}_{n+1}$ | ${K}_{1}=-{K}_{4}={{\bf{K}}}_{n+1}$ |
For the equation (26 ) with different (σ, δ), its 1SS are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{4{cdk}{{\rm{e}}}^{2{k}^{2}y+8{k}^{4}t}}{{c}^{2}{{\rm{e}}}^{-2{kx}}+{d}^{2}{{\rm{e}}}^{2{kx}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{4{cdk}{{\rm{e}}}^{2{k}^{2}y+8{k}^{4}t}}{{c}^{2}{{\rm{e}}}^{-2{kx}}-{d}^{2}{{\rm{e}}}^{2{kx}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{4k{{\rm{e}}}^{2{k}^{2}y+8{k}^{4}t}}{-{{\rm{e}}}^{-2{kx}}-{{\rm{e}}}^{2{kx}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{4{\rm{i}}k{{\rm{e}}}^{2{k}^{2}y+8{k}^{4}t}}{-{{\rm{e}}}^{-2{kx}}-{{\rm{e}}}^{2{kx}}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{2},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.As an example we consider the dynamics of ${q}_{(\sigma ,\delta )}={q}_{(1,-1)}$, which is governed by equation65a ) can be rewritten as
$\begin{eqnarray}{q}_{{t}_{2}}={q}_{{xxy}}+4\widetilde{q}{{qq}}_{y}+2{q}_{x}{\partial }_{x}^{-1}{\left(q\widetilde{q}\right)}_{y}+2q{\partial }_{x}^{-1}({q}_{x}{\widetilde{q}}_{y}-{\widetilde{q}}_{x}{q}_{y}),\end{eqnarray}$
where $\widetilde{q}=q(x,-y,-t)$. Its 1SS ( $\begin{eqnarray}q(x,y,t)=\displaystyle \frac{2{cdk}\,{{\rm{e}}}^{2{k}^{2}y+8{k}^{4}t}}{| {cd}| }{\rm{{\rm{sech}} }}\left(-2{kx}+\mathrm{ln}| \displaystyle \frac{c}{d}| \right).\end{eqnarray}$
Depicted as figure 2(a), this is a stationary wave with an initial phase $\mathrm{ln}\left|\tfrac{c}{d}\right|$ and an amplitude that exponentially increases with time t and y, and the top trajectory is $x(t)=\tfrac{\mathrm{ln}\left|\tfrac{c}{d}\right|}{2k}$. It is interesting that the amplitude is y and t-dependent, which looks like in nonisospectral case (see [35]) where amplitudes are changed due to time-dependent eigenvalues (spectral parameters). However, here the amplitude changes might be caused by the reversed y and t in the nonlocal reduction, rather than nonspectral parameters as the eigenvalues (see A in table 1) are still constant.
Figure 2. (a). Shape and motion of 1SS ( |
The 2SS of equation (66 ) can be written as
$\begin{eqnarray}q(x,y,t)=\displaystyle \frac{G}{F},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}G & = & -4{c}_{1}{d}_{1}{k}_{1}\left({k}_{1}^{2}-{k}_{2}^{2}\right)\left({c}_{2}^{2}\right.\\ & & \left.+{d}_{2}^{2}{{\rm{e}}}^{4{k}_{2}x}\right){{\rm{e}}}^{8{k}_{1}^{4}t+8{k}_{2}^{4}t+2{k}_{1}^{2}y+2{k}_{2}^{2}y+2{k}_{1}\left(4{k}_{1}^{3}t+{k}_{1}y+x\right)}\\ & & +4\left({k}_{1}^{2}-{k}_{2}^{2}\right){{\rm{e}}}^{8{k}_{1}^{4}t+8{k}_{2}^{4}t+2{k}_{1}^{2}y+2{k}_{2}^{2}y}\left({c}_{2}{c}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}\left(4{k}_{2}^{3}t+{k}_{2}y+x\right)}\right.\\ & & \left.+{c}_{2}{d}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{8{k}_{2}^{4}t+2{k}_{2}x+4{k}_{1}x+2{k}_{2}^{2}y}\right),\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}F & = & -{d}_{1}^{2}\left({c}_{2}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}\right.\\ & & \left.+{d}_{2}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}{{\rm{e}}}^{4{k}_{2}x}\right){{\rm{e}}}^{\left(8{k}_{1}^{4}t+8{k}_{2}^{4}t+4{k}_{1}x+2{k}_{1}^{2}y+2{k}_{2}^{2}y\right)}\\ & & +4{c}_{1}{c}_{2}{d}_{1}{d}_{2}{k}_{1}{k}_{2}{{\rm{e}}}^{2\left({k}_{1}+{k}_{2}\right)x}\left({{\rm{e}}}^{4{k}_{1}^{2}\left(4{k}_{1}^{2}t+y\right)}+{{\rm{e}}}^{4{k}_{2}^{2}\left(4{k}_{2}^{2}t+y\right)}\right)\\ & & -{c}_{1}^{2}\left({{\rm{e}}}^{8{k}_{1}^{4}t+8{k}_{2}^{4}t+2{k}_{1}^{2}y+2{k}_{2}^{2}y}\right)\\ & & \times \left({c}_{2}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}+{d}_{2}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}{{\rm{e}}}^{4{k}_{2}x}\right).\end{array}\end{eqnarray*}$
4.1.4. Reduction (28 )
For the reduction (
$\begin{eqnarray}{q}_{{t}_{2l}}={K}_{\mathrm{1,2}l}{| }_{(28)},\,\,l=1,2,\cdots \end{eqnarray}$
allow a solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi $ and $\psi $ are defined by ( $\begin{eqnarray}\psi (x,y,t)=T{\varphi }^{* }(\sigma x,-y,-t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+* },\end{eqnarray}$
in which $T$ is a constant matrix satisfied (When T and A are block matrices (36 ), solutions to (54 ) are already given by table 2. In the case that ${{\bf{K}}}_{n+1}$ is a diagonal matrix (37 ), the vector φ in (5 ) can be written as39 ), we get
$\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\eta ({k}_{1})},{c}_{2}{{\rm{e}}}^{\eta ({k}_{2})},\cdots ,{c}_{n+1}{{\rm{e}}}^{\eta ({k}_{n+1})},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\eta (-\sigma {k}_{1}^{* })},{d}_{2}{{\rm{e}}}^{\eta (-\sigma {k}_{2}^{* })},\cdots ,{d}_{n+1}{{\rm{e}}}^{\eta (-\sigma {k}_{n+1}^{* })}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\eta ({k}_{i})=-{k}_{i}x+{k}_{i}^{2}y+\sum _{j=1}^{\infty }{2}^{2j}{k}_{i}^{2j+2}{t}_{2j}.\end{eqnarray}$
When ${{\bf{K}}}_{n+1}={J}_{n+1}(k)$ ( $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\eta (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\eta (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\eta (k)}),\right.\\ & & {\left.d{{\rm{e}}}^{\eta (-\sigma {k}^{* })},\displaystyle \frac{{\partial }_{{k}^{* }}}{1!}(d{{\rm{e}}}^{\eta (-\sigma {k}^{* })}),\cdots ,\displaystyle \frac{{\partial }_{{k}^{* }}^{n}}{n!}(d{{\rm{e}}}^{\eta (-\sigma {k}^{* })}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
For equation (29 ) with different (σ, δ), its 1SS are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{k}^{* 2}y-8{k}^{* 4}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-8{k}^{4}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{k}^{* 2}y-8{k}^{* 4}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-8{k}^{4}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{2{cd}(k-{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x-2{k}^{* 2}y-8{k}^{* 4}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-8{k}^{4}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{2{cd}(k-{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x-2{k}^{* 2}y-8{k}^{* 4}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{k}^{2}y-8{k}^{4}t}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{2},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.4.2. Classical cases
For the classical reduction (
$\begin{eqnarray}{q}_{{t}_{2l+1}}={K}_{\mathrm{1,2}l+1}{| }_{(17)},\,\,l=0,1,2,\cdots \end{eqnarray}$
allow a solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\hat{\varphi }}^{(n+1)};{\hat{\psi }}^{(n-1)}| }{| {\hat{\varphi }}^{(n)};{\hat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi $ and $\psi $ are defined by ( $\begin{eqnarray}\psi (x,y,t)=T{\varphi }^{* }(x,y,t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+* },\end{eqnarray}$
in which $T$ is a constant matrix determined through $\begin{eqnarray}{AT}+{{TA}}^{* }=0,\,\,\end{eqnarray}$
$\begin{eqnarray}{{TT}}^{* }=\delta I,\,\,\delta =\pm 1.\end{eqnarray}$
When T and A are block matrices (36 ), solutions to equations (79 ) are given in table 2 with σ = 1. When ${{\bf{K}}}_{n+1}$ is a diagonal matrix (37 ), the vector φ in (77 ) can be given as39 ), we have18 )
$\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\theta ({k}_{1})},{c}_{2}{{\rm{e}}}^{\theta ({k}_{2})},\cdots ,{c}_{n+1}{{\rm{e}}}^{\theta ({k}_{n+1})},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\theta (-{k}_{1}^{* })},{d}_{2}{{\rm{e}}}^{\theta (-{k}_{2}^{* })},\cdots ,{d}_{n+1}{{\rm{e}}}^{\theta (-{k}_{n+1}^{* })}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\theta ({k}_{j})=-{k}_{j}x+{\rm{i}}{k}_{j}^{2}y+\sum _{l=0}^{\infty }{2}^{2l+1}{k}_{j}^{2l+3}{t}_{2l+1}.\end{eqnarray}$
When ${{\bf{K}}}_{n+1}={J}_{n+1}(k)$ ( $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\theta (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\theta (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\theta (k)}),\right.\\ & & {\left.d{{\rm{e}}}^{\theta (-{k}^{* })},\displaystyle \frac{{\partial }_{{k}^{* }}}{1!}(d{{\rm{e}}}^{\theta (-{k}^{* })}),\cdots ,\displaystyle \frac{{\partial }_{{k}^{* }}^{n}}{n!}(d{{\rm{e}}}^{\theta (-{k}^{* })}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
In addition, we present the 1SS of the equation ( $\begin{eqnarray}{q}_{\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{\rm{i}}{k}^{* 2}y+4{k}^{* 3}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{\rm{i}}{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\delta =-1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{\rm{i}}{k}^{* 2}y+4{k}^{* 3}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{\rm{i}}{k}^{2}y-4{k}^{3}t}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{1},\,{C}^{+}={(c,d)}^{{\rm{T}}}$. For the reduction (
$\begin{eqnarray}{q}_{{t}_{2l}}={\rm{i}}{K}_{\mathrm{1,2}l}{| }_{(19)},\,\,l=1,2,\cdots \end{eqnarray}$
allow a solution $\begin{eqnarray}q(x,y,t)=2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$ expressed as (When T and A are block matrices (36 ), solutions to (79 ) are already given by table 2. In the case that ${{\bf{K}}}_{n+1}$ is a diagonal matrix (37 ), the vector φ in (85 ) can be written as39 ), we get20 ), its 1SS with different δ are given as
$\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left({c}_{1}{{\rm{e}}}^{\eta ({k}_{1})},{c}_{2}{{\rm{e}}}^{\eta ({k}_{2})},\cdots ,{c}_{n+1}{{\rm{e}}}^{\eta ({k}_{n+1})},\right.\\ & & {\left.{d}_{1}{{\rm{e}}}^{\eta (-{k}_{1}^{* })},{d}_{2}{{\rm{e}}}^{\eta (-{k}_{2}^{* })},\cdots ,{d}_{n+1}{{\rm{e}}}^{\eta (-{k}_{n+1}^{* })}\right)}^{{\rm{T}}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\eta ({k}_{j})=-{k}_{j}x+{\rm{i}}{k}_{j}^{2}y+\sum _{l=1}^{\infty }{2}^{2l}{\rm{i}}{k}_{j}^{2l+2}{t}_{2l}.\end{eqnarray}$
When ${{\bf{K}}}_{n+1}={J}_{n+1}(k)$ ( $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \left(c{{\rm{e}}}^{\eta (k)},\displaystyle \frac{{\partial }_{k}}{1!}(c{{\rm{e}}}^{\eta (k)}),\cdots ,\displaystyle \frac{{\partial }_{k}^{n}}{n!}(c{{\rm{e}}}^{\eta (k)}),\right.\\ & & {\left.d{{\rm{e}}}^{\eta (-{k}^{* })},\displaystyle \frac{{\partial }_{{k}^{* }}}{1!}(d{{\rm{e}}}^{\eta (-{k}^{* })}),\cdots ,\displaystyle \frac{{\partial }_{{k}^{* }}^{n}}{n!}(d{{\rm{e}}}^{\eta (-{k}^{* })}\right)}^{{\rm{T}}}.\end{array}\end{eqnarray}$
For equation ( $\begin{eqnarray}{q}_{\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{\rm{i}}{k}^{* 2}y-8{\rm{i}}{k}^{* 4}t}-| d{| }^{2}{{\rm{e}}}^{2{kx}-2{\rm{i}}{k}^{2}y-8{\rm{i}}{k}^{4}t}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\delta =-1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-2{\rm{i}}{k}^{* 2}y-8{\rm{i}}{k}^{* 4}t}+| d{| }^{2}{{\rm{e}}}^{2{kx}-2{\rm{i}}{k}^{2}y-8{\rm{i}}{k}^{4}t}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{2},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.5. Negative order AKNS hierarchy
In this section, we list double Wronskian solutions for the negative order AKNS hierarchy and consider several cases of reductions.
5.1. Solutions
For the negative order AKNS hierarchy [33]3 ). The hierarchy can be rewritten as [36]
$\begin{eqnarray}\left(\begin{array}{c}{q}_{{t}_{n}}\\ {r}_{{t}_{n}}\end{array}\right)={K}_{n}=\left(\begin{array}{c}{K}_{1,n}\\ {K}_{2,n}\end{array}\right)={L}^{-n}\left(\begin{array}{c}-q\\ r\end{array}\right),\,\,n=1,2,3,\cdots ,\end{eqnarray}$
where L is defined by ( $\begin{eqnarray}q={q}_{x,{t}_{1}}-2q{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{1}},\end{eqnarray}$
$\begin{eqnarray}r={r}_{x,{t}_{1}}-2r{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{1}},\end{eqnarray}$
$\begin{eqnarray}{q}_{{t}_{n-1}}=-{q}_{x,{t}_{n}}+2q{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{n}},\end{eqnarray}$
$\begin{eqnarray}{r}_{{t}_{n-1}}={r}_{x,{t}_{n}}-2r{\partial }_{x}^{-1}{\left({qr}\right)}_{{t}_{n}},\,\,(n=2,3,\cdots ).\end{eqnarray}$
By the transformation $\begin{eqnarray}q=-\displaystyle \frac{g}{f},\,\,r=\displaystyle \frac{h}{f},\end{eqnarray}$
its bilinear form turns out to be [36] $\begin{eqnarray}{D}_{x}^{2}f\cdot f=2{gh},\end{eqnarray}$
$\begin{eqnarray}{D}_{x}{D}_{{t}_{1}}g\cdot f={gf},\end{eqnarray}$
$\begin{eqnarray}{D}_{x}{D}_{{t}_{1}}h\cdot f={hf},\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{n-1}}+{D}_{x}{D}_{{t}_{n}})g\cdot f=0,\end{eqnarray}$
$\begin{eqnarray}({D}_{{t}_{n-1}}-{D}_{x}{D}_{{t}_{n}})h\cdot f=0,\,\,(n=2,3\cdots ),\end{eqnarray}$
which allow double Wronskian solution [32] $\begin{eqnarray}\begin{array}{rcl}f & = & | {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(m)}| ,\,\,g=2| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(m-1)}| ,\\ h & = & 2| {\widehat{\varphi }}^{(n-1)};{\widehat{\psi }}^{(m+1)}| .\end{array}\end{eqnarray}$
where φ and ψ are defined as $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \exp \left(-{Ax}-\sum _{n=1}^{\infty }{2}^{-(n+1)}{A}^{-n}{t}_{n}\right){C}^{+},\\ \psi & = & \exp \left({Ax}+\sum _{n=1}^{\infty }{2}^{-(n+1)}{A}^{-n}{t}_{n}\right){C}^{-},\end{array}\end{eqnarray}$
$A\in {{\mathbb{C}}}_{(n+m+2)\times (n+m+2)}$ and ${C}^{\pm }$ are $(n+m+2)$th-order constant column vector.5.2. Reductions and solutions
The odd members in the hierarchy (91 ) admit a real reduction91 ) also admit a complex reduction
$\begin{eqnarray}r(x,t)=\delta q(\sigma x,\sigma t),\,\,\sigma ,\delta =\pm 1,\end{eqnarray}$
under which we have $\begin{eqnarray}{q}_{{t}_{2l+1}}={K}_{\mathrm{1,2}l+1}{| }_{(96)},\,\,l=0,1,2,\cdot \cdot \cdot \end{eqnarray}$
and the representative one $\begin{eqnarray}q={q}_{x,{t}_{1}}-2\delta q{\partial }_{x}^{-1}{\left({qq}(\sigma x,\sigma t\right)}_{{t}_{1}}.\end{eqnarray}$
In addition, the odd members in the hierarchy ( $\begin{eqnarray}r(x,t)=\delta {q}^{* }(\sigma x,\sigma t),\,\,\sigma ,\delta =\pm 1,\end{eqnarray}$
under which we have $\begin{eqnarray}{q}_{{t}_{2l+1}}={K}_{\mathrm{1,2}l+1}{| }_{(99)},\,\,l=0,1,2,\cdot \cdot \cdot \end{eqnarray}$
and $\begin{eqnarray}q={q}_{x,{t}_{1}}-2\delta q{\partial }_{x}^{-1}{\left({{qq}}^{* }(\sigma x,\sigma t\right)}_{{t}_{1}}\end{eqnarray}$
as the first member.For those even members in the hierarchy (91 ), first, they allow a real reduction
$\begin{eqnarray}r(x,t)=\delta q(\sigma x,-t),\,\,\sigma ,\delta =\pm 1,\end{eqnarray}$
which generate $\begin{eqnarray}{q}_{{t}_{2l}}={K}_{\mathrm{1,2}l}{| }_{(102)},\,\,l=1,2,\cdot \cdot \cdot ,\end{eqnarray}$
the simplest one is $\begin{eqnarray}\begin{array}{rcl}q & = & -{q}_{{{xxt}}_{2}}+4\delta \widehat{q}{{qq}}_{{t}_{2}}+2\delta {q}_{x}{\partial }_{x}^{-1}{\left(q\widehat{q}\right)}_{{t}_{2}}\\ & & +2\delta q{\partial }_{x}^{-1}({q}_{x}{\widehat{q}}_{{t}_{2}}-{\widehat{q}}_{x}{q}_{{t}_{2}}),\,\,\delta ,\sigma =\pm 1,\end{array}\end{eqnarray}$
where $\widehat{q}=q(\sigma x,-t)$. And second, a complex reduction $\begin{eqnarray}r(x,t)=\delta {q}^{* }(\sigma x,t),\,\,\sigma ,\delta =\pm 1,\,\,{t}_{2j}\to {\rm{i}}{t}_{2j},\end{eqnarray}$
leads to $\begin{eqnarray}\ {\rm{i}}{q}_{{t}_{2l}}=-{K}_{\mathrm{1,2}l}{| }_{(105)},\,\,l=1,2,\cdot \cdot \cdot ,\end{eqnarray}$
in which the representative one is $\begin{eqnarray}\begin{array}{rcl}\ {\rm{i}}q & = & -{q}_{{{xxt}}_{2}}+4\delta {\widehat{q}}^{* }{{qq}}_{{t}_{2}}+2\delta {q}_{x}{\partial }_{x}^{-1}{\left(q{\widehat{q}}^{* }\right)}_{{t}_{2}}\\ & & +2\delta q{\partial }_{x}^{-1}({q}_{x}{\widehat{q}}_{{t}_{2}}^{* }-{\widehat{q}}_{x}^{* }{q}_{{t}_{2}}),\,\,\delta ,\sigma =\pm 1,\end{array}\end{eqnarray}$
where $\widehat{q}=q(\sigma x,t)$.In the following we list out double Wronskian solutions and we skip proof.
The system (
$\begin{eqnarray}q(x,t)=-2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$, are defined by $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \exp \left(-{Ax}-\sum _{j=0}^{\infty }{2}^{-(2j+2)}{A}^{-(2j+1)}{t}_{2j+1}\right){C}^{+},\\ \psi & = & \exp \left({Ax}+\sum _{j=0}^{\infty }{2}^{-(2j+2)}{A}^{-(2j+1)}{t}_{2j+1}\right){C}^{-}\end{array}\end{eqnarray}$
and satisfy $\begin{eqnarray}\psi (x,t)=T\varphi (\sigma x,\sigma t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+},\end{eqnarray}$
in which $T$ is a constant matrix determined by (For equation (98 ) with different (σ, δ), its 1SS are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=-\displaystyle \frac{4{cdk}}{{c}^{2}{{\rm{e}}}^{-2{kx}-\tfrac{t}{2k}}+{d}^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{4{cdk}}{{d}^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}-{c}^{2}{{\rm{e}}}^{-2{kx}-\tfrac{t}{2k}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{4k}{{{\rm{e}}}^{-2{kx}-\tfrac{t}{2k}}+{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{4k{\rm{i}}}{{{\rm{e}}}^{-2{kx}-\tfrac{t}{2k}}+{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{1},\,{C}^{+}=(c,d)$. The system (
$\begin{eqnarray}q(x,t)=-2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$, are defined by ( $\begin{eqnarray}\psi (x,t)=T{\varphi }^{* }(\sigma x,\sigma t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+* },\end{eqnarray}$
in which $T$ is a constant matrix determined by (1SS of the equation (101 ) with different (σ, δ) are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{-2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-\tfrac{t}{2{k}^{* }}}+| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}-| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x-\tfrac{t}{2{k}^{* }}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{2{cd}({k}^{* }-k)}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x+\tfrac{t}{2{k}^{* }}}-| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{2{cd}({k}^{* }-k)}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x+\tfrac{t}{2{k}^{* }}}+| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{2k}}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{1},\,{C}^{+}={(c,d)}^{{\rm{T}}}$. The hierarchy (
$\begin{eqnarray}q(x,t)=-2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where $\varphi ={({\varphi }_{1},{\varphi }_{2},\cdots ,{\varphi }_{2n+2})}^{{\rm{T}}}$, $\psi ={({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{2n+2})}^{{\rm{T}}}$ are defined by $\begin{eqnarray}\begin{array}{rcl}\varphi & = & \exp \left(-{Ax}-\sum _{j=1}^{\infty }{2}^{-(2j+1)}{A}^{-2j}{t}_{2j}\right){C}^{+},\\ \psi & = & \exp \left({Ax}+\sum _{j=1}^{\infty }{2}^{-(2j+1)}{A}^{-2j}{t}_{2j}\right){C}^{-}\end{array}\end{eqnarray}$
and satisfy $\begin{eqnarray}\psi (x,t)=T\varphi (\sigma x,-t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+},\end{eqnarray}$
in which $T$ is a constant matrix determined by (For equation (104 ), its 1SS with different (σ, δ) are
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{-4{cdk}}{{c}^{2}{{\rm{e}}}^{-2{kx}+\tfrac{t}{4{k}^{2}}}+{d}^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{4{k}^{2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{4{cdk}}{{d}^{2}{{\rm{e}}}^{2{kx}+\tfrac{t}{4{k}^{2}}}-{c}^{2}{{\rm{e}}}^{-2{kx}+\tfrac{t}{4{k}^{2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{4k}{{{\rm{e}}}^{-2{kx}+\tfrac{t}{4{k}^{2}}}+{{\rm{e}}}^{2{kx}+\tfrac{t}{4{k}^{2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{-4{\rm{i}}k}{{{\rm{e}}}^{-2{kx}+\tfrac{t}{4{k}^{2}}}+{{\rm{e}}}^{2{kx}+\tfrac{t}{4{k}^{2}}}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{2},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.As an example, let us illustrate dynamics of ${q}_{(\sigma ,\delta )}={q}_{(1,-1)}$ which is governed by the equation118a ) describes a single soliton
$\begin{eqnarray}\begin{array}{rcl}q & = & -{q}_{{{xxt}}_{2}}-4\widehat{q}{{qq}}_{{t}_{2}}-2{q}_{x}{\partial }_{x}^{-1}{\left(q\widehat{q}\right)}_{{t}_{2}}\\ & & -2q{\partial }_{x}^{-1}({q}_{x}{\widehat{q}}_{{t}_{2}}-{\widehat{q}}_{x}{q}_{{t}_{2}}),\end{array}\end{eqnarray}$
where $\widehat{q}=q(x,-t)$, ( $\begin{eqnarray}q(x,t)=\displaystyle \frac{2{cdk}}{| {cd}| }{{\rm{e}}}^{-\tfrac{t}{4{k}^{2}}}{\rm{{\rm{sech}} }}\left(-2{kx}+\mathrm{ln}\left|\displaystyle \frac{c}{d}\right|\right),\end{eqnarray}$
with an initial phase $\mathrm{ln}\left|\tfrac{c}{d}\right|$, top trajectory $x(t)=\tfrac{\mathrm{ln}\left|\tfrac{c}{d}\right|}{2k}$, and an amplitude that decreases with time, see figure 3(a).
Figure 3. (a). Shape and motion of 1SS ( |
The 2SS of equation (119 ) is
$\begin{eqnarray}q(x,t)=\displaystyle \frac{{G}^{{\prime} }}{{F}^{{\prime} }},\end{eqnarray}$
where $\begin{eqnarray*}\begin{array}{rcl}{G}^{{\prime} } & = & -4\left({k}_{1}^{2}-{k}_{2}^{2}\right)\left(-{c}_{1}{d}_{1}{k}_{1}{{\rm{e}}}^{2{k}_{1}x}\left({c}_{2}^{2}+{d}_{2}^{2}{{\rm{e}}}^{\tfrac{t}{4{k}_{2}^{2}}+4{k}_{2}x}\right)\right.\\ & & \left.+{c}_{2}{d}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{\tfrac{t}{4{k}_{1}^{2}}+4{k}_{1}x+2{k}_{2}x}\right)\\ & & -4\left({k}_{1}^{2}-{k}_{2}^{2}\right){c}_{2}{c}_{1}^{2}{d}_{2}{k}_{2}{{\rm{e}}}^{2{k}_{2}x},\end{array}\end{eqnarray*}$
$\begin{eqnarray*}\begin{array}{rcl}{F}^{{\prime} } & = & {c}_{1}^{2}\left({c}_{2}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}+{d}_{2}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}{{\rm{e}}}^{\tfrac{t}{4{k}_{2}^{2}}+4{k}_{2}x}\right)\\ & & -4{c}_{2}{c}_{1}{d}_{1}{d}_{2}{k}_{1}{k}_{2}\left({{\rm{e}}}^{\tfrac{t}{4{k}_{1}^{2}}}+{{\rm{e}}}^{\tfrac{t}{4{k}_{2}^{2}}}\right){{\rm{e}}}^{2\left({k}_{1}+{k}_{2}\right)x}\\ & & +{d}_{1}^{2}{{\rm{e}}}^{\tfrac{t}{4{k}_{1}^{2}}+4{k}_{1}x}\left({c}_{2}^{2}\left({k}_{1}+{k}_{2}\right){}^{2}+{d}_{2}^{2}\left({k}_{1}-{k}_{2}\right){}^{2}{{\rm{e}}}^{\tfrac{t}{4{k}_{2}^{2}}+4{k}_{2}x}\right),\end{array}\end{eqnarray*}$
which we depict in figure 3(b). The hierarchy (
$\begin{eqnarray}q(x,t)=-2\displaystyle \frac{| {\widehat{\varphi }}^{(n+1)};{\widehat{\psi }}^{(n-1)}| }{| {\widehat{\varphi }}^{(n)};{\widehat{\psi }}^{(n)}| },\end{eqnarray}$
where φ and ψ are $(2n+2)$th order column vectors (i.e. $m=n$ in ( $\begin{eqnarray}\psi (x,t)=T{\varphi }^{* }(\sigma x,t),\end{eqnarray}$
$\begin{eqnarray}{C}^{-}={{TC}}^{+* },\end{eqnarray}$
in which $T$ is a constant matrix determined by (The equation (107 ) with different (σ, δ) has the following 1SS:
$\begin{eqnarray}{q}_{\sigma =1,\delta =-1}=\displaystyle \frac{-2{cd}(k+{k}^{* })}{| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x+\tfrac{{\rm{i}}t}{4{k}^{* 2}}}+| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{{\rm{i}}t}{4{k}^{2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =1,\delta =1}=\displaystyle \frac{2{cd}(k+{k}^{* })}{| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{{\rm{i}}t}{4{k}^{2}}}-| c{| }^{2}{{\rm{e}}}^{-2{k}^{* }x+\tfrac{{\rm{i}}t}{4{k}^{* 2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =-1}=\displaystyle \frac{2{cd}({k}^{* }-k)}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x+\tfrac{{\rm{i}}t}{4{k}^{* 2}}}-| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{{\rm{i}}t}{4{k}^{2}}}},\end{eqnarray}$
$\begin{eqnarray}{q}_{\sigma =-1,\delta =1}=\displaystyle \frac{2{cd}({k}^{* }-k)}{| c{| }^{2}{{\rm{e}}}^{2{k}^{* }x+\tfrac{{\rm{i}}t}{4{k}^{* 2}}}+| d{| }^{2}{{\rm{e}}}^{2{kx}+\tfrac{{\rm{i}}t}{4{k}^{2}}}},\end{eqnarray}$
where $k={k}_{1},\,t={t}_{2},\,{C}^{+}={(c,d)}^{{\rm{T}}}$.6. Conclusions
We have discussed possible nonlocal reductions of the (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy. By means of the reduction technique (see [15–18]), we derived N-soliton solutions and multiple-pole solutions in double Wronskian form of the reduced hierarchies from those of the unreduced hierarchies. Note that these hierarchies contain integration w.r.t. x, the integration operator with form of (30 ) is necessary and helpful in considering nonlocal reductions. In addition, this reduction method is based on bilinear forms and double Wronskians, As we can see, it allows us to obtain N-soliton solutions and multiple-pole solutions for the whole reduced hierarchy. Compared with [14, 16, 19] where only some single equations in the (2+1)-D BAKNS hierarchy and the negative order AKNS hierarchy were considered, here we solved the two hierarchies and obtained more solutions. Finally, it is remarkable that for some real nonlocal equations, amplitudes of solutions are related to the independent variables that are reversed in the real nonlocal reductions, e.g. (44 ), (67 ) and (120 ).