一类(1+1)维变系数复方程的可积性研究

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

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

(1.1)

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

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

(1.2)

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

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

(1.3)

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

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

(2.1)

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

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

(2.2)

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

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

(2.3)

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

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

(2.4)

定义 2.4 $P$ -多项式定义为

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

(2.5)

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

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

(2.6)

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

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

(2.7)

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

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

(2.8)

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

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

(2.9)

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

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

(2.10)

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

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

(2.11)

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

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

(3.1)

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

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

(3.2)

引入辅助变量 $w$ 得到

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

(3.3)

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

$i{{\cal Y}_t}(v,w) + \alpha {{\cal Y}_{2x}}(v,w) + i\beta {{\cal Y}_{3x}}(v,w) = 0,$

(3.4a)

${P_{2x}}(w - v) = \gamma {e^{v + {v^*}}}.$

(3.4b)

最后, 结合性质 $2.2$ , 做变量变换

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

(3.5)

可以得到方程(1.1)的双线性形式

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

(3.6a)

$D_x^2f \cdot f = \gamma g{g^*}.$

(3.6b)

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

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

(3.7)

$\begin{equation}{P_{2x}}(w' - v') - {P_{2x}}(w - v) - \gamma {e^{v' + {{v'}^*}}} + \gamma {e^{v + {v^*}}} = 0.\end{equation}$

(3.8)

应用(2.4) 和(2.6), (3.7)和(3.8)式可以写成

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

(3.9)

$\begin{equation}{\left( {w' - w} \right)_{2x}} - {\left( {v' - v} \right)_{2x}} = \gamma {e^{v' + {{v'}^*}}} - \gamma {e^{v + {v^*}}}.\end{equation}$

(3.10)

为求出方程(1.1)的双线性Bäcklund变换, 考虑引入混合变量

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

(3.11)

我们得到以下关系式

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

(3.12)

那么(3.9) 和(3.10)式改写为

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

(3.13)

$\begin{equation}2{\left( {{v_2}} \right)_{xx}} = \gamma e^{{v^*_4} - {v_3} }\left( {e^{ {v_2} - {v^*_1}}} - {e^{ {v_1} - {v_2}}} \right).\end{equation}$

(3.14)

选取约束条件

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

(3.15)

通过共轭关系, 把(3.15)式代入(3.14)式得到

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

(3.16)

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

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

(3.17)

即

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

(3.18a)

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

(3.18b)

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

$i{{\cal Y}_t}\left( {v_1},w_1 \right) + \alpha {{\cal Y}_{2x}}\left( {{v_1},{w_1}} \right) + i\beta \left[ {{{\cal Y}_{3x}}\left( {{v_1},{w_1}} \right) + \frac{{3{\gamma ^2}}}{{4}}{{\cal Y}_x}\left( {v_1},w_1 \right)} \right] = 0,$

(3.19a)

$i{{\cal Y}_t}\left( {v_2},w_2 \right) + \alpha {{\cal Y}_{2x}}\left( {{v_2},{w_2}} \right) + i\beta \left[ {{{\cal Y}_{3x}}\left( {{v_2},{w_2}} \right) + \frac{{3{\gamma ^2}}}{{4}}{{\cal Y}_x}\left( {v_2},w_2 \right)} \right] = 0,$

(3.19b)

${{\cal Y}_x}\left( {v_3},w_3 \right) = \frac{\gamma }{{2}}{e^{{v_1} - {v_2}}},$

(3.19c)

${{\cal Y}_x}\left( {v_4},w_4 \right) = \frac{\gamma }{{2}}{e^{{v_2} - {v_1}}},$

(3.19d)

${{\cal Y}_x}\left( {v_2},w_2 \right) = {e^{{v^*_4} - {v_3}}}.%, \\ \label{YSv7}$

(3.19e)

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

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

(3.20)

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

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

(3.21a)

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

(3.21b)

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

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

(3.22)

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.

参考文献

原文顺序

文献年度倒序

文中引用次数倒序

被引期刊影响因子

[1]

, Duan

C N

, Yu

F J

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

Phys Lett A, 2019, 383: 1578-1582

DOI:10.1016/j.physleta.2019.02.031 [本文引用: 1]

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation. (C) 2019 Elsevier B.V.

[2]

Zhang

Y P

, Liu

, Wei

G M

Lax pair, auto-Bäcklund transformation and conservation law for a generalized variable-coefficient KdV equation with external-force term

Appl Math Lett, 2015, 45: 58-63

DOI:10.1016/j.aml.2015.01.007 URL [本文引用: 1]

[3]

Jia

T T

, Gao

Y T

, Yu

, et al.

Lax pairs, infinite conservation laws, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation

Appl Math Lett, 2021, 114: 106702

DOI:10.1016/j.aml.2020.106702 URL [本文引用: 1]

[4]

, Hua

Y F

, Chen

S J

, et al.

Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, Bäcklund transformation, Lax pair and infinitely many conservation laws

Commun Nonlinear Sci, 2021, 95: 105612

DOI:10.1016/j.cnsns.2020.105612 URL [本文引用: 1]

[5]

Ablowitz

M J

, Clarkson

P A

. Soliton,Nonlinear Evolution Equations and Inverse Scatting. Cambridge: Cambridge University Press, 1991

[本文引用: 1]

[6]

Fang

, Hu

B B

, Zhang

Inverse scattering transform for the generalized derivative nonlinear Schrödinger equation via matrix Riemann-Hilbert problem

Math Probl Eng, 2022: Article ID 3967328

[本文引用: 1]

[7]

Butler

, Joshi

An inverse scattering transform for the lattice potential KdV equation

Inverse Probl, 2010, 26: 115012

DOI:10.1088/0266-5611/26/11/115012 URL [本文引用: 1]

[8]

Gao

X D

, Zhang

Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

Therm Sci, 2017, 21: S153-S160

[本文引用: 1]

[9]

谷超豪, 胡和生, 周子翔. 孤立子理论中的达布变换及其几何应用. 上海: 上海科学技术出版社, 1999

[本文引用: 1]

C H

, Hu

H S

, Zhou

Z X

. Darboux Transformation in Soliton Theory and Geometric Application. Shanghai: Shanghai Science and Technology Press, 1999

[本文引用: 1]

[10]

Mikhailov

A V

, Papamikos

, Wang

J P

Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere

Lett Math Phys, 2016, 106(7): 973-996

DOI:10.1007/s11005-016-0855-5 URL [本文引用: 1]

[11]

Feng

L L

, Yu

F J

, Li

Darboux transformation for coupled non-linear Schrödinger equation and its breather solutions

Z Naturforsch, 2016, 72(1): 1-7

DOI:10.1515/znb-2016-0178 URL [本文引用: 1]

The title compounds have raised considerable medical and broad public interest in that sildenafil is used as an agent against male erectile dysfunction; iso-sildenafil is not in clinical use. A comparison of their structural and electronic properties therefore seems of interest. The electron densities of iso-sildenafil and the cationic and neutral forms of sildenafil were examined by the application of the invariom formalism relying on diffraction data reported in the literature. The electron-density distributions obtained were subjected to topological analysis using the quantum theory of atoms in molecules (QTAIM) formalism to yield bond topological and atomic properties. Moreover, molecular Hirshfeld surfaces and electrostatic potentials (ESPs) were calculated. A number of structural and electronic differences were thus identified between sildenafil and the iso-analog. In both sildenafil structures, the phenyl ring and the pyrazolopyrimidine fragment are practically coplanar (planar conformation), whereas in the iso-analog they exhibit an angle of 44° (inclined form). Related to differences in molecular structure are completely different hydrogen bonding patterns and differences in the ESPs, the latter ones being influenced by different methylation at the pyrazolopyrimidine fragment. Iso-sildenafil is present as a hydrogen-bond dimer in the crystal, and the ESP of this dimer is dominated by a surrounding positive potential.

[12]

Miura

M R

. Bäcklund Transformation. Berlin: Springer, 1978

[本文引用: 1]

[13]

Rogers

, Shadwick

W F

. Bäcklund Transformations and Their Applications. New York: Academic Press, 1982

[本文引用: 1]

[14]

Huang

New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method

Nonlinear Dynam, 2013, 72: 87-90

DOI:10.1007/s11071-012-0692-8 URL [本文引用: 1]

[15]

Zeng

Z F

, Liu

J G

Multiple soliton solutions, soliton-type solutions and hyperbolic solutions for the Benjamin-Bona-Mahony equation with variable coefficients in rotating fluids and one-dimensional transmitted waves

Int J Nonlin Sci Num, 2016, 17(5): 195-203

DOI:10.1515/ijnsns-2015-0122 URL [本文引用: 1]

With the help of symbolic computation, the Benjamin–Bona–Mahony (BBM) equation with variable coefficients is presented, which was proposed for the first time by Benjamin as the regularized long-wave equation and originally derived as approximation for surface water waves in a uniform channel. By employing the improved \n \n \n \n (\n \n G\n \n \n ′\n \n \n \n /\n \n G\n )\n \n $(G^' /G)$ \n \n -expansion method, the truncated Painlevé expansion method, we derive new auto-Bäcklund transformation, hyperbolic solutions, a variety of traveling wave solutions, soliton-type solutions and two solitary wave solutions of the BBM equation. These obtained solutions possess abundant structures. The figures corresponding to these solutions are illustrated to show the particular localized excitations and the interactions between two solitary waves.

[16]

H L

, Chen

Q Y

, Song

J F

Bäcklund transformation, residual symmetry and exact interaction solutions of an extended (2+1)-dimensional Korteweg-de Vries equation

Appl Math Lett, 2022, 124: 107640

DOI:10.1016/j.aml.2021.107640 URL [本文引用: 1]

[17]

Wei

G M

, Gao

Y T

, Hu

, et al.

Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries(KdV) equation

Eur Phys J B, 2006, 53: 343-350

DOI:10.1140/epjb/e2006-00378-3 URL [本文引用: 1]

[18]

Y X

, Tian

, Qu

Q X

, et al.

Bäcklund transformations, kink soliton, breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics

Chinese J Phys, 2021, 73: 600-612

DOI:10.1016/j.cjph.2021.07.001 URL [本文引用: 1]

[19]

Hirota

. The Direct Method in Soliton Theory. Berlin: Springer-verlag, 2004

[本文引用: 1]

[20]

陈登远

. 孤子引论. 北京: 科学出版社, 2006

[本文引用: 2]

Chen

D Y

. Soliton Theory. Beijing: Science Press, 2006

[本文引用: 2]

[21]

Shan

W R

, Yan

T Z

, Lv

, et al.

Analytic study on the Sawada-Kotera equation with a nonvanishing boundary condition in fluids

Commun Nonlinear Sci Numer Simulat, 2013, 18: 1568-1575

DOI:10.1016/j.cnsns.2012.11.001 URL [本文引用: 1]

[22]

许晓革, 张小媛, 孟祥花.

Bell 多项式在变系数Gargner-KP方程中的应用

量子电子学报, 2019, 33(6): 671-679

[本文引用: 3]

X G

, Zhang

X Y

, Meng

X H

Application of Bell polynomials in variable-coefficient Gargner-KP equation

Chinese J Quantum Elect, 2019, 33(6): 671-679

[本文引用: 3]

[23]

Qin

, Tian

, Wang

Y F

, et al.

Bell-polynomial approach and Wronskian determinant solutions for three sets of differential-difference nonlinear evolution equations with symbolic computation

Z Angew Math Phys, 2017, 68: 111

DOI:10.1007/s00033-017-0853-1 URL [本文引用: 1]

[24]

Tang

Y N

, Yuen

M W

, Zhang

L J

Double Wronskian solutions to the (2+1)-dimensional Broer-Kaup-Kupershmidt equation

Appl Math Lett, 2020, 105: 106285

DOI:10.1016/j.aml.2020.106285 URL [本文引用: 1]

[25]

Bell

E T

Exponential polynomials

Ann Math, 1934, 35: 258-277

DOI:10.2307/1968431 URL [本文引用: 1]

[26]

Gilson

, Lambert

, Nimmo

, et al.

On the combinatorics of the Hirota D-operators

Proceedings Roya Soci London Seri A-MPS, 1996, 452(1945): 223-234

[本文引用: 1]

[27]

Pelinovsky

, Springael

, Lambert

, et al.

On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization

J Phys A: Math Gen, 1997, 30(24): 8705-8717

DOI:10.1088/0305-4470/30/24/029 URL [本文引用: 1]

[28]

Wang

, Xia

T C

Bell polynomial approach to an extended Korteweg-de Vries equation

Math Meth Appl Sci, 2013, 37(10): 1476-1487

DOI:10.1002/mma.2908 URL [本文引用: 1]

[29]

Fan

E G

, Chow

K W

Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation

J Math Phys, 2011, 52: 023504

DOI:10.1063/1.3545804 URL [本文引用: 1]

In this paper, the binary Bell polynomials are applied to succinctly construct bilinear formulism, bilinear Bäcklund transformations, Lax pairs, and Darboux covariant Lax pairs for the (2+1)-dimensional breaking soliton equation. An extra auxiliary variable is introduced to get the bilinear formulism. The infinitely local conservation laws of the equation are found by virtue of its Lax equation and a generalized Miura transformation. All conserved densities and fluxes are given with explicit recursion formulas.

[30]

Hon

Y C

, Fan

E G

Binary Bell polynomial approach to the non-isospectral and variable-coefficient KP equations

Ima J Appl Math, 2012, 77: 236-251

DOI:10.1093/imamat/hxr023 URL [本文引用: 1]

[31]

Fan

E G

The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials

Phys Lett A, 2011, 375: 493-497

DOI:10.1016/j.physleta.2010.11.038 URL [本文引用: 1]

[32]

, Xiao

J H

, Yan

T Z

, et al.

Integrability and soliton interaction of a resonant nonlinear Schrödinger equation via binary Bell polynomials

Nonlinear Anal-Real, 2013, 14: 1669-1679

DOI:10.1016/j.nonrwa.2012.11.003 URL [本文引用: 1]

[33]

X R

, Chen

A direct procedure on the integrability of nonisospectral and variable-coefficient mkdv equation

J Nonlinear Math Phy, 2012, 19: 16-26

DOI:10.1142/S1402925112500027 URL [本文引用: 1]

[34]

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, 2019, 39A(3): 451-460

[本文引用: 2]

Hao

X H

, Cheng

Z L

The integrability of the KdV-Shallow Water Waves equation

Acta Math Sci, 2019, 39A(3): 451-460

[本文引用: 2]

[35]

Y H

, Li

R M

, Xue

, et al.

A generalized complex mKdV equation: Darboux transformations and explicit solutions

Wave Motion, 2020, 98: 102639

[本文引用: 1]

[36]

Zha

Q L

, Li

Z B

Darboux transformation and multi-solitons for complex mKdV equation

Chinese Phys Lett, 2008, 25: 8-11

DOI:10.1088/0256-307X/25/1/003 URL [本文引用: 1]

[37]

Zha

Q L

Nth-order rogue wave solutions of the complex modified Korteweg-de Vries equation

Phys Scripta, 2013, 87: 065401

DOI:10.1088/0031-8949/87/06/065401 URL [本文引用: 1]

[38]

D C

, Seadawy

, Arshada

Applications of extended simple equation method on unstable nonlinear Schrödinger equations

Optik, 2017, 140: 136-144

DOI:10.1016/j.ijleo.2017.04.032 URL [本文引用: 1]

[39]

高静

扩展KP方程的周期波解以及可积性质

潍坊学院学报, 2015, 15(2): 1-6

[本文引用: 1]

Gao

Strong stability of non-autonomous system with respect to partial variables

J Weifang Univ, 2015, 15(2): 1-6

[本文引用: 1]

[40]

罗天琦, 黄欣.

Bell多项式在(2+1)维Nizhnik方程组中的应用

四川师范大学学报(自然科学版), 2015, 38(6): 861-866

[本文引用: 1]

Luo

T Q

, Huang

Bell Polynomials application in (2+1)-dimensional Nizhnik equations

J Sichuan Norm Univ (Nat Sci), 2015, 38(6): 861-866

[本文引用: 1]

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

2019

... 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. ...

Lax pair, auto-B?cklund transformation and conservation law for a generalized variable-coefficient KdV equation with external-force term

2015

Lax pairs, infinite conservation laws, Darboux transformation, bilinear forms and solitonic interactions for a combined Calogero-Bogoyavlenskii-Schiff-type equation

2021

Integrability characteristics of a novel (2+1)-dimensional nonlinear model: Painlevé analysis, soliton solutions, B?cklund transformation, Lax pair and infinitely many conservation laws

2021

1991

Inverse scattering transform for the generalized derivative nonlinear Schr?dinger equation via matrix Riemann-Hilbert problem

2022

An inverse scattering transform for the lattice potential KdV equation

2010

Inverse scattering transform for a new non-isospectral integrable non-linear AKNS model

2017

1999

Darboux transformation for the vector sine-Gordon equation and integrable equations on a sphere

2016

Darboux transformation for coupled non-linear Schr?dinger equation and its breather solutions

2016

1978

1982

New no-traveling wave solutions for the Liouville equation by B?cklund transformation method

2013

Multiple soliton solutions, soliton-type solutions and hyperbolic solutions for the Benjamin-Bona-Mahony equation with variable coefficients in rotating fluids and one-dimensional transmitted waves

2016

B?cklund transformation, residual symmetry and exact interaction solutions of an extended (2+1)-dimensional Korteweg-de Vries equation

2022

Painlevé analysis, auto-B?cklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries(KdV) equation

2006

B?cklund transformations, kink soliton, breather- and travelling-wave solutions for a (3+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluid dynamics

2021

2004

2006

... 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. ...

2006

Analytic study on the Sawada-Kotera equation with a nonvanishing boundary condition in fluids

2013

Bell 多项式在变系数Gargner-KP方程中的应用

2019

... [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. ...

... 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]}. ...

Bell 多项式在变系数Gargner-KP方程中的应用

2019

... [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. ...

... 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]}. ...

Bell-polynomial approach and Wronskian determinant solutions for three sets of differential-difference nonlinear evolution equations with symbolic computation

2017

Double Wronskian solutions to the (2+1)-dimensional Broer-Kaup-Kupershmidt equation

2020

Exponential polynomials

1934

On the combinatorics of the Hirota D-operators

1945

On modified NLS, Kaup and NLBq equations: differential transformations and bilinearization

1997

Bell polynomial approach to an extended Korteweg-de Vries equation

2013

Darboux covariant Lax pairs and infinite conservation laws of the (2+1)-dimensional breaking soliton equation

2011

Binary Bell polynomial approach to the non-isospectral and variable-coefficient KP equations

2012

The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials

2011

Integrability and soliton interaction of a resonant nonlinear Schr?dinger equation via binary Bell polynomials

2013

A direct procedure on the integrability of nonisospectral and variable-coefficient mkdv equation

2012

一类广义浅水波KdV方程的可积性研究

2019

... 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]}. ...

一类广义浅水波KdV方程的可积性研究

2019

... 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]}. ...

A generalized complex mKdV equation: Darboux transformations and explicit solutions

2020

... 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 ...

Darboux transformation and multi-solitons for complex mKdV equation

2008

$\alpha=1$

$\beta=0$

$\gamma=1$

时, 该方程为经典的薛定谔方程 ...

Nth-order rogue wave solutions of the complex modified Korteweg-de Vries equation

2013

$\alpha=1$

$\beta=0$

$\gamma=1$

时, 该方程为经典的薛定谔方程 ...

Applications of extended simple equation method on unstable nonlinear Schr?dinger equations

2017

扩展KP方程的周期波解以及可积性质

2015

... 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]}. ...

扩展KP方程的周期波解以及可积性质

2015

... 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]}. ...

Bell多项式在(2+1)维Nizhnik方程组中的应用

2015

... 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]}. ...

Bell多项式在(2+1)维Nizhnik方程组中的应用

2015

... 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]}. ...

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