近年来, 数学、物理等各个领域都在研究孤子, 其中求解孤子方程的精确解是应用数学领域中的热门话题之一.目前已有了多种求解孤子方程精确解的方法, 如:双线性导数法^[1]、反散射方法^{[2, 3]}、Darboux变换^{[4, 5]}、tanh方法^{[6, 7]}等.

这些方法中, 双线性导数法是由著名的日本数学、物理学家Ryogo Hirota提出, 他是在研究非线性偏微分方程的解的过程中, 利用摄动法得到一种双线性方程, 并定义出一种新的微分算子——Hirota双线性算子

$ \begin{eqnarray} &&D_{x}^m D_{t}^n f \cdot g\\ & = &(\frac{\partial}{\partial x}-\frac{\partial}{\partial x^{'}}) {}^m (\frac{\partial}{\partial t}-\frac{\partial}{\partial t^{'}}) {}^n f(x, t) g(x^{'}, t^{'})\mid_{x^{'} = x, t^{'} = t}\\ & = &\frac{\partial^{m}}{\partial x^{'m}} \frac{\partial^{n}}{\partial t^{'n}} f(x+x^{'}, t+t^{'}) g(x-x^{'}, t-t^{'}) \mid_{x^{'} = 0, t^{'} = 0}\, . \end{eqnarray} $

如果某一方程具有双线性形式, 那么该方程就能具备可积性.在Hirota双线性算子基础上, 马文秀^[8-11]教授提出了一般的双线性微分算子

$ \begin{eqnarray} &&D_{p, x}^m D_{p, t}^n f \cdot f\\ & = &(\frac{\partial}{\partial x}+\alpha_{p} \frac{\partial}{\partial x^{'}}) {}^m (\frac{\partial}{\partial t}+\alpha_{p} \frac{\partial}{\partial t^{'}}) {}^n f(x, t) f(x^{'}, t^{'}) \mid_{x^{'} = x, t^{'} = t}\\ & = &\sum\limits_{i = 0}^m \sum\limits_{j = 0}^n (\begin{array}{c} m\\ i\\ \end{array}) (\begin{array}{c} n \\ j \\ \end{array}) \alpha_{p}^i \alpha_{p}^j \frac{\partial^{m-i}}{\partial x^{m-i}} \frac{\partial^{i}}{\partial x^{'i}} \frac{\partial^{n-j}}{\partial t^{n-j}}\frac{\partial^{j}}{\partial t^{'j}} f(x, t) f(x^{'}, t^{'}) \mid_{x^{'} = x, t^{'} = t}\\ & = &\sum\limits_{i = 0}^m \sum\limits_{j = 0}^n (\begin{array}{c} m \\ i \\ \end{array}) (\begin{array}{c} n \\ j \\ \end{array}) \alpha_{p}^i \alpha_{p}^j \frac{\partial^{m+n-i-j}f(x, t)}{\partial x^{m-i} \partial t^{n-j}} \frac{\partial^{i+j}f(x, t)}{\partial x^{i} \partial t^{j}}, \, \, \, \, \, \, \, \, m, n\geq0\, , \end{eqnarray} $

(1.1)

其中$ p $是素数且$ p\geq2 $, 并且(1.1)式中的$ \alpha_{p}^s $满足

$ \begin{eqnarray} \alpha_{p}^s = (-1)^{r_{p}(s)}, \, \, \, \, s = r_{p}(s) \bmod p\, \end{eqnarray} $

(1.2)

且$ \begin{aligned} \alpha_{p}^i \alpha_{p}^j \neq \alpha_{p}^{i+j}, \, \, \, \, i, j\geq0\, . \end{aligned} $

本文中, 借助这个一般双线性算子(1.1), 从(2+1)维浅水波方程^{[12, 13]}

$ \begin{eqnarray} u_{xxy}+3 u_{x} u_{y}-u_{y}-u_{t} = 0\, \end{eqnarray} $

(1.3)

的双线性形式, 构造出$ p = 3 $对应的一个广义浅水波类方程.再通过求解该方程的一般双线性方程的多项式解, 构造了该广义浅水波类方程的有理解.

(2+1)维浅水波方程(1.3)通过变换$ u = 2(\ln f)_{x} $得到其双线性形式

$ \begin{eqnarray} &&(D_{x}^3 D_{y}-D_{x} D_{y}-D_{x} D_{t}) f\cdot f \\ & = &6 f_{xx} f_{xy}-6 f_{x} f_{xxy}+2 f_{xxxy} f-2 f_{xxx} f_{y}-2 f f_{xy}+2 f_{x} f_{y}-2 f f_{xt}+2 f_{x} f_{t} \\ & = &0. \end{eqnarray} $

(2.1)

通过计算可以证明, (2.1)即为$ p = 2 $时$ (D_{2, x}^3 D_{2, y}-D_{2, x} D_{2, y}-D_{2, x} D_{2, t}) f\cdot f $的形式.

根据一般的双线性算子(1.1), 求得

$ \begin{eqnarray} &&D_{3, x}^3 D_{3, y} f\cdot f = 6 f_{xx} f_{xy}-6 f_{xxy} f_{x}, \, \, \, \, \, \, \, \, D_{3, x} D_{3, y} f\cdot f = 2 f_{xy} f-2 f_{x} f_{y}, \\ &&D_{3, x} D_{3, t} f\cdot f = 2 f_{xt} f-2 f_{x} f_{t}, \end{eqnarray} $

其中$ \alpha_{3} = -1, \, \, \alpha_{3}^2 = \alpha_{3}^3 = 1, \, \, \alpha_{3}^4 = -1, \, \, \alpha_{3}^5 = \alpha_{3}^6 = 1, \, \cdots , $所以有

$ \begin{eqnarray} &&(D_{3, x}^3 D_{3, y}-D_{3, x} D_{3, y}-D_{3, x} D_{3, t}) f\cdot f \\ & = &6 f_{xx} f_{xy}-6 f_{xxy} f_{x}-2 f_{xy} f+2 f_{x} f_{y}-2 f_{xt} f+2 f_{x} f_{t} \\ & = &0. \end{eqnarray} $

(2.2)

利用贝尔多项式理论^[14-16], 选取变换

$ \begin{eqnarray} u = (\ln f)_{x}, \end{eqnarray} $

(2.3)

可得

$ \begin{eqnarray} \frac{(D_{3, x}^3 D_{3, y}-D_{3, x} D_{3, y}-D_{3, x} D_{3, t}) f\cdot f}{f^2} = -2(3 u u_{xy}+3 u^2 u_{y}-3 u_{x} u_{y}+u_{y}+u_{t}), \end{eqnarray} $

(2.4)

则有广义(2+1)维浅水波类方程

$ \begin{eqnarray} 3 u u_{xy}+3 u^2 u_{y}-3 u_{x} u_{y}+u_{y}+u_{t} = 0. \end{eqnarray} $

(2.5)

比较浅水波方程(1.3)和广义浅水波类方程(2.5), 可以发现(2.5)的双线性形式(2.2)比(1.3)的双线性形式(2.1)更简单一些, 但(2.5)比(1.3)更具有非线性.

根据变换(2.3), 若$ f $是方程(2.2)的解, 则有$ u $为方程(2.5)的解.

借助数学软件Mathematica, 令

$ \begin{eqnarray} f(x, y, t) = \sum\limits_{i = 0}^2 \sum\limits_{j = 0}^2 \sum\limits_{k = 0}^1 c_{i, j, k} x^i y^j t^k, \end{eqnarray} $

(3.1)

代入(2.2)式, 可以得到它的一系列的多项式解

$ \begin{eqnarray} f_{1}& = &\frac{t x^2 y c_{2, 0, 1} \left(c_{2, 0, 1}+c_{2, 1, 0}\right)}{c_{2, 0, 0}}+\frac{t x y c_{1, 0, 0} c_{2, 0, 1} \left(c_{2, 0, 1}+c_{2, 1, 0}\right)}{c_{2, 0, 0}^2}+t x^2 c_{2, 0, 1}+\frac{t x c_{1, 0, 0} c_{2, 0, 1}}{c_{2, 0, 0}} \\ &&+\frac{t y c_{0, 0, 1} \left(c_{2, 0, 1}+c_{2, 1, 0}\right)}{c_{2, 0, 0}}-\frac{x^2 y^2 \left(c_{2, 0, 1}^2+c_{2, 1, 0} c_{2, 0, 1}\right)}{c_{2, 0, 0}}-\frac{x y^2 c_{1, 0, 0} c_{2, 0, 1} \left(c_{2, 0, 1}+c_{2, 1, 0}\right)}{c_{2, 0, 0}^2} \\ &&+x^2 y c_{2, 1, 0}+\frac{x y c_{1, 0, 0} c_{2, 1, 0}}{c_{2, 0, 0}}+x^2 c_{2, 0, 0}+x c_{1, 0, 0}-\frac{y^2 c_{0, 0, 1} \left(c_{2, 0, 1}+c_{2, 1, 0}\right)}{c_{2, 0, 0}} \\ &&+\frac{y \left(-c_{0, 0, 1} c_{2, 0, 0}+c_{0, 0, 0} c_{2, 0, 1}+c_{0, 0, 0} c_{2, 1, 0}\right)}{c_{2, 0, 0}}+t c_{0, 0, 1}+c_{0, 0, 0}, \end{eqnarray} $

(3.2)

$ \begin{eqnarray} f_{2}& = &t x^2 y c_{2, 1, 1}+\frac{t x y c_{1, 1, 0} c_{2, 1, 1}}{c_{2, 1, 0}}-t x^2 c_{2, 1, 0}-t x c_{1, 1, 0}-\frac{t y c_{0, 0, 1} c_{2, 1, 1}}{c_{2, 1, 0}}-x^2 y^2 c_{2, 1, 1} \\ &&-\frac{x y^2 c_{1, 1, 0} c_{2, 1, 1}}{c_{2, 1, 0}}+x^2 y c_{2, 1, 0}+x y c_{1, 1, 0}+\frac{y^2 c_{0, 0, 1} c_{2, 1, 1}}{c_{2, 1, 0}}-\frac{y \left(c_{0, 0, 1} c_{2, 1, 0}+c_{0, 0, 0} c_{2, 1, 1}\right)}{c_{2, 1, 0}} \\ &&+t c_{0, 0, 1}+c_{0, 0, 0}, \end{eqnarray} $

(3.3)

$ \begin{eqnarray} f_{3}& = &\frac{t x y c_{1, 0, 1} \left(c_{1, 0, 1}+c_{1, 1, 0}\right)}{c_{1, 0, 0}}+t x c_{1, 0, 1}+\frac{t y c_{0, 0, 1} \left(c_{1, 0, 1}+c_{1, 1, 0}\right)}{c_{1, 0, 0}}-\frac{x y^2 \left(c_{1, 0, 1}^2+c_{1, 1, 0} c_{1, 0, 1}\right)}{c_{1, 0, 0}} \\ &&+x y c_{1, 1, 0}-\frac{y^2 c_{0, 0, 1} \left(c_{1, 0, 1}+c_{1, 1, 0}\right)}{c_{1, 0, 0}}+\frac{y \left(-c_{0, 0, 1} c_{1, 0, 0}+c_{0, 0, 0} c_{1, 0, 1}+c_{0, 0, 0} c_{1, 1, 0}\right)}{c_{1, 0, 0}} \\ &&+x c_{1, 0, 0}+t c_{0, 0, 1}+c_{0, 0, 0}, \end{eqnarray} $

(3.4)

$ \begin{eqnarray} f_{4}& = &t x^2 y^2 c_{2, 2, 1}+\frac{t x y^2 c_{1, 2, 0} c_{2, 2, 1}}{c_{2, 2, 0}}+\frac{t y^2 c_{0, 2, 0} c_{2, 2, 1}}{c_{2, 2, 0}}+x^2 y^2 c_{2, 2, 0}+x y^2 c_{1, 2, 0}+y^2 c_{0, 2, 0}.\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \end{eqnarray} $

(3.5)

对应地, 根据变换$ u = (\ln f)_{x} $, 可以求得广义(2+1)维浅水波类方程(2.5)的4类有理解.

第一类有理解

$ \begin{eqnarray} u_{1} = \frac{p}{q}, \end{eqnarray} $

(3.6)

其中

$ \begin{eqnarray} p& = &2 t x c_{2, 0, 1} c_{2, 0, 0}-2 x y c_{2, 0, 1} c_{2, 0, 0}+2 x c_{2, 0, 0}^2-y c_{1, 0, 0} c_{2, 0, 1}+t c_{1, 0, 0} c_{2, 0, 1}+c_{1, 0, 0} c_{2, 0, 0}, \, \, \, \, \, \, \, \\ q& = &t x^2 c_{2, 0, 0} c_{2, 0, 1}-x^2 y c_{2, 0, 0} c_{2, 0, 1}-x y c_{1, 0, 0} c_{2, 0, 1}+t x c_{1, 0, 0} c_{2, 0, 1}+x^2 c_{2, 0, 0}^2+x c_{1, 0, 0} c_{2, 0, 0}\\ &&-y c_{0, 0, 1} c_{2, 0, 0}+t c_{0, 0, 1} c_{2, 0, 0}+c_{0, 0, 0} c_{2, 0, 0}. \end{eqnarray} $

第二类有理解

$ \begin{eqnarray} u_{2} = - \frac{2 t x c_{2, 1, 0}-2 x y c_{2, 1, 0}-y c_{1, 1, 0}+t c_{1, 1, 0}}{x^2 y c_{2, 1, 0}-t x^2 c_{2, 1, 0}+x y c_{1, 1, 0}-t x c_{1, 1, 0}-y c_{0, 0, 1}+t c_{0, 0, 1}+c_{0, 0, 0}}. \end{eqnarray} $

(3.7)

第三类有理解

$ \begin{eqnarray} u_{3} = \frac{t c_{1, 0, 1}-y c_{1, 0, 1}+c_{1, 0, 0}}{t x c_{1, 0, 1}-x y c_{1, 0, 1}+x c_{1, 0, 0}-y c_{0, 0, 1}+t c_{0, 0, 1}+c_{0, 0, 0}}. \end{eqnarray} $

(3.8)

第四类有理解

$ \begin{eqnarray} u_{4} = \frac{2 x c_{2, 2, 0}+c_{1, 2, 0}}{x \left(x c_{2, 2, 0}+c_{1, 2, 0}\right)+c_{0, 2, 0}}. \end{eqnarray} $

(3.9)

本文中, 在(2+1)维浅水波方程(1.3)的基础上, 利用一般的双线性微分算子(1.1), 当素数$ p = 3 $时, 得到了具有一般双线性形式的微分方程——广义(2+1)维浅水波类方程(2.5).借助广义(2+1)维浅水波类方程(2.5)的一般双线性形式, 利用数学软件Mathematica, 得到了方程(2.5)的4类有理解.

当参数被选取为

$ \begin{eqnarray} c_{i, j, k} = 1+i^2+j^2+k^2, \, \, \, \, \, \, \, 0\leq i, j \leq 2, \, \, \, \, \, 0\leq k \leq 1\, \end{eqnarray} $

(4.1)

时, 解(3.6) – (3.8)分别为

$ \begin{eqnarray} u_{1} = \frac{60 t x-60 x y+12 t+50 x-12 y+10}{30 t x^2+12 t x-30 x^2 y-12 x y+25 x^2+10 x-10 y+10 t+5}, \end{eqnarray} $

(4.2)

$ \begin{eqnarray} u_{2} = -\frac{12 t x-12 x y+3 t-3 y}{6 x^2 y-6 t x^2+3 x y-3 t x+2 t-2 y+1}, \end{eqnarray} $

(4.3)

$ \begin{eqnarray} u_{3} = \frac{3 t-3 y+2}{3 t x-3 x y+2 x+2 t-2 y+1}. \end{eqnarray} $

(4.4)

即得到广义(2+1)维浅水波类方程(2.5)的3类特殊有理解.解(4.2), (4.3)和(4.4)在$ t = 1 $时刻的三维图、密度图和等高线图如图 1–3所示.从这些图能看出, $ y = 0 $直线附近解曲面变化很大, 且当$ x $趋于无穷大时这些解都趋向于0, 即解曲面趋向水平面.