数学物理学报, 2023, 43(4): 994-1002

一类(1+1)维变系数复方程的可积性研究

张金玉,, 王丹, 耿勇, 杨苗苗, 王晓丽,*

齐鲁工业大学(山东省科学院)数学与统计学院 济南 250353

The Integrability to a (1+1) Dimensional Variable Coefficient Complex Equation

Zhang Jinyu,, Wang Dan, Geng Yong, Yang Miaomiao, Wang Xiaoli,*

School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353

通讯作者: *王晓丽, E-mail: wxlspu@qlu.edu.cn

收稿日期: 2022-08-26   修回日期: 2023-04-10  

基金资助: 国家自然科学基金(12275017)
山东省自然科学基金(ZR2019PA020)
山东省自然科学基金(ZR2020MA049)
齐鲁工业大学(山东省科学院)(2022PX092)

Received: 2022-08-26   Revised: 2023-04-10  

Fund supported: NSFC(12275017)
NSFSD(ZR2019PA020)
NSFSD(ZR2020MA049)
QLU(2022PX092)

作者简介 About authors

张金玉,E-mail:zhangjinyu0611@163.com

摘要

该文基于双Bell多项式与Hirota双线性算子之间的关系, 研究了一类(1+1)维变系数复方程的可积性. 首先通过适当的变换, 构造出方程的双线性表达式、双线性Bäcklund变换, 又通过Hopf-Cole变 换得到方程的Lax对, 从而证明该方程具有Lax可积性.

关键词： (1+1)维变系数复方程; Bell多项式; Hirota双线性形式; Bäcklund变换; Lax对

Abstract

In this paper, we study the integrability of a (1+1)-dimensional variable coefficient complex equation based on the relationship between multi-dimensional binary bell polynomial and Hirota bilinear operator. Firstly, the bilinear form and bilinear Bäcklund transformation of the equation are constructed by appropriate transformation. Then the lax pair is obtained by Hopf-Cole transformation, which proves that the equation is Lax integrable.

Keywords： (1+1)-dimensional variable coefficient complex equation; Bell polynomials; Hirota bilinear form; Bäcklund transformation; Lax pair

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本文引用格式

张金玉, 王丹, 耿勇, 杨苗苗, 王晓丽. 一类(1+1)维变系数复方程的可积性研究[J]. 数学物理学报, 2023, 43(4): 994-1002

Zhang Jinyu, Wang Dan, Geng Yong, Yang Miaomiao, Wang Xiaoli. The Integrability to a (1+1) Dimensional Variable Coefficient Complex Equation[J]. Acta Mathematica Scientia, 2023, 43(4): 994-1002

1 引言

非线性科学是继量子理论和相对论之后20世纪自然科学的重大发展. 孤子理论作为非线性科学重要组成部分已经广泛应用于流体力学、等离子体物理、非线性光学等物理学领域内, 引起了许多数学家和物理学家的关注. 可积性是孤子理论的一个重要研究内容, 研究非线性方程的可积性有助于人们分析和理解它所反映的自然现象[1⇓⇓-4]. 非线性方程的可积性定义目前尚无定论, 但通常可以通过精确解、对称性、Bäcklund变换、Lax 对等来描述. 迄今为止, 人们已经提出并发展了许多研究非线性方程可积性的方法, 例如反散射变换法[5⇓⇓-8]、Darboux变换法[9⇓-11]、Bäcklund变换法[12⇓-14]、截断Painlevé展开法[15⇓⇓-18]、Hirota双线性法[19⇓⇓⇓⇓-24]等. 其中, Hirota 双线性法是研究方程可积性一种直接有效的方法, 这一方法与Bell多项式紧密联系. Bell多项式是1934年Bell首次提出的[25], 随后Lambert[26-27]等建立了Bell 多项式和Hirota 双线性算子之间的联系, 为求解方程的双线性形式提供了很大便利. 近年来, 范恩贵教授等[22,28⇓⇓⇓⇓⇓-34]基于Bell多项式方法研究了非等谱和变系数非线性演化方程的双线性Bäcklund 变换、Lax对, 验证了方程的可积性.

本文的目的是基于Bell多项式研究一类(1+1)维变系数复方程

$\begin{equation}\label{ZYFC} i{u_t} + \alpha {u_{xx}} + i\beta {u_{xxx}} + \alpha \gamma {\left| u \right|^2}u + 3i\beta \gamma {\left| u \right|^2}{u_x} = 0, \end{equation} $

其中$u$ 是关于变量 $x$ 和 $t$ 的函数, ${\left| u \right|^2} = u{u^*}$, $*$ 表示复共轭, $\alpha$, $ \beta$, $ \gamma$ 是任意常数. 该方程可用于描绘等离子体物理、流体动力学、非线性光学中的超短脉冲、非线性传输、非线性晶格的动力学演化等众多物理现象. 当 $\alpha=0$, $ \beta=1$, $ \gamma=2 $ 时, 该方程是著名的自聚焦复mKdV方程

$\begin{equation}\label{MKDV} {u_t} + {u_{xxx}} + 6{\left| u \right|^2}{u_x} = 0. \end{equation} $

复mKdV方程用来描述短脉冲在光纤中的传播[35]. 扎其劳教授等[36-37]研究了方程(1.2)的Lax 对、无穷守恒律, 利用Darboux矩阵方法构造了该方程的广义Darboux 变换, 并得到了N阶怪波解. 当$\alpha=1$, $\beta=0$, $\gamma=1$ 时, 该方程为经典的薛定谔方程

$\begin{equation}\label{SCHODINGER} i{u_t} + {u_{xx}} + {\left| u \right|^2}u = 0. \end{equation} $

薛定谔方程通常用来描述非线性介质中缓变波包和一般小振幅的演化[38]. 前人已经研究了该方程的Bäcklund 变换、Lax 对、无穷守恒律和孤子解等可积性[20]. 本文主要目的是运用Bell多项式和Hirota双线性方法研究方程(1.1)的双线性形式、双线性Bäcklund变换和Lax对.

2 双线性算子与Bell多项式的定义及性质

在本节中, 我们将简单地介绍Hirota双线性算子和Bell多项式的定义和性质[22,34,39-40].

定义 2.1 Hirota双线性算子定义为

$\begin{aligned} & D_{x_{1}}^{p_{1}} D_{x_{2}}^{p_{2}} \cdots D_{x_{n}}^{p_{n}} D_{t}^{r} f(X, t) \cdot g\left(X^{\prime}, t^{\prime}\right) \\ = & \left(\frac{\partial}{\partial x_{1}}-\frac{\partial}{\partial x_{1}^{\prime}}\right)^{p_{1}}\left(\frac{\partial}{\partial x_{2}}-\frac{\partial}{\partial x_{2}^{\prime}}\right)^{p_{2}} \cdots \\ & \times\left.\left(\frac{\partial}{\partial x_{n}}-\frac{\partial}{\partial x_{n}^{\prime}}\right)^{p_{n}}\left(\frac{\partial}{\partial t}-\frac{\partial}{\partial t^{\prime}}\right)^{r} f(X, t) g\left(X^{\prime}, t^{\prime}\right)\right|_{X^{\prime}=X, t^{\prime}=t},\end{aligned}$

其中${p_1},{p_2}, \cdots,{p_n},r$ 是非负整数, $f$ 是关于 $X$ 和 $t$ 的函数, $g$ 是关于 $X'$ 和 $t'$ 的函数, $X = \left( {{x_1},{x_2}, \cdots,{x_n}} \right)$, $X' = \left( {x_1^\prime,x_2^\prime, \cdots,x_n^\prime } \right)$.

定义 2.2 Bell多项式又称为$Y$ -多项式, 定义为

$\begin{equation} {Y_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( \phi \right) \equiv {Y_{{n_1}, \cdots,{n_l}}}\left( \phi \right)\left|_ {{\phi _{{r_1}{x_1}, \cdots,{r_l}{x_l}}}} \right.= {e^{ - \phi }}\partial _{{x_1}}^{{r_1}} \cdots \partial _{{x_l}}^{{r_l}}{e^\phi }, \end{equation}$

其中$\phi = \phi \left( {{x_1}, \cdots,{x_n}} \right)$ 是定义在 $C^{\infty}$ 上的 $n$ 元函数, $l=1,\cdots, n$, ${\phi _{{r_1}{x_1}, \cdots,{r_l}{x_l}}} = \partial _{{x_1}}^{{r_1}} \cdots \partial _{{x_l}}^{{r_l}}\phi$$\left({{r_1} = 0, \cdots,{n_1}; \cdots ;{r_l} = 0, \cdots,{n_l}} \right).$

例如, 当 $\phi = \phi \left( {x,t} \right)$ 时, 对应的二维$Y$ -多项式为

${Y_{x,t}} = {\phi _{x,t}} + {\phi _x}{\phi _t}, {Y_{2x,t}} = {\phi _{2x,t}} + {\phi _{2x}}{\phi _t} + 2{\phi _{x,t}}{\phi _x} + \phi _x^2{\phi _t}.$

定义 2.3 双Bell多项式又称为${\cal Y}$ -多项式, 定义为

$\begin{equation}{{\cal Y}_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( {v,w} \right) \equiv {Y_{{n_1}, \cdots,{n_l}}}\left( \phi \right)\left|_ {{\phi _{{r_1}{x_1}, \cdots,{r_l}{x_l}}}} \right.,\end{equation}$

其中

${{\phi _{{r_1}{x_1}, \cdots,{r_l}{x_l}}}}= \left\{ \begin{array}{l}{v_{{r_1}{x_1}, \cdots,{r_l}{x_l}}}, uad \mbox{如果}\ {r_1} + \cdots + {r_l}\ \mbox{为奇数},\\{w_{{r_1}{x_1}, \cdots,{r_l}{x_l}}}, uad \mbox{如果}\ {r_1} + \cdots + {r_l}\ \mbox{为偶数}.\end{array} \right.$

这里 $v=v\left( {{x_1}, \cdots,{x_n}} \right)$ 和 $w=w\left( {{x_1}, \cdots,{x_n}} \right)$ 是 $C^\infty$ 上的 $n$ 元函数.

例如, 当 $v = v\left( {x, t} \right)$ 和 $w = w\left( {x, t} \right)$ 时, 对应的${\cal Y}$ -多项式为

$\begin{array}{l}\mathcal{Y}_{x}(v, w)=v_{x}, \quad \mathcal{Y}_{2 x}(v, w)=v_{x}^{2}+w_{2 x}, \\ \mathcal{Y}_{3 x}(v, w)=v_{x}^{3}+3 v_{x} w_{2 x}+v_{3 x}, \quad \mathcal{Y}_{x, t}(v, w)=v_{x} v_{t}+w_{x, t}\end{array}$

定义 2.4$P$ -多项式定义为

$\begin{equation}{P_{{n_1}{x_1},\cdots,{n_l}{x_l}}}\left( w \right)={{\cal Y}_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( {v=0,w} \right).\end{equation}$

例如, 当 $w = w\left( {x, t} \right)$ 时, 对应的$P$ -多项式为

$\begin{matrix}\label{LZ2}&{P_{2x}}\left( w \right) = {w_{2x}},uad {P_{x,t}}\left(w \right) = {w_{xt}}, uad {P_{4x}}\left( w \right) = {w_{4x}} + 3w_{2x}^2.\end{matrix}$

性质 2.1$Y$ -多项式在Hopf-Cole变换 $v = \ln \psi$ 之下可线性化

$\begin{equation}{Y_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( v \right)\left| {_{v = \ln \psi }} \right. = \frac{{{\psi _{{n_1}{x_1}, \cdots,{n_l}{x_l}}}}}{\psi }.\end{equation}$

性质 2.2${\cal Y}$ -多项式与Hirota双线性$D$ -算子之间的关系为

$\begin{equation}{{\cal Y}_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left[ v = \ln \left( {G/F} \right),w = \ln \left( {GF} \right) \right] = {\left( {GF} \right)^{ - 1}}D_{{x_1}}^{{n_1}} \cdots D_{{x_l}}^{{n_l}}G \cdot F,\end{equation}$

其中 ${n_1} + {n_2} + \cdots + {n_l} \ge 1$.

当 $G=F$ 时, $v=0$, $w= 2\ln F$, 方程(2.8)可以写为

$\begin{equation}\begin{array}{l}{P_{{n_1}{x_1},\cdots,{n_l}{x_l}}}\left( 2\ln F\right)={F^{ - 2}}D_{{x_1}}^{{n_1}} \cdots D_{{x_l}}^{{n_l}}{F\cdot F}.\end{array}\end{equation}$

性质 2.3${\cal Y}$ -多项式可以分为$P$ -多项式和$Y$ -多项式的组合形式

$\begin{matrix} && \ {{\cal Y}_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( {v,w} \right)\left| {_{v = \ln \frac{G}{F},w = \ln GF}} \right. \\ &=& {{\cal Y}_{{n_1}{x_1}, \cdots,{n_l}{x_l}}}\left( {v,v + q} \right)\left| {_{v = \ln \frac{G}{F},q = 2\ln F}} \right.\\ &= &\sum\limits_{{n_1} + \cdots + {n_l}} {\sum\limits_{{r_1} = 0}^{{n_1}} { \cdots \sum\limits_{{r_l} = 0}^{{n_l}} {\prod\limits_{i = q}^l {\left( \begin{array}{l}{n_i}\\{r_i}\end{array} \right)} } } } {P_{{r_1}{x_1}, \cdots,{r_l}{x_l}}}(q)\times {Y_{\left( {{n_1} - {r_1}} \right){x_1}, \cdots,\left( {{n_l} - {r_l}} \right){x_l}}}(v),\end{matrix}$

其中 ${n_1} + {n_2} + \cdots + {n_l}$ 为偶数.

例如, 当 $v = v\left( {x, t} \right)=\ln \psi$, $q = q\left( {x, t} \right)$ 时, ${\cal Y}$-多项式可以分为$P$ -多项式和$Y$ -多项式的组合

$\begin{matrix} &&{{\cal Y}_x}\left( v,v+q\right) =P_0\left( q\right)Y_x\left( v\right), uad {{\cal Y}_{2x}}\left( {v,v+q} \right) = P_0\left( q\right)Y_{2x}\left( v\right)+P_{2x}\left( q\right)Y_{0}\left( v\right),\\&&{{\cal Y}_t}\left( v,v+q \right) = P_0\left( q\right)Y_t\left( v\right), uad \ {{\cal Y}_{3x}}\left( {v,v+q} \right) = P_0\left( q\right)Y_{3x}\left( v\right)+3P_{2x}\left( q\right)Y_{x}\left( v\right),\end{matrix}$

其中 $P_0(q)=1$, $Y_{0}(v)=1$.

注 2.1 性质 $2.1$ 和 $2.3$ 在推导方程的Lax对时起到重要作用.

3 方程的可积性研究

3.1 双线性表达式

定理 3.1 作变换 $u=\frac{g}{f}$ 可得方程(1.1)的双线性表达式为

$\left( {i{D_t} + \alpha D_x^2 + i\beta D_x^3} \right)g \cdot f = 0,qD_x^2f \cdot f = \gamma g{g^*},$

其中 $f$ 是实函数, $g$ 是复函数, $g^*$ 是 $g$ 的共轭.

证 做变换

$\begin{equation}\label{Shedeu}u = {e^v},\end{equation}$

其中 $v$ 是关于 $x$ 和 $t$ 的函数, 将其代入方程(1.1), 得到

$\begin{equation}i{v_t} + \alpha \left( v_x^2 + {v_{xx}} \right) + i\beta \left( {v_x^3 + 3{v_x}{v_{xx}} + {v_{3x}}} \right) + \alpha \gamma {e^{v + {v^*}}} + 3i\beta \gamma {v_x}{e^{v + {v^*}}} = 0,\end{equation}$

引入辅助变量 $w$ 得到

$\begin{array}{l}i v_{t}+\alpha\left(v_{x}^{2}+w_{x x}\right)-\alpha(w-v)_{x x}+i \beta\left(v_{x}^{3}+3 v_{x} w_{x x}+v_{3 x}\right) \\ -3 i \beta v_{x}(w-v)_{x x}+\alpha \gamma e^{v+v^{*}}+3 i \beta \gamma v_{x} e^{v+v^{*}}=0\end{array}$

其中 $w$ 是关于 $x$ 和 $t$ 的函数. 应用公式(2.4)和(2.6), 方程(3.3)可写为如下 ${\cal Y}$ -多项式与$P$ -多项式的形式

$i{{\cal Y}_t}(v,w) + \alpha {{\cal Y}_{2x}}(v,w) + i\beta {{\cal Y}_{3x}}(v,w) = 0,$
${P_{2x}}(w - v) = \gamma {e^{v + {v^*}}}.$

最后, 结合性质 $2.2$, 做变量变换

$v = \ln \left( {\frac{g}{f}} \right),uad w = \ln \left( {gf} \right),$

可以得到方程(1.1)的双线性形式

$\left( i{D_t} + \alpha D_x^2 + i\beta D_x^3 \right)g \cdot f = 0,$
$D_x^2f \cdot f = \gamma g{g^*}.$

证毕.

3.2 双线性Bäcklund 变换

定理 3.2 令 $u=\frac{g}{f}$, $u^\prime=\frac{g^\prime}{f^\prime}$ 为方程(1.1)的两个不同解, 可以得出方程(1.1)的双线性Bäcklund变换为

\begin{eqnarray*} &&\left[ i{D_t} + \alpha D_x^2 + i\beta \left( {D_x^3 + \frac{{3{\gamma ^2}}}{{4}}{D_x}} \right) \right]g' \cdot g = 0,\\ &&\left[ i{D_t} + \alpha D_x^2 + i\beta \left( {D_x^3 + \frac{{3{\gamma ^2}}}{{4}}{D_x}} \right) \right]f' \cdot f = 0,\\ &&{D_x}f' \cdot g = \frac{\gamma }{{2}}g'f,\\ &&{D_x}g' \cdot f = \frac{\gamma }{{2}}f'g,\\ &&{D_x}f' \cdot f = {{g'}^*}g,\end{eqnarray*}

其中 $f, f'$ 为实函数, $g, g'$ 为复函数.

证 假定 $v = \ln \frac{g}{f},\ w = \ln \left( {gf} \right),\ v' = \ln \frac{{g'}}{{f'}},\ w^\prime = \ln \left( {g'f'} \right)$ 是方程(3.4)的两个不同解, 则我们有

$\begin{equation}i{{\cal Y}_t}(v',w') + \alpha {{\cal Y}_{2x}}(v',w') + i\beta {{\cal Y}_{3x}}(v',w') - \left[ {i{{\cal Y}_t}(v,w) + \alpha {{\cal Y}_{2x}}(v,w) + i\beta {{\cal Y}_{3x}}(v,w)} \right] = 0,\end{equation}$
$\begin{equation}{P_{2x}}(w' - v') - {P_{2x}}(w - v) - \gamma {e^{v' + {{v'}^*}}} + \gamma {e^{v + {v^*}}} = 0.\end{equation}$

应用(2.4) 和(2.6), (3.7)和(3.8)式可以写成

$\begin{array}{l}i\left(v^{\prime}-v\right)_{t}+\alpha\left[\left(v_{x}^{\prime}+v_{x}\right)\left(v_{x}^{\prime}-v_{x}\right)+\left(w^{\prime}-w\right)_{x x}\right]+i \beta\left[\left(v^{\prime}-v\right)_{3 x}+\frac{1}{4}\left(v_{x}^{\prime}-v_{x}\right)^{3}\right. \\ \left.+\frac{3}{4}\left(v_{x}^{\prime}+v_{x}\right)^{2}\left(v_{x}^{\prime}-v_{x}\right)+\frac{3}{2}\left(v^{\prime}+v\right)_{x}\left(w^{\prime}-w\right)_{2 x}+\frac{3}{2}\left(v^{\prime}-v\right)_{x}\left(w^{\prime}+w\right)_{2 x}\right]=0\end{array}$
$\begin{equation}{\left( {w' - w} \right)_{2x}} - {\left( {v' - v} \right)_{2x}} = \gamma {e^{v' + {{v'}^*}}} - \gamma {e^{v + {v^*}}}.\end{equation}$

为求出方程(1.1)的双线性Bäcklund变换, 考虑引入混合变量

$\begin{array}{l}v_{1}=\ln \left(\frac{g^{\prime}}{g}\right), v_{2}=\ln \left(\frac{f^{\prime}}{f}\right), v_{3}=\ln \left(\frac{f^{\prime}}{g}\right), v_{4}=\ln \left(\frac{g^{\prime}}{f}\right) \\ w_{1}=\ln \left(g^{\prime} g\right), w_{2}=\ln \left(f^{\prime} f\right), w_{3}=\ln \left(f^{\prime} g\right), w_{4}=\ln \left(g^{\prime} f\right)\end{array}$

我们得到以下关系式

$\begin{matrix}&&v' - v = {v_1} - {v_2} = {w_4} - {w_3},uad v' + v = {w_1} - {w_2} = {v_4} - {v_3},\\ &&w' - w = {v_1} + {v_2} = {v_3} + {v_4},uad w' + w = {w_1} + {w_2} = {w_3} + {w_4},\\ &&v' + {{v'}^*} = {v^*_4} - {v_3} + {v_1} - {v_2},uad v + {v^*} = {v^*_4} - {v_3} + {v_2} - {v^*_1},\\&&v=v_2-v_3={v_4}-{v_1}.\end{matrix}$

那么(3.9) 和(3.10)式改写为

$\begin{array}{l}i\left(v_{1}-v_{2}\right)_{t}+\alpha\left[\left(v_{4}-v_{3}\right)_{x}\left(v_{1}-v_{2}\right)_{x}+\left(v_{3}+v_{4}\right)_{2 x}\right]+i \beta\left\{\left(v_{1}-v_{2}\right)_{3 x}+\frac{1}{4}\left[\left(v_{1}-v_{2}\right)_{x}\right]^{3}\right. \\ \left.+\frac{3}{4}\left[\left(v_{4}-v_{3}\right)_{x}\right]^{2}\left(v_{1}-v_{2}\right)_{x}+\frac{3}{2}\left(v_{4}-v_{3}\right)_{x}\left(v_{3}+v_{4}\right)_{2 x}+\frac{3}{2}\left(v_{1}-v_{2}\right)_{x}\left(w_{1}+w_{2}\right)_{2 x}\right\}=0,\end{array}$
$\begin{equation}2{\left( {{v_2}} \right)_{xx}} = \gamma e^{{v^*_4} - {v_3} }\left( {e^{ {v_2} - {v^*_1}}} - {e^{ {v_1} - {v_2}}} \right).\end{equation}$

选取约束条件

$\begin{equation}{\left( {{v_3}} \right)_x} = \frac{\gamma}{{2}}{e^{{v_1} - {v_2}}},uad {\left( {{v_4}} \right)_x} = \frac{\gamma}{{2}}{e^{{v_2} - {v_1}}},\end{equation}$

通过共轭关系, 把(3.15)式代入(3.14)式得到

$\begin{matrix}\label{JLv2x}{\left( {{v_2}} \right)_x} = {e^{{v^*_4} - {v_3} }}.\end{matrix}$

运用(3.11)和(3.12)式定义的变量关系, 在条件(3.15) 的约束下, 经过大量计算, 方程(3.13)化为

$\begin{array}{l}i\left(v_{1}-v_{2}\right)_{t}+\alpha\left[\left(v_{1}\right)_{x}^{2}+\left(w_{1}\right)_{2 x}-\left(v_{2}\right)_{x}^{2}-\left(w_{2}\right)_{2 x}\right]+i \beta\left\{\left(v_{1}\right)_{3 x}+3\left(v_{1}\right)_{x}\left(w_{1}\right)_{2 x}+\left(v_{1}\right)_{x}^{3}\right. \\ \left.-\left[\left(v_{2}\right)_{3 x}+3\left(v_{2}\right)_{x}\left(w_{2}\right)_{2 x}+\left(v_{2}\right)_{x}^{3}\right]+\frac{3 \gamma^{2}}{4}\left(v_{1}-v_{2}\right)_{x}\right\}=0.\end{array}$

即

$i{v_{1,t}} + \alpha \left( {v_{1,x}^2 + {w_{1,xx}}} \right) + i\beta \left[ \left( {v_{1,x}^3 + 3{v_{1,x}}{w_{1,xx}} + {v_{1,3x}}} \right) + \frac{{3{\gamma ^2}}}{{4}}{v_{1,x}}\right] = 0,$
$i{v_{2,t}} + \alpha \left( {v_{2,x}^2 + {w_{2,xx}}} \right) + i\beta \left[ \left( {v_{2,x}^3 + 3{v_{2,x}}{w_{2,xx}} + {v_{2,3x}}} \right) + \frac{{3{\gamma ^2}}}{{4}}{v_{2,x}}\right] = 0.$

把(3.18)、(3.15)和(3.16)式写成 ${\cal Y}$ -多项式形式

$i{{\cal Y}_t}\left( {v_1},w_1 \right) + \alpha {{\cal Y}_{2x}}\left( {{v_1},{w_1}} \right) + i\beta \left[ {{{\cal Y}_{3x}}\left( {{v_1},{w_1}} \right) + \frac{{3{\gamma ^2}}}{{4}}{{\cal Y}_x}\left( {v_1},w_1 \right)} \right] = 0, $
$i{{\cal Y}_t}\left( {v_2},w_2 \right) + \alpha {{\cal Y}_{2x}}\left( {{v_2},{w_2}} \right) + i\beta \left[ {{{\cal Y}_{3x}}\left( {{v_2},{w_2}} \right) + \frac{{3{\gamma ^2}}}{{4}}{{\cal Y}_x}\left( {v_2},w_2 \right)} \right] = 0,$
${{\cal Y}_x}\left( {v_3},w_3 \right) = \frac{\gamma }{{2}}{e^{{v_1} - {v_2}}}, $
${{\cal Y}_x}\left( {v_4},w_4 \right) = \frac{\gamma }{{2}}{e^{{v_2} - {v_1}}},$
${{\cal Y}_x}\left( {v_2},w_2 \right) = {e^{{v^*_4} - {v_3}}}.%, \\ \label{YSv7}$

结合性质2.2可得方程(1.1)的双线性Bäcklund变换

$\begin{array}{l}{\left[i D_{t}+\alpha D_{x}^{2}+i \beta\left(D_{x}^{3}+\frac{3 \gamma^{2}}{4} D_{x}\right)\right] g^{\prime} \cdot g=0,} \\ {\left[i D_{t}+\alpha D_{x}^{2}+i \beta\left(D_{x}^{3}+\frac{3 \gamma^{2}}{4} D_{x}\right)\right] f^{\prime} \cdot f=0,} \\ D_{x} f^{\prime} \cdot g=\frac{\gamma}{2} g^{\prime} f, \\ D_{x} g^{\prime} \cdot f=\frac{\gamma}{2} f^{\prime} g, \\ D_{x} f^{\prime} \cdot f=g^{\prime *} g.\end{array}$

证毕.

3.3 Lax对

定理 3.3 方程(1.1)的Lax对为

$\begin{matrix}\label{LAXpair}\left( \begin{array}{l}{\psi _1}\\{\psi _2}\end{array} \right)_x = M\left( \begin{array}{l}{\psi _1}\\{\psi _2}\end{array} \right),\left( \begin{array}{l}{\psi _1}\\{\psi _2}\end{array} \right)_t = N\left( \begin{array}{l}{\psi _1}\\{\psi _2}\end{array} \right),\end{matrix}$

其中

$\begin{array}{l}M=\left(\begin{array}{cc}-\frac{u_{x}}{u} & \frac{\gamma}{2} \\ \frac{\gamma}{2} & \frac{u_{x}}{u}\end{array}\right), \quad N=\left(\begin{array}{cc}A & B \\ C & -A\end{array}\right), \\ A=-i \alpha\left(\frac{u_{x x}}{u}+\gamma|u|^{2}+\frac{\gamma^{2}}{4}\right)+\beta\left(\frac{u_{3 x}}{u}+3 \gamma u_{x} u^{*}+\frac{\gamma^{2} u_{x}}{u}\right), \\ B=-\frac{\beta \gamma}{2}\left(5 \frac{u_{x x}}{u}-4 \frac{u_{x}^{2}}{u^{2}}+3 \gamma|u|^{2}+\gamma^{2}\right), \\ C=-\frac{\beta \gamma}{2}\left(\frac{u_{x x}}{u}+3 \gamma|u|^{2}+\gamma^{2}\right).\end{array}$

证 设 $\left( {v',w'} \right),\left( {v,w} \right)$ 是方程(3.4)的两个不同解, 运用(3.12)式中的式子 ${v_4} = {v_1} + v$ 和 ${v_3} = {v_2} - v$, 将其代入(3.15)式可得

$\label{V1x} \left({{v_1}} \right)_x = - {v_x} + \frac{\gamma }{{2}}{e^{{v_2} - {v_1}}},$
$\label{V2x} \left( {{v_2}} \right)_x = {v_x} + \frac{\gamma }{2}{e^{{v_1} - {v_2}}}.$

借助Hopf-Cole变换 ${v_1} = \ln {\psi _1}$ 和 ${v_2} = \ln {\psi _2}$, 分别代入(3.21)和(3.18)式可得方程的Lax对为

$\begin{matrix}\label{Laxpair} \left( \begin{array}{l} {\psi _1}\\ {\psi _2} \end{array} \right)_x = M\left( \begin{array}{l} {\psi _1}\\ {\psi _2} \end{array} \right),\left( \begin{array}{l} {\psi _1}\\ {\psi _2} \end{array} \right)_t = N\left( \begin{array}{l} {\psi _1}\\ {\psi _2} \end{array} \right), \end{matrix} $

其中

$\begin{array}{l}M=\left(\begin{array}{cc}-v_{x} & \frac{\gamma}{2} \\ \frac{\gamma}{2} & v_{x}\end{array}\right), \quad N=\left(\begin{array}{cc}A & B \\ C & -A\end{array}\right) \\ A=-i \alpha\left(v_{x}^{2}+w_{x x}+\frac{\gamma^{2}}{4}\right)+\beta\left(v_{3 x}+3 v_{x} w_{x x}+v_{x}^{3}+\gamma^{2} v_{x}\right) \\ B=-\frac{\beta \gamma}{2}\left(v_{x}^{2}+3 w_{x x}+2 v_{x x}+\gamma^{2}\right) \\ C=-\frac{\beta \gamma}{2}\left(v_{x}^{2}+3 w_{x x}-2 v_{x x}+\gamma^{2}\right)\end{array}$

结合(3.4)式可验证相容性条件 ${M_t} - {N_x} + \left[ {M,N} \right] = 0$ 成立. 由(3.1)和(3.4b)式分别得出 $v=\ln u$, ${w_{xx}} = \frac{{{u_{xx}}}}{u} - \frac{{u_x^2}}{{{u^2}}} + \gamma {\left| u \right|^2}$, 将其代入(3.22)式可得Lax对(3.20)式. 证毕.

4 结论

本文研究了一类(1+1)维变系数复方程, 当取定系数后, 该方程可以返回到著名的复mKdV方程和经典的薛定谔方程. 我们基于Bell多项式求出了该方程的双线性形式、双线性Bäcklund变换和Lax对, 验证了该方程的Lax可积性. 非线性方程的可积性是孤立子研究的一个核心问题之一. 研究非线性方程的可积性为解释等离子体、光学、流体力学中的物理现象开辟了一条新途径.

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