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