数学物理学报, 2019, 39(3): 451-460 doi:

论文

一类广义浅水波KdV方程的可积性研究

郝晓红,1, 程智龙,2

The Integrability of the KdV-Shallow Water Waves Equation

Hao Xiaohong,1, Cheng Zhilong,2

通讯作者: 程智龙, E-mail: zhilong0793@sina.cn

收稿日期: 2017-04-10  

基金资助: 安徽省自然科学研究项目.  KJ2016A071

Received: 2017-04-10  

Fund supported: the Natural Science Foundation of Anhui Province.  KJ2016A071

作者简介 About authors

郝晓红,haoxiaohong200866@163.com , E-mail:haoxiaohong200866@163.com

摘要

该文应用双Bell多项式,系统研究了一类广义浅水波KdV方程的可积性.先构造出双线性表达式、Bäklund变换,再通过Bäklund变换线性化得到孤子解与Lax对.最后通过级数展开式代入得到无穷守恒律,从而证明此方程具有可积性.

关键词: Bäklund变换 ; Lax对 ; 无穷守恒律

Abstract

In this paper, the binary Bell polynomials to construct bilinear forma, bilinear Bäcklund transformation, Lax pair of the KdV-shallow water waves equation. Through bilinear Bäcklund transformation, some soliton solutions are presented. Moreover, the infinite conservation laws are also derived by Bell polynomials, all conserved densities and fluxes are given with explicit recursion formulas.

Keywords: Bäcklund transformation ; Lax pair ; Infinite conservation laws

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本文引用格式

郝晓红, 程智龙. 一类广义浅水波KdV方程的可积性研究. 数学物理学报[J], 2019, 39(3): 451-460 doi:

Hao Xiaohong, Cheng Zhilong. The Integrability of the KdV-Shallow Water Waves Equation. Acta Mathematica Scientia[J], 2019, 39(3): 451-460 doi:

1 引言

近几十年以来,通过双线性方法[1-3, 5], Bäcklund变换[6-10, 19],黎曼$\theta$函数法[5-8]等研究非线性偏微分方程中可积性[9, 12]与精确解[4-9, 15-17, 19]越来越受到一些专家学者的关注.

1996年, Lambert, Gilson和Nimmo建立了Bell多项式与Hirota双线性算子之间的关系[14],通过转换关系得到双线性Bäcklund变换,这个方法很有效地避免了求Bäcklund变换过程中使用交换公式繁琐的计算,简洁实用.并且直接对其做线性化还可以得到方程的Lax对[9, 12],最后将级数展开式代入计算求解得到方程的无穷守恒律.

本文重点应用Bell多项式研究一类广义浅水波KdV方程

$\begin{equation} r_{t}-\alpha r_{xxt}-4\alpha r r_{t}-2 \alpha r_{x}\int_x^{\infty}r_{t}{\rm d}x+ \alpha r_x+\beta r_{xxx}+6\beta rr_x=0, \end{equation}$

其中$\alpha$$\beta$是任意的非零常数.方程(1.1)包涵一些特殊的物理方程:

(ⅰ)当$\alpha$=0, $\beta$=1时

$ \begin{equation} r_{t}+r_{xxx}+6rr_x=0, \end{equation} $

即为KdV方程[1-3];

(ⅱ)当$\alpha$=1, $\beta$=0时

$ \begin{equation} r_{t}-r_{xxt}-4r r_{t}-2 r_{x}\int_x^{\infty}r_{t}{\rm d}x+r_x=0, \end{equation} $

即为浅水波方程[1, 4]. Ablowitz等用反散射法研究了此类方程[13]. Gilson, Nimmo和Willox考虑其Wronskian形式的孤子解[18]. Shang和Hong通过其次平衡法与扩展双曲函数法得到大量的精确解与Bäcklund变换形式[19], Wazwaz则应用简化形式的Hirota双线性法求解出$N$-孤子解[4].

本文重点研究方程(1.1)的可积性,即为双线性表达式、Bäcklund变换、Lax对与无穷守恒律.本文结构如下:第二部分给出必要Bell多项式的定义与性质;第三部分应用Bell多项式法研究其可积性;最后部分给出结论与参考文献.

2 多维双Bell多项式

首先我们先给出Bell多项式的定义以及性质:

定义2.1   设$f=f(x_{1}, x_{2}, \cdots, x_{n})$是具有$n$个变量的${\cal C}^{\infty}$函数,则称

$ \begin{equation}Y_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(f)\equiv Y_{n_{1}, \cdots, n_{l}}(f_{r_{1}x_{1}, \cdots, r_{l}x_{l}})= e^{-f}\partial_{x_{1}}^{n_{1}}\cdots\partial_{x_{l}}^{n^{l}}e^{f} \end{equation} $

为Bell多项式[9, 14].其中$f_{r_{1}x_{1}, \cdots, r_{l}x_{l}}=\partial_{x_{1}}^{n_{1}}\cdots\partial_{x_{l}}^{n^{l}}f (r_{1}=0, \cdots, n_{1};\cdots; r_{l}=0, \cdots, n_{l}.)$.$f=f(x, t)$时,对应的2-维Bell多项式,举例为

$\begin{array}{l}&&Y_{x}{(f)}=f_{x}, \qquad Y_{2x}{(f)}=f_{2x}+f_{x}^2, \qquadY_{x, t}{(f)}=f_{x, t}+f_{x}f_{t}, \\&&Y_{2x, t}{(f)}=f_{2x, t}+f_{2x}f_{t}+2f_{x, t}f_{x}+f_{x}^{2}f_{t}, \\&&Y_{3x}{(f)}=f_{3x}+3f_{2x}f_{x}+f_{x}^{3}, \qquad\cdots .\end{array}$

定义2.2  基于上述Bell多项式定义,多维的双Bell多项式定义为

${\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v, w) = Y_{n_{1}, \cdots, n_{l}}(f)|_{f_{r_{1}x_{1}, \cdots, r_{l}x_{l}}}\\ = \left\{ \begin{array}{ll}v_{r_{1}x_{1}, \cdots, r_{l}x_{l}}, & r_{1}+r_{2}+ \cdots +r_{l}\ \mbox{为奇数, } \\w_{r_{1}x_{1}, \cdots, r_{l}x_{l}}, & n_{1}+n_{2}+\cdots+n_{l}\ \mbox{为偶数, }\end{array} \right.$

则双Bell多项式可以表示为函数具有$v$$w$的形式,举例为

$\begin{array}{l}&&{\cal Y}_{x}(v)=v_{x}, {\cal Y}_{2x}(v, w)=w_{2x}+v_{x}^{2}, {\cal Y}_{x, t}(v, w)=w_{xt}+v_{x}v_{t}, \\&&{\cal Y}_{2x, t}(v, w)=v_{2x, t}+w_{2x}v_{t}+2w_{xt}v_x+v_x^2v_t, \\&&{\cal Y}_{3x}(v, w)=v_{3x}+3v_{x}w_{2x}+v_{x}^{3}, \cdots .\end{array}$

定义2.3  ${\cal Y}$多项式和Hirota双线性$D$算子之间的关系可由如下等式得出

$\begin{equation}{\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v=\ln F/G, w=\lnFG)=(FG)^{-1}D_{x_{1}}^{n_{1}}\cdots D_{x_{l}}^{n_{l}}F\cdot G, \end{equation}$

其中$n_{1}+n_{2}+\cdots+n_{l}\geq 1$.算子$D_{x_1}, \cdots, D_{x_l}$为Hirota双线性算子定义为

特别的,当$F=G$时, $(2.5)$式则化为

$\begin{array}{ll}(F)^{-2}D_{x_{1}}^{n_{1}}\cdots D_{x_{l}}^{n_{l}}F\cdot F = {\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(0, q=2\ln F)\\ = \left\{\begin{array}{ll} 0, &n_{1}+n_{2}+ \cdots+n_{l} \ \mbox{为奇数, } \\ P_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(q), & n_{1}+n_{2}+ \cdots+n_{l}\ \mbox{为偶数.} \end{array}\right.\end{array}$

称之为$P$多项式,此类多项式具有易于被识别的偶数部分划分结构的特点,如

$\begin{array}{l}&&P_{2x}(q)=q_{2x}, \qquad P_{x, t}(q)=q_{xt}, \\&&P_{4x}(q)=q_{4x}+3q_{2x}^{2}, \qquadP_{6x}(q)=q_{6x}+15q_{2x}q_{4x}+15q_{2x}^3 \cdots .\end{array}$

定义2.4   双Bell多项式${\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v, w)$可分离成$P$多项式和$Y$多项式的组合形式

$\begin{array}{l} (FG)^{-1}D_{x_{1}}^{n_{1}}\cdots D_{x_{l}}^{n_{l}}F\cdot G\\ = {\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v, w)|_{v=\ln F/G, w=\ln FG} ={\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v, v+q)|_{v=\ln F/G, q=2\ln G}\\ = \sum\limits_{n_{1}+\cdots+n_{l}=even}^{}\sum\limits_{r_{1} =0}^{n_{1}}\cdots\sum\limits_{r_{l}=0}^{n_{l}}\prod\limits_{i=1}^{l}(n_{i}, r_{i})' P_{r_{1}x_{1}, \cdots, r_{l}x_{l}}(q)Y_{(n_{1}-r_{1})x_{1}, \cdots, (n_{l}-r_{l})x_{l}}(v), \end{array}$

注意

$\begin{equation}Y_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v)|_{v=\ln\psi}=\frac{\psi_{n_{1}x_{1}, \cdots, n_{l}x_{l}}}{\psi}, \end{equation}$

这意味着一个Bell多项式${\cal Y}_{n_{1}x_{1}, \cdots, n_{l}x_{l}}(v, w)$可以通过Hopf-Cole变换$v=\ln\psi$ (即$\psi=F/G)$进行线性化.性质(2.8)、(2.9)在推导孤子方程的Lax对时起到重要作用.

接着应用Bell多项式求解无穷守恒律时,只需作一个新的变量变换

$\begin{equation}\eta=\frac{(q_{x_{k}}' -q_{x_{k}})}{2}, \end{equation}$

这里$q$$q' $满足$q=w-v$, $q' =w+v$, $x_{k}$$x_{1}, \cdots, x_{l}$中任意随机变量.则由条件

$\begin{equation}C(q', q)=E(q' )-E(q)=0\end{equation}$

出发可以得到一组约束条件,只要把这一组约束条件化为Bell多项式以及关于某个变量的微分的形式,即一个Riccati方程

$\begin{equation}\eta_{x_{k}}+f(\eta)=0\end{equation}$

和一个离散型的方程

$\begin{equation}\partial_{x_{1}}F_{1}(\eta)+\cdots+\partial_{x_{l}}F_{l}(\eta)=0.\end{equation}$

然后,通过将级数展开式代入计算就可以从中导出非线性方程的守恒律.

3 一类广义浅水波KdV方程

$r(x, t)=u_x(x, t)$,方程(1.1)化成

$\begin{equation}u_{xt}-\alpha u_{xxxt}-4\alpha u_xu_{xt}-2\alpha u_{xx}u_{t}+ \alpha u_{xx}+\beta u_{xxxx}+6 \beta u_xu_{xx}=0, \end{equation}$

在本部分,通过Bell多项式的应用,给出方程(3.1)双线性表达式, Bäklund变换, Lax对和无穷守恒律.

3.1 双线性表达式

定理3.1  作$u=q_x, $变换,方程(3.1)线性化为

$\begin{array}{l}&&(D_{x}^4+D_{x}D_{z})G\cdot G=0, \\&&[D_{x}D_{t}+\alpha(-\frac{2}{3}D_{t}D_{x}^{3}+\frac{1}{3}D_{t}D_{z}+D_{x}^{2})+\betaD_{x}^{4}]G\cdot G=0, \end{array}$

其中$z$为辅助变量.

  为了得到方程(3.1)的双线性化表达式,首先引入一个辅助变量$q$,且令

$\begin{equation}u=cq_{x}, \end{equation}$

其中$c$是一个待选择的可以与方程(3.1) $P$ -多项式相关联的常数.将方程(3.3)代入方程(3.1),得到

$\begin{equation}q_{2x, t}+\alpha(-\frac{2}{3}q_{4x, t}-2cq_{3x}q_{xt}-4cq_{2x}q_{2x, t}-\frac{1}{3}q_{4x, t}+q_{3x})+\beta(q_{5x}+6cq_{2x}q_{3x})=0, \end{equation}$

并将其对$x$积分一次可得

$ \begin{equation}E(q)\equivq_{xt}+\alpha\big(-\frac{2}{3}(q_{3x, t}+3cq_{2x}q_{xt})-\frac{1}{3}\partial_x^{-1}\partial_t(q_{4x}+3cq_{2x}^2)+q_{2x}\big)+\beta(q_{4x}+3cq_{2x}^2)=0, \end{equation}$

应用性质(2.7),得$ c=1$,则方程(3.5)可化为

$\begin{equation}E(q)\equivq_{xt}+\alpha\big(-\frac{2}{3}(q_{3x, t}+3q_{2x}q_{xt})-\frac{1}{3}\partial_x^{-1}\partial_t(q_{4x}+3q_{2x}^2)+q_{2x}\big)+\beta(q_{4x}+3q_{2x}^2)=0.\end{equation}$

为了将方程(3.6)写成双线性形式,需要消除$\partial_x^{-1}$积分项,为此引入一个辅助变量$z$,并附加一个约束条件

$\begin{equation}q_{4x}+3q_{2x}^2=-q_{xz}, \end{equation} $

那么,方程(3.6)可化为

$\begin{equation}E(q)\equivq_{xt}+\alpha\big(-\frac{2}{3}(q_{3x, t}+3q_{2x}q_{xt})+\frac{1}{3}q_{tz}+q_{2x}\big)+\beta(q_{4x}+3q_{2x}^2)=0.\end{equation}$

依据性质(2.5),方程(3.7)与方程(3.8)可被转化成如下$P$ -多项式组合形式

$\begin{array}{l}&&P_{4x}(q)+P_{xz}(q)=0, \\&&P_{xt}(q)+\alpha(-\frac{2}{3}P_{3x, t}(q)+\frac{1}{3}P_{tz}(q)+P_{2x}(q))+\beta P_{4x}(q)=0.\end{array}$

最后,由性质(2.6),在因变量变换

$\begin{equation}q=2\ln G\Longleftrightarrow u=cq_x=2(\ln G)_x\end{equation}$

作用下, (3.9)式可得方程(3.1)的双线性形式:

$\begin{array}{l}&&(D_{x}^4+D_{x}D_{z})G\cdot G=0, \\&&[D_{x}D_{t}+\alpha(-\frac{2}{3}D_{t}D_{x}^{3}+\frac{1}{3}D_{t}D_{z}+D_{x}^{2})+\betaD_{x}^{4}]G\cdot G=0. \end{array}$

证毕.

3.2 Bäcklund变换,孤子解与Lax对

定理3.2   假设$F$为双线性方程(3.11)的一个解,如果$G$满足

$\begin{array}{l}&&(D_{x}^2-\lambda)F\cdot G=0, \\&&[(1-3\alpha\lambda)D_{t}+(\alpha+3\beta\lambda)D_{x}-\alphaD_{x}^{2}D_{t}+\beta D_{x}^{3}]F\cdotG=0, \end{array}$

$G$即为方程(3.11)的另解,其中$\lambda$为任意参数.

  令$q$$q' $为方程(3.6)的两个不同的解

$\begin{equation} q=2\ln F, \qquad q' =2\ln G, \end{equation}$

相应地,引入两个新的变量

$ \begin{equation}w=\frac{q' +q}{2}=\ln (FG), \qquadv=\frac{q' -q}{2}=\ln(\frac{F}{G}), \end{equation} $

则二场条件为

$ \begin{array}{l}E(q' )-E(q) = E(w+v)-E(w-v)\\ = 2v_{xt}+\alpha[-2v_{3x, t}-4w_{2x}v_{x, t}-4w_{x, t}v_{2x} \\ -4\partial_x^{-1}(w_{2x}v_{2x, t}+w_{2x, t}v_{2x})+2v_{xx}]+\beta(2v_{4x}+12w_{xx}v_{xx})\\ = 2\partial_x[{\cal Y}_t(v)-\alpha{\cal Y}_{2x, t}{(v, w)}+\beta {\cal Y}_{3x}{(v, w)}]+R{(v, w)}=0, \end{array} $

其中

这个二场条件可以认为是在适当限制条件下便于求得Bäcklund变换.

为了将二场条件(3.15)写成一对限制条件,可加入一个限制条件,则$R(v, w)$可以表示成${\cal Y}$-多项式的$x$-导数结合形式.可选

$\begin{equation}{\cal Y}_{2x}{(v, w)}=w_{2x}+v_x^2=\lambda, \end{equation}$

其中$\lambda$为任意参数.由(3.16)式, $R(v, w)$可化为

$\begin{array}{l}R{(v, w)} = \alpha[2\lambdav_{xt}-4w_{2x}v_{xt}+4w_{2x, t}v_x-4\partial_x^{-1}(w_{2x}v_{2x, t}+w_{2x, t}v_{2x})+2v_{xx}]\\+6\beta(w_{2x}v_{xx}-v_xw_{3x}-v_x^2v_{2x})\\ = -6\alpha\lambda v_{xt}+2\alpha v_{2x}+6\beta\lambda v_{2x}, \end{array} $

此处应用$w_{2x, t}=-2v_xv_{xt}$$w_{2x}=\lambda-v_x^2.$

结合(3.15)-(3.17)式,可得到${\cal Y}$ -多项式系统

$\begin{array}{l}&&{\cal Y}_{2x}(v, w)-\lambda=0, \\&&\partial_x[{(1-3\alpha\lambda)}{\cal Y}_{t}{(v)}+(\alpha+3\beta\lambda){\cal Y}_{x}{(v)}-\alpha{\cal Y}_{2x, t}{(v, w)}+\beta{\cal Y}_{3x}{(v, w)}]=0, \end{array}$

其中第二个方程保留导数形式,这在之后的求解方程的守恒律时是非常重要.应用性质(2.5),从方程(3.15)可立即得到双线性Bäcklund变换

$\begin{array}{l}&&(D_{x}^2-\lambda)F\cdot G=0, \\&&[(1-3\alpha\lambda)D_{t}+(\alpha+3\beta\lambda)D_{x}-\alphaD_{x}^{2}D_{t}+\beta D_{x}^{3}]F\cdotG=0, \end{array}$

其中我们对系统(3.18)中第二个方程$x$进行积分.

通过此Bäcklund变换,我们可以很容易的求出其孤子解.接下来我们以一孤子解与二孤子解为例.

从平凡解$u(x, t)=0$出发,选择$F=1$,代入方程(3.19),可得方程(3.19)的另解,令$\lambda=\frac{k_1^2}{4}$,解得

$ \begin{equation}G_1=e^{\frac{\xi_1}{2}}+e^{-\frac{\xi_1}{2}}, \quad\xi_1=k_1x-\frac{\alpha k_1+\beta k_1^3}{1-\alphak_1^2}t+\xi_1^{(0)}, \end{equation} $

其中$k_1, \xi_1^{(0)}$为任意实数.方程(3.20)化为

$ \begin{equation}G_1=e^{-\frac{\xi_1}{2}}(1+e^{\xi_1}), \end{equation} $

依据(3.13)式有$q'=2\ln G$, (3.3)式有$u=q_x$, (3.1)式有$r=u_x$得一孤子解为

$\begin{equation}r=2\ln[1+e^{k_1x-\frac{\alpha k_1+\beta k_1^3}{1-\alphak_1^2}t+\xi_1^{(0)}}]_{xx}, \end{equation}$

再令$F=e^{\frac{\xi_1}{2}}+e^{-\frac{\xi_1}{2}}$,可得解

$\begin{equation}G_2=(k_1-k_2)(e^{\frac{\xi_1+\xi_2}{2}}+e^{-\frac{\xi_1+\xi_2}{2}})-(k_1+k_2)(e^{\frac{\xi_1-\xi_2}{2}}+e^{-\frac{\xi_1-\xi_2}{2}}), \quad\lambda=\frac{k_2^2}{4}, \end{equation}$

其中$\xi_j=k_j x-\frac{\alpha k_j+\beta k_j^3}{1-\alpha k_j^2}t+\xi_j^{(0)}, j=1, 2.$

作变换$\xi_j=\eta_j+\frac{1}{2}A_{12}$其中$e^{A_{12}}=\frac{(k_1-k_2)^2}{(k_1+k_2)^2}, j=1, 2$,则(3.23)式可化为

$\begin{equation}G_2=1+e^{\eta_1}+e^{\eta_2}+e^{\eta_1+\eta_2+A_{12}}, \end{equation}$

其对应方程的二孤子解

$\begin{equation}r=2\ln [1+e^{\eta_1}+e^{\eta_2}+e^{\eta_1+\eta_2+A_{12}}]_{xx}.\end{equation}$

接下来求解三孤子解,将$F=(k_1-k_2)(e^{\frac{\xi_1+\xi_2}{2}}+e^{-\frac{\xi_1+\xi_2}{2}})-(k_1+k_2)(e^{\frac{\xi_1-\xi_2}{2}}+e^{-\frac{\xi_1-\xi_2}{2}})$代入方程组(3.19),可得

$\begin{array}{l}G_3 = a(e^{\frac{\xi_1+\xi_2+\xi_3}{2}}+e^{-(\frac{\xi_1+\xi_2+\xi_3}{2}})+b(e^{\frac{-\xi_1+\xi_2+\xi_3}{2}}+e^{-\frac{-\xi_1+\xi_2+\xi_3}{2}})\\+c(e^{\frac{\xi_1-\xi_2+\xi_3}{2}}+e^{-\frac{\xi_1-\xi_2+\xi_3}{2}})+d(e^{\frac{\xi_1+\xi_2-\xi_3}{2}}+e^{-\frac{\xi_1+\xi_2-\xi_3}{2}}), \end{array}$

其中

$\xi_j=k_j x-\frac{\alpha k_j+\beta k_j^3}{1-\alpha k_j^2}t+\xi_j^{(0)}\ (j=1, 2, 3)$.

以此类推,得出$N$-孤子解为

$\begin{array}{l}&&G_n=\sum\limits_{\epsilon={\pm1}}\prod \limits_{1\leqj <l}^{n}\epsilon_l(\epsilon_jk_j-\epsilon_lk_l)e^{\frac{1}{2}\sum \limits _{j=1}^{n}\epsilon_j\xi_j}, \\&&\xi_j=k_j x-\frac{\alpha k_j+\beta k_j^3}{1-\alphak_j^2}t+\xi_j^{(0)}\ (j=1, 2, \cdots, n), \quad\lambda=\frac{k_n^2}{4}, \end{array}$

其中$\sum \limits_{\epsilon={\pm1}}$表示的是$\epsilon _j=1, -1\ (j=1, 2, \cdots, n) $的所有可能形式.

接下来,我们求解方程(3.1)的Lax对.

定理3.3   方程(3.1) Lax对为

$\begin{array}{l}&&L\psi=\psi_{xx}+(u_{x}-\lambda)\psi=0, \\&&M\psi=(1-3\alpha\lambda-u_{x})\psi_{t}+(\alpha+3\beta\lambda+3\betau_{x}-2\alpha u_{t})\psi_{x}-\alpha\psi_{xxt}+\beta\psi_{xxx}=0, \end{array}$

其中$\lambda$任意参数.

  由Hopf-Cole变换$v=\ln \psi$,由公式(2.8)和(2.9)可得

$\begin{array}{l}&&{\cal Y}_{x}(v)=\frac{\psi_{x}}{\psi}, \qquad{\cal Y}_{t}=\frac{\psi_{t}}{\psi}, \qquad{\cal Y}_{2x}(v, w)=q_{2x}+\frac{\psi_{2x}}{\psi}, \\&&{\cal Y}_{3x}(v, w)=3q_{2x}\frac{\psi_{x}}{\psi}+\frac{\psi_{3x}}{\psi}, \qquad{\cal Y}_{2x, t}(v, w)=2q_{xt}\frac{\psi_{x}}{\psi}+q_{2x}\frac{\psi_{t}}{\psi}+\frac{\psi_{2x, t}}{\psi}.\end{array}$

结合(3.29)式, (3.19)式可被线性化为带有参数$\lambda$的系统

$\begin{array}{l}L\psi=0, M\psi=0, \end{array}$

其中

$\begin{array}{l}&&L=\partial_{x}^2+q_{2x}-\lambda, \\&&M=(1-3\alpha\lambda-q_{2x})\partial_{t}+(\alpha+3\beta\lambda+3\betaq_{2x}-2\alpha q_{xt})\partial_{x}-\alpha\partial_t\partial_{x}^2+\beta \partial_{x}^3, \end{array}$

$q_{x}$代替$ u$

$\begin{array}{l}&&L\psi=\psi_{xx}+(u_{x}-\lambda)\psi=0, \\&&M\psi=(1-3\alpha\lambda-u_{x})\psi_{t}+(\alpha+3\beta\lambda+3\betau_{x}-2\alpha u_{t})\psi_{x}-\alpha\psi_{xxt}+\beta\psi_{xxx}=0.\end{array}$

得出Lax对,且容易验证相容性条件

$\begin{equation} [L, M]=q_{2x, t}+\alpha(-q_{4x, t}-4 q_{2x}q_{2x, t}-2 q_{3x}q_{xt}+q_{3x})+\beta(q_{5x}+6q_{2x}q_{3x})=0. \end{equation}$

证毕.

3.3 无穷守恒律

定理3.4   方程(3.1)具有如下无穷守恒律

$\begin{equation}I_{n, t}+F_{n, x}=0, n=1, 2, 3, \cdots , \end{equation}$

其中守恒密度$I_n$递推公式为

$$\begin{array}{l}&&I_{1}=-\frac{1}{2}u_{x}, \qquadI_{2}=-\frac{1}{2}I_{1, x}=\frac{1}{4}u_{2x}, \\& &\cdots\cdots\\&&I_{n}=-\frac{1}{2}\bigg[I_{n-1, x}+\sum\limits_{k=1}^nI_{k}I_{n-1-k}\bigg], \ n=2, 3, 4, \cdots. \end{array}

连带流$F_n$递推关系为

$\begin{array}{l}&&F_{1}=-\frac{1}{2}[\alpha(u_{x}-u_{xxt}-2\partial_x^{-1}(u_xu_{xt})-2u_tu_x)+\beta(u_{3x}+3u_x^2)], \\&&\cdots\cdots\\&&F_{n}=\alpha\bigg[I_n- I_{n, xt}+4\partial_x^{-1}\bigg(\sum\limits_{k=1}^{n}I_{k}I_{n+1-k, t}\bigg)+4\partial_x^{-1}\bigg(\sum\limits_{k=1}^nI_{k, t}\bigg)I_{n+1-k}\\&& +4\partial_x^{-1}\bigg(\sum\limits_{i+j+k=n}I_{i}I_{j, t}I_{k}\bigg)\bigg]\\&& +\beta\bigg(I_{n, 2x}-2\sum\limits_{i+j+k=n}I_{i}I_{j}I_{k}-6\sum\limits_{k=1}^nI_kI_{n+1-k}\bigg), \ n=2, 3, 4, \cdots .\end{array}$

  首先从公式(3.19)出发,由关系$\partial_x{\cal Y}_t(v)=\partial_t{\cal Y}_x(v)=v_{xt}$,则(3.19)式可化为

$\begin{array}{l}&&{\cal Y}_{2x}(v, w)-\lambda=0, \\&&\partial_t(1-3\alpha\lambda){\cal Y}_{x}+\partial_x[{(\alpha+3\beta\lambda){\cal Y}_{x}{(v)}-\alpha{\cal Y}_{2x, t}{(v, w)}+\beta{\cal Y}_{3x}{(v, w)}}]=0.\end{array}$

引入一个势函数

$\begin{equation}\eta=\frac{q_{x}' -q_{x}}{2}, \end{equation}$

那么从关系式(3.15)可得到

$\begin{equation}v_{x}=\eta, w_{x}=q_{x}+\eta, \end{equation} $

将(3.39)式代入(3.37)式,可得到一个Riccati -类型的方程

$\begin{equation}\eta_{x}+\eta^{2}+q_{2x}=\lambda\end{equation} $

和一个离散型方程

$\begin{equation}(1-4\alpha\lambda)\eta_{t}+\partial_{x}\{(\alpha+6\beta\lambda)\eta-\alpha\eta_{xt}+4\alpha[\partial_x^{-1}(\eta\eta_t)]\eta+\beta\eta_{2x}-2\beta\eta^3\}=0, \end{equation}$

其中我们取$\lambda=\varepsilon^2$.

进一步,在$\eta=\varepsilon+\tilde{\eta} $变换下, (3.40)式和(3.41)式可化为

$\begin{equation}\tilde{\eta}_x +\tilde{\eta}^2+2\varepsilon\tilde{\eta}+q_{xx}=0\end{equation}$

$\begin{equation}\tilde{\eta}_t+\partial_x\{\alpha\tilde{\eta}-\alpha\tilde{\eta}_{xt}+4\alpha\partial_x^{-1}(\tilde{\eta}\tilde{\eta}_t)\varepsilon+4\alpha\partial_x^{-1}(\varepsilon\tilde{\eta}_t+\tilde{\eta}\tilde{\eta}_t)\tilde{\eta}+\beta\tilde{\eta}_{2x}-2\beta\tilde{\eta}^3-6\beta\varepsilon\tilde{\eta}^2\}=0.\end{equation} $

将级数展开式

$ \begin{equation}\tilde{\eta}=\sum\limits_{n=1}^{\infty}I_{n}(q, q_{x}, \cdots)\varepsilon^{-n}\end{equation} $

代入(3.42)式且令上式中$\varepsilon$的幂次系数相等,就可以得到守恒密度$I_{n}$的递推关系式

$\begin{array}{l}&&I_{1}=-\frac{1}{2}u_{x}, \qquadI_{2}=-\frac{1}{2}I_{1, x}=\frac{1}{4}u_{2x}, \\&&I_{3}=-\frac{1}{2}(I_{2, x}+I_{1}^{2})=-\frac{1}{8}(u_{xxx}+u_x^2), \\&&I_{4}=-\frac{1}{2}(I_{3, x}+2I_{1}I_{2})=\frac{1}{16}(u_{4x}+4u_xu_{2x}), \\&&\cdots\cdots\\&&I_{n}=-\frac{1}{2}[I_{n-1, x}+\sum\limits_{k=1}^nI_{k}I_{n-1-k}], n=2, 3, 4, \cdots .\end{array}$

再将展开式(3.44)代入(3.43)式,则有

$\begin{array}{l}&&\sum\limits_{n=1}^{\infty}I_{n, t}\varepsilon^{-n}+\partial_{x}\bigg[\alpha\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}-\alpha\sum\limits_{n=1}^{\infty}I_{n, xt}\varepsilon^{-n}+4\alpha\partial_x^{-1}\bigg(\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}\sum\limits_{n=1}^{\infty}I_{n, t}\varepsilon^{-n}\bigg)\cdot\varepsilon\\&&+4\alpha\partial_x^{-1}\bigg(\varepsilon\cdot\sum\limits_{n=1}^{\infty}I_{n, t}\varepsilon^{-n}+\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}\sum\limits_{n=1}^{\infty}I_{n, t}\varepsilon^{-n}\bigg)\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}+\beta\sum\limits_{n=1}^{\infty}I_{n, 2x}\varepsilon^{-n}\\&&-2\beta\bigg(\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}\bigg)^{3}-6\beta\varepsilon\bigg(\sum\limits_{n=1}^{\infty}I_{n}\varepsilon^{-n}\bigg)^2\bigg]=0, \end{array}$

即为无穷守恒律

$\begin{equation}I_{n, t}+F_{n, x}=0, n=1, 2, 3, \cdots, \end{equation}$

在无穷守恒律(3.47)中, (3.45)式给出守恒密度$I_{n}$,连带流$F_n$即为

$\begin{array}{l}&&F_{1}=-\frac{1}{2}[\alpha(u_{x}-u_{xxt}-2\partial_x^{-1}(u_xu_{xt})-2u_tu_x)+\beta(u_{3x}+3u_x^2)], \\&&F_{2}=\frac{1}{4}[\alpha(u_{2x}-u_{3x, t}-4(u_xu_{xt})-2u_tu_{2x})+\beta(u_{4x}+6u_xu_{2x})], \\&&\cdots\cdots\\&&F_{n}=\alpha\bigg[I_n- I_{n, xt}+4\partial_x^{-1}\bigg(\sum\limits_{k=1}^{n}I_{k}I_{n+1-k, t}\bigg)+4\partial_x^{-1}\bigg(\sum\limits_{k=1}^nI_{k, t}\bigg)I_{n+1-k}\\&& +4\partial_x^{-1}\bigg(\sum\limits_{i+j+k=n}I_{i}I_{j, t}I_{k}\bigg)\bigg]\\&& +\beta\bigg(I_{n, 2x}-2\sum\limits_{i+j+k=n}I_{i}I_{j}I_{k}-6\sum\limits_{k=1}^nI_kI_{n+1-k}\bigg), \ n=2, 3, 4, \cdots .\end{array} $

据此可以验证(3.47)式的第一个方程即为方程(3.1).

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