布雷瑟顿方程

布雷瑟顿方程(Bretherton equation)是一个非线性偏微分方程：^[1]

${\displaystyle u_{tt}+u_{xx}+u_{xxxx}-\alpha *u^{3}=0}$

解析解

利用Maple软件包TWSolution可得布雷瑟顿方程两个WeirstrassP函数行波解和六个Jacobi椭圆函数行波解。

${\displaystyle {u(x,t)=-(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})-2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}}$

${\displaystyle {u(x,t)=(1/30)*(_{C}5^{2}+_{C}4^{2})*{\sqrt {(}}30)/({\sqrt {(}}\alpha )*_{C}4^{2})+2*{\sqrt {(}}30)*_{C}4^{2}*WeierstrassP(_{C}3+_{C}4*x+_{C}5*t,-(1/180)*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{8},(1/5400)*(_{C}5^{2}+_{C}4^{2})*(_{C}5^{4}+2*_{C}5^{2}*_{C}4^{2}+_{C}4^{4})/_{C}4^{1}2)/{\sqrt {(}}\alpha )}}$

${\displaystyle p[3]:=-4.4229081351691113421-0.93307517430300270455e-1*I+(18.208764436791314548+0.*I)*JacobiDN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

${\displaystyle p[4]:=-5.1919544739096094160+5.3436314406870407268*I+(18.208764436791314548+0.*I)*JacobiNS(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

${\displaystyle p[5]:=-2.5743416547838020433-3.2877514938561477464*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiND(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

${\displaystyle p[6]:=-5.5945307529851545322+1.6067031040792677714*I+(2.3878725879366620662-40.119130323671954306*I)*JacobiNC(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

${\displaystyle p[7]:=-5.1805015655073574818+2.2007257608074972052*I+(21.217747598613725681-27.288660748456732356*I)*JacobiSN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

${\displaystyle p[8]:=-5.7120768471542134149-.77489365858670457610*I+(21.217747598613725681-27.288660748456732356*I)*JacobiCN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t,1.1984050731412980386-.42555954136146448708*I)^{1}.5}$

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行波图

Bretherton equation traveling wave WeierstrassP plot 1

Bretherton equation traveling wave WeierstrassP plot 1

Bretherton equation traveling wave Jacobi function plot

Bretherton equation traveling wave Jacobi function plot

Bretherton equation traveling wave Jacobi function plot

Bretherton equation traveling wave Jacobi function plot

Bretherton equation traveling wave Jacobi function plot

Bretherton equation traveling wave Jacobi function plot

参考文献

1. 李志斌编著 《非线性数学物理方程的行波解》 152页 科学出版社 2008

1. *谷超豪 《孤立子理论中的达布变换及其几何应用》 上海科学技术出版社
2. *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
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