修正离散KP系列的流方程

修正离散KP系列的流方程

王晓翊¹, 程纪鹏^,², 李红云³^,⁴

The Flow Equations of the Modified Discrete KP Hierarchy

Wang Xiaoyi¹, Cheng Jipeng^,², Li Hongyun³^,⁴

通讯作者: 程纪鹏, E-mail:chengjp@cumt.edu.cn

收稿日期: 2018-02-17

Received: 2018-02-17

摘要

该文主要研究修正离散KP系列的流方程问题，给出流方程的一般表示形式.

关键词： 离散可积系统 ; 修正离散KP系列 ; 流方程

Abstract

In this paper, we mainly study the flow equations of the modified discrete KP hierarchy, and derive a general formula of the flow equation.

Keywords： Discrete integrable system ; The modified discrete KP hierarchy ; Flow equation

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王晓翊, 程纪鹏, 李红云. 修正离散KP系列的流方程. 数学物理学报[J], 2019, 39(2): 386-392 doi:

Wang Xiaoyi, Cheng Jipeng, Li Hongyun. The Flow Equations of the Modified Discrete KP Hierarchy. Acta Mathematica Scientia[J], 2019, 39(2): 386-392 doi:

离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子 $\partial_x$ 及变量 $x$ 分别替换成差分算子 $\Delta$ 和离散变量 $n$ .到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究.

与经典可积系统类似,流方程问题^[15-18]是离散可积系统中非常重要且基本的问题,如何将离散可积系统的流方程表示出来是离散可积系统研究的核心问题之一.本文主要研究修正离散KP系列的流方程问题.

首先,移动算子 $\Lambda$ 和差分算子 $\Delta$ 定义如下

$\begin{eqnarray}\Lambda(v(n))&=&v(n+1), \end{eqnarray}$

(1)

$\begin{eqnarray}\Delta(v(n))&=&v(n+1)-v(n)=(\Lambda-I)(v(n)).\end{eqnarray}$

(2)

对 $\forall j\in Z$ ,算子 $\Delta$ 满足Leibniz法则

$\begin{eqnarray}\Delta^jv&=&\sum\limits_{i=0}^\infty C_j^i \Delta^i(v)(n+j-i)\Delta^{j-i}, \end{eqnarray}$

(3)

$\begin{eqnarray}C_j^i&=&\frac{j(j-1)(j-2)\cdots(j-i+1)}{i!}.\end{eqnarray}$

(4)

修正离散KP系列^[11-14]的Lax方程为

$\begin{eqnarray}L_{t_m}&=&[(L^m)_{\geqslant1}, L], \end{eqnarray}$

(5)

其中, $L$ 为Lax算子,形式如下

$\begin{eqnarray}L=\sum\limits_{i=0}^\infty v_i\Delta^{-i+1}=v_0 \Delta+v_1+v_2\Delta^{-1}+\cdots, \end{eqnarray}$

(6)

这里, $v_i=v_i(n, x, t)=v_i(n, x, t_1, t_2, \cdots)$ ,并且记

$\begin{eqnarray}(L^m)_{\geqslant1}&=&\sum\limits_{s=0}^{m-1}b_{sm}(n)\Delta^{m-s}.\end{eqnarray}$

(7)

从观察

$\begin{equation} [(L^m)_{\geqslant1}, L]_{>1}=-[(L^m)_{<1}, L]_{>1}=0 \end{equation}$

(8)

出发,下面将给出 $b_{sm}(n)$ 的递推公式.为此，先计算 $[(L^m)_{\geq 1}, L]$ ,将(6)和(7)式代入即可.

$\begin{eqnarray} [(L^m)_{\geq 1}, L]&=&\sum\limits_{i=0}^\infty\sum\limits_{s=0}^{m-1}[b_{sm}(n)\Delta^{m-s}, v_i\Delta^{-i+1}]\nonumber\\ &=&\sum\limits_{i=0}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^\infty(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_i))\\&&-v_iC_{-i+1}^p\Delta^p (\Lambda^{-i+1-p}(b_{sm}(n))))\Delta^{m-s-p-i+1}\nonumber\\ &=&\sum\limits_{l=0}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^l(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{l-p}))\\&&-v_{l-p}C_{p-l+1}^p\Delta^p (\Lambda^{1-l}(b_{sm}(n))))\Delta^{m-s-l+1}\nonumber\\ &=&\sum\limits_{k=0}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p (\Lambda^{1-k+s}(b_{sm}(n))))\Delta^{m-k+1}\nonumber\\ &=&\sum\limits_{k=0}^{m-1}\sum\limits_{s=0}^k\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p (\Lambda^{1-k+s}(b_{sm}(n))))\Delta^{m-k+1}\nonumber\\ &&+\sum\limits_{k=m}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p (\Lambda^{1-k+s}(b_{sm}(n))))\Delta^{m-k+1}.\end{eqnarray}$

(9)

这里第三个等式中记 $l=i+p$ ,在第四个等式中记 $k=s+l$ .因此

$\begin{eqnarray} [(L^m)_{\geq 1}, L]_{>1}&=&\sum\limits_{k=0}^{m-1}\sum\limits_{s=0}^k\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\nonumber\\ &&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p(\Lambda^{1-k+s}(b_{sm}(n))))\Delta^{m-k+1}. \end{eqnarray}$

(10)

所以,由(8)和(10)式可知,对 $k=0, 1, 2, \cdots, m-1$ 有

$\begin{equation} \sum\limits_{s=0}^k\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p \Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))-v_{k-s-p}C_{p-k+s+1}^p\Delta^p (\Lambda^{1-k+s}(b_{sm}(n))))=0. \end{equation}$

(11)

特别地,当 $k=0$ 时,有

$\begin{equation} b_{0m}(n)\Lambda^m(v_0)-v_0\Lambda(b_{0m}(n))=0, \end{equation}$

(12)

对(12)式两边取自然对数,整理得

$\begin{equation} ln(b_{0m}(n+1))-ln(b_{0m}(n))=ln(v_0(n+m))-ln(v_0(n)), \end{equation}$

(13)

即

$\begin{equation} \Delta(ln(b_{0m}(n)))=\Delta((I+\Lambda+\Lambda^2+\cdots+\Lambda^{m-1})(ln(v_0(n)))), \end{equation}$

(14)

从而

$\begin{equation} b_{0m}(n)=v_0(n)v_0(n+1)\cdots v_0(n+m-1), \ m=1, 2, 3, \cdots. \end{equation}$

(15)

对于 $k>0$ 的情况,根据(11)式整理可得

$\begin{eqnarray} -b_{km}(n)\Lambda^{m-k}(v_0)+v_0\Lambda(b_{km}(n))&=&\sum\limits_{s=0}^{k-1}\sum\limits_{p=0}^{k-s}(b_{sm}(n)C_{m-s}^p \Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\nonumber\\ &&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p (\Lambda^{1-k+s}(b_{sm}(n)))). \end{eqnarray}$

(16)

进一步可以将(16)的左边部分计算如下

$\begin{eqnarray} v_0(n)\Lambda(b_{km}(n))-b_{km}(n)\Lambda^{m-k}(v_0(n))&=&(v_0(n)\Lambda-v_0(n+m-k))b_{km}(n)\nonumber\\ &=&(v_0(n)\Delta+v_0(n)-v_0(n+m-k))b_{km}(n), \end{eqnarray}$

(17)

为了求出 $b_{km}(n)$ ,先看如下的引理.

引理 1 假设 $v_0(n)\Delta+v_0(n)-v_0(n+m-k)=f(n)\Delta g(n)$ ,则

$\begin{eqnarray} f(n)&=&v_0(n)v_0(n+1)\cdots v_0(n+m-k), \end{eqnarray}$

(18)

$\begin{eqnarray}g(n)&=&\frac{1}{v_0(n)v_0(n+1)\cdots v_0(n+m-k-1)}. \end{eqnarray}$

(19)

证由于 $f(n)\Delta g(n)=f(n)g(n+1)\Delta+f(n)\Delta(g(n))$ ,与假设条件比较,得

$\begin{eqnarray}f(n)g(n+1)=v_0(n), \quadf(n)\Delta (g(n))=v_0(n)-v_0(n+m-k), \nonumber\end{eqnarray}$

两式相除得

$\begin{equation}\frac{g(n)}{g(n+1)}=\frac{v_0(n+m-k)}{v_0(n)}.\nonumber\end{equation}$

(20)

对上式两边取自然对数,有

$\begin{eqnarray}-\Delta(ln(g(n)))&=&(\Lambda^{m-k}-I)(ln(v_0(n)))\nonumber\\&=&(\Lambda-I)(I+\Lambda+\Lambda^2+\cdots+\Lambda^{m-k-1})(ln(v_0(n)))\nonumber\\&=&\Delta\left((I+\Lambda+\cdots+\Lambda^{m-k-1})(ln(v_0(n)))\right), \nonumber\end{eqnarray}$

从而

$\begin{equation}g(n)=\frac{1}{v_0(n)v_0(n+1)\cdots v_0(n+m-k-1)}, \nonumber\end{equation}$

(21)

将 $g(n)$ 代入 $f(n)g(n+1)=v_0(n)$ 中,得

$\begin{equation}f(n)=\frac {v_0(n)}{g(n+1)}=v_0(n)v_0(n+1)\cdots v_0(n+m-k).\nonumber\end{equation}$

(22)

证毕.

由(16)式, (17)式及引理1,可得

$\begin{eqnarray}(v_0(n)\Delta+v_0(n)-v_0(n+m-k))b_{km}(n)&=&(f(n)\Delta g(n))b_{km}(n)\nonumber\\ &=&\sum\limits_{s=0}^{k-1}\sum\limits_{p=0}^{k-s}\Big(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p(\Lambda^{1-k+s}(b_{sm}(n)))\Big), \end{eqnarray}$

(23)

从而

$\begin{eqnarray}b_{km}(n)&=&\frac{1}{g(n)}\Delta^{-1}\bigg(\frac{1}{f(n)}\sum\limits_{s=0}^{k-1}\sum\limits_{p=0}^{k-s}\Big(b_{sm}(n)C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\nonumber\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p(\Lambda^{1-k+s}(b_{sm}(n)))\Big)\bigg).\end{eqnarray}$

(24)

综合上述,给出 $b_{km}(n)$ 的表示形式.

定理 2

$\begin{eqnarray}b_{0m}(n)&=&v_0(n)v_0(n+1)\cdots v_0(n+m-1), \nonumber\\b_{km}(n)&=&v_0(n)v_0(n+1)\cdots v_0(n+m-k-1)\Delta^{-1}\bigg(\frac{1}{v_0(n)\cdots v_0(n+m-k)}\nonumber\\&&\sum\limits_{s=0}^{k-1}\sum\limits_{p=0}^{k-s}\Big(b_{sm}C_{m-s}^p\Delta^p\big(\Lambda^{m-s-p}(v_{k-s-p})\big)-v_{k-s-p}C_{p-k+s+1}^p\Delta^p\big(\Lambda^{1-k+s}(b_{sm})\big)\Big)\bigg)\nonumber\\&&k=1, 2, \cdots, m-1.\end{eqnarray}$

(25)

由定理2罗列一些结果如下

$\begin{eqnarray}(L)_{\geqslant1}&=&v_0(n)\Delta, \end{eqnarray}$

(26)

$\begin{eqnarray}(L^2)_{\geqslant1}&=&v_0(n)v_0(n+1)\Delta^2+v_0(n)\Big(v_1(n+1)+v_1(n)+v_0(n+1)-v_0(n)\Big)\Delta, \end{eqnarray}$

(27)

$\begin{eqnarray}(L^3)_{\geqslant1}&=&v_0(n)v_0(n+1)v_0(n+2)\Delta^3+v_0(n)v_0(n+1)\Big(v_1(n+2)\nonumber\\&&+v_1(n+1)+v_1(n)+2v_0(n+2)-v_0(n+1)-v_0(n)\Big)\Delta^2\nonumber\\&&+v_0(n)\Big(v_2(n+2)v_0(n+1)+v_2(n+1)v_0(n)+v_2(n)v_0(n-1)\nonumber\\&&+2v_1(n+2)v_0(n+1)+v_1(n+1)v_1(n+1)+v_1(n+1)v_1(n)\nonumber\\&&-v_1(n+1)v_0(n)+v_1(n)v_1(n)+v_1(n)v_0(n+1)-2v_1(n)v_0(n)\nonumber\\&&+v_0(n+2)v_0(n+1)-v_0(n+1)v_0(n+1)-v_0(n+1)v_0(n)\nonumber\\&&+v_0(n)v_0(n)\Big)\Delta, \end{eqnarray}$

(28)

$\begin{eqnarray}(L^4)_{\geqslant1}&=&v_0(n)v_0(n+1)v_0(n+2)v_0(n+3)\Delta^4+v_0(n)v_0(n+1)v_0(n+2)\nonumber\\&&\Big(v_1(n+3)+v_1(n+2)+v_1(n+1)+v_1(n)+3v_0(n+3)-v_0(n+2)\nonumber\\&&-v_0(n+1)-v_0(n)\Big)\Delta^3+v_0(n)v_0(n+1)\Big(v_2(n+3)v_0(n+2)\nonumber\\&&+v_2(n+2)v_0(n+1)+v_2(n+1)v_0(n)+v_2(n)v_0(n-1)\nonumber\\&&+3v_1(n+3)v_0(n+2)+v_1(n+2)v_1(n+2)+v_1(n+2)v_1(n+1)\nonumber\\&&+v_1(n+2)v_1(n)+v_1(n+2)v_0(n+2)-v_1(n+2)v_0(n+1)\nonumber\\&&-v_1(n+2)v_0(n)+v_1(n+1)v_1(n+1)+v_1(n+1)v_1(n)\nonumber\\&&+2v_1(n+1)v_0(n+2)-2v_1(n+1)v_0(n+1)-v_1(n+1)v_0(n)\nonumber\\&&+v_1(n)v_1(n)+2v_1(n)v_0(n+2)-v_1(n)v_0(n+1)-2v_1(n)v_0(n)\nonumber\\&&+3v_0(n+3)v_0(n+2)-2v_0(n+2)v_0(n+2)-2v_0(n+2)v_0(n+1)\nonumber\\&&-2v_0(n+2)v_0(n)+v_0(n+1)v_0(n+1)+v_0(n+1)v_0(n)\nonumber\\&&+v_0(n)v_0(n)\Big)\Delta^2+v_0(n)\Big(v_3(n+3)v_0(n+2)v_0(n+1)\nonumber\\&&+v_3(n+2)v_0(n+1)v_0(n)+v_3(n+1)v_0(n)v_0(n-1)\nonumber\\&&+v_3(n)v_0(n-1)v_0(n-2)+3v_2(n+3)v_0(n+2)v_0(n+1)\nonumber\\&&+v_2(n+2)v_1(n+2)v_0(n+1)+2v_2(n+2)v_1(n+1)v_0(n+1)\nonumber\\&&+v_2(n+2)v_1(n)v_0(n+1)-v_2(n+2)v_0(n+2)v_0(n+1)\nonumber\\&&+2v_2(n+1)v_1(n+1)v_0(n)+2v_2(n+1)v_1(n)v_0(n)\nonumber\\&&-2v_2(n+1)v_0(n)v_0(n)+v_2(n+1)v_0(n)v_0(n-1)\nonumber\\&&+v_2(n)v_1(n+1)v_0(n-1)+2v_2(n)v_1(n)v_0(n-1)\nonumber\\&&+v_2(n)v_1(n-1)v_0(n-1)+v_2(n)v_0(n+1)v_0(n-1)\nonumber\\&&-2v_2(n)v_0(n)v_0(n-1)-v_2(n)v_0(n-1)v_0(n-1)\nonumber\\&&+v_2(n)v_0(n-1)v_0(n-2)+3v_1(n+3)v_0(n+2)v_0(n+1)\nonumber\\&&+2v_1(n+2)v_1(n+2)v_0(n+1)+2v_1(n+2)v_1(n+1)v_0(n+1)\nonumber\\&&+2v_1(n+2)v_1(n)v_0(n+1)-v_1(n+2)v_0(n+3)v_0(n+1)\nonumber\\&&-2v_1(n+2)v_0(n+1)v_0(n+1)-2v_1(n+2)v_0(n+1)v_0(n)\nonumber\\&&+v_1(n+1)v_1(n+1)v_1(n+1)+v_1(n+1)v_1(n+1)v_1(n)\nonumber\\&&-v_1(n+1)v_1(n+1)v_0(n+1)-v_1(n+1)v_1(n+1)v_0(n)\nonumber\\&&+v_1(n+1)v_1(n)v_1(n)-2v_1(n+1)v_1(n)v_0(n)\nonumber\\&&+v_1(n+1)v_0(n+2)v_0(n+1)-v_1(n+1)v_0(n+1)v_0(n+1)\nonumber\\&&+v_1(n+1)v_0(n)v_0(n)+v_1(n)v_1(n)v_1(n)+v_1(n)v_1(n)v_0(n+1)\nonumber\\&&-3v_1(n)v_1(n)v_0(n)+v_1(n)v_0(n+2)v_0(n+1)\nonumber\\&&-v_1(n)v_0(n+1)v_0(n+1)-2v_1(n)v_0(n+1)v_0(n)\nonumber\\&&+3v_1(n)v_0(n)v_0(n)+v_0(n+3)v_0(n+2)v_0(n+1)\nonumber\\&&-v_0(n+2)v_0(n+2)v_0(n+1)-v_0(n+2)v_0(n+1)v_0(n+1)\nonumber\\&&-v_0(n+2)v_0(n+1)v_0(n)+v_0(n+1)v_0(n+1)v_0(n+1)\nonumber\\&&+v_0(n+1)v_0(n+1)v_0(n)+v_0(n+1)v_0(n)v_0(n)\nonumber\\&&-v_0(n)v_0(n)v_0(n)\Big)\Delta.\end{eqnarray}$

(29)

有了以上的结论和结果,下面将给出修正离散KP系列的流方程的表示形式.根据(8)和(9)式,有

$\begin{eqnarray}[(L^m)_{\geq 1}, L]&=&\sum\limits_{k=m}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^{k-s}(b_{sm}C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-v_{k-s-p}C_{p-k+s+1}^p\Delta^p(\Lambda^{1-k+s}(b_{sm})))\Delta^{m-k+1}\nonumber\\&=&\sum\limits_{k=m}^\infty\sum\limits_{s=0}^{m-1}\sum\limits_{p=0}^{k-s}(b_{sm}C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-(-1)^pC_{k-s-2}^p\Delta^p(\Lambda^{1-k+s}(b_{sm})))\Delta^{m-k+1}\nonumber\\&=&\sum\limits_{k=m}^\infty\sum\limits_{s=0}^{m-1}(b_{sm}C_{m-s}^{k-s}\Delta^{k-s}(\Lambda^{m-s}(v_0))+b_{sm}C_{m-s}^{k-s-1}\Delta^{k-s-1}(\Lambda^{m-k-1}(v_1)))\nonumber\\&&+\sum\limits_{p=0}^{k-s-2}(b_{sm}C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s-p}))\\&&-(-1)^pC_{k-s-2}^p\Delta^p(\Lambda^{1-k+s}(b_{sm})))\Delta^{m-k+1}\nonumber\\&=&\sum\limits_{k\geqslant1}^\infty\sum\limits_{s=0}^{m-1}(b_{sm}C_{m-s}^{k-s}\Delta^{k-s}(\Lambda^{m-s}(v_0))+b_{sm}C_{m-s}^{k-s-1}\Delta^{k-s-1}(\Lambda^{m-k-1}(v_1)))\nonumber\\&&+\sum\limits_{p=0}^{k-s+m-3}(b_{sm}C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s+m-p-1}))\\&&+(-1)^pC_{k-s+m-3}^p\Delta^p(\Lambda^{2-k-m+s}(b_{sm})))\Delta^{-k+2}, \end{eqnarray}$

(30)

比较(30), (5)和(6)式,得到关于流方程的表示形式.

定理 3 对于修正离散KP系列,其流方程表示如下

$\begin{eqnarray}v_{k, t_m}&=&\sum\limits_{s=0}^{m-1}(b_{sm}C_{m-s}^{k-s}\Delta^{k-s}(\Lambda^{m-s}(v_0))+b_{sm}C_{m-s}^{k-s-1}\Delta^{k-s-1}(\Lambda^{m-k-1}(v_1)))\nonumber\\&&+\sum\limits_{p=0}^{k-s+m-3}(b_{sm}C_{m-s}^p\Delta^p(\Lambda^{m-s-p}(v_{k-s+m-p-1}))\\&&+(-1)^pC_{k-s+m-3}^p\Delta^p(\Lambda^{2-k-m+s}(b_{sm}))).\end{eqnarray}$

(31)

依据(31)和(27)式,现给出 $t_3$ 流如下

$\begin{eqnarray}v_{0, t_3}&=&v_0(n)v_0(n+1)v_0(n+2)[v_0(n+3)+v_2(n+3)]+v_0(n-1)v_0(n)v_0(n+1), \quad \end{eqnarray}$

(32)

$\begin{eqnarray}v_{1, t_3}&=&2v_0(n-2)v_0(n-1)v_0(n)-v_0(n-1)v_0(n)v_0(n+1)\nonumber\\&&+v_0(n)v_0(n+1)v_0(n+2)[-3v_0(n+3)+3v_0(n+4)+v_1(n+1)\nonumber\\&&-3v_2(n+2)+3v_2(n+3)+v_3(n+3)]\nonumber\\&&+v_0(n-1)v_0(n)[-v_0(n-1)-v_0(n)+2v_0(n+1)+v_1(n-1)\nonumber\\&&+v_1(n)+v_1(n+1)]+v_0(n)v_0(n+1)[v_0(n+2)+v_2(n+2)][-v_0(n)\nonumber\\&&-v_0(n+1)+2v_0(n+2)+v_1(n)+v_1(n+1)+v_1(n+2)].\end{eqnarray}$

(33)

本论文主要得到了修正离散KP系列的流方程递推公式,对于进一步研究修正离散KP系列有很大帮助.

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Commutative. Lax pairs, symmetries and an Adilic flag manifold

2000

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

Discrete Lax equations and difference calculus

1985

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

The Wronskian solution of the constrained discrete Kadomtsev-Petviashvili hierarchy

2016

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

The string aquation and the τ-function witt constraints for the discrete Kadomtsev-Petviashvili hierarchy

2013

The determinant representtion of the gauge transformation for the discrete KP hierarchy

2010

The gauge transformation of the constrained semi-discrete KP hierarchy

2013

The Virasoro action on the tau function for the constrained discrete KP hierarchy

2013

The compatibility of additional symmetry and geuge transformations for the constrained discrete Kadomtsev-Petviashvili hierarchy

2015

Sato B?cklund transformation, additional symmetries and ASvM formula for the discrete KP hierarchy

2010

Virasoro type algebraic structure hidden in the constrained discrete KP hierarchy

2013

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

... 修正离散KP系列^[11-14]的Lax方程为 ...

Gauge equivalence and ι-reductions of the differential-difference KP equation

2000

The successive application of the gauge transformation for the modified semidiscrete KP hierarchy

2016

Gauge transformations of constrained discrete modified KP systems with self-consistent sources

2017

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

... 修正离散KP系列^[11-14]的Lax方程为 ...

General formulae for two pseudo-differential operaters

2008

... 与经典可积系统类似,流方程问题^[15-18]是离散可积系统中非常重要且基本的问题,如何将离散可积系统的流方程表示出来是离散可积系统研究的核心问题之一.本文主要研究修正离散KP系列的流方程问题. ...

Addendum to "some general formulas in the Sato theory"

2003

Some general formulas in the Sato theory

2003

A general formula of flow equations for Harry-Dym hierarchy

2011

... 离散可积系统^[1-2]是现代数学物理中重要的研究课题之一,是将经典可积系统中的微分算子

$\partial_x$

及变量

$x$

分别替换成差分算子

$\Delta$

和离散变量

$n$

.到目前为止,重要的离散可积系统,如离散KP系列^{[3-10, 18]},修正离散KP系列^[11-14]等都得到了研究. ...

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