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

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。

:{u(x, t) = -(1/30)(_C5^2+_C4^2)\sqrt(30)/(\sqrt(\alpha)_C4^2)-2\sqrt(30)_C4^2WeierstrassP(_C3+_C4x+_C5t, -(1/180)(_C5^4+2_C5^2_C4^2+_C4^4)/_C4^8, (1/5400)(_C5^2+_C4^2)(_C5^4+2_C5^2*_C4^2+_C4^4)/_C4^12)/\sqrt(\alpha)}
:{u(x, t) = (1/30)(_C5^2+_C4^2)\sqrt(30)/(\sqrt(\alpha)_C4^2)+2\sqrt(30)_C4^2WeierstrassP(_C3+_C4x+_C5t, -(1/180)(_C5^4+2_C5^2_C4^2+_C4^4)/_C4^8, (1/5400)(_C5^2+_C4^2)(_C5^4+2_C5^2*_C4^2+_C4^4)/_C4^12)/\sqrt(\alpha)}
:p[3] := -4.4229081351691113421-0.93307517430300270455e-1I+(18.208764436791314548+0.I)JacobiDN(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
:p[4] := -5.1919544739096094160+5.3436314406870407268I+(18.208764436791314548+0.I)JacobiNS(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
:p[5] := -2.5743416547838020433-3.2877514938561477464I+(2.3878725879366620662-40.119130323671954306I)JacobiND(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
:p[6] := -5.5945307529851545322+1.6067031040792677714I+(2.3878725879366620662-40.119130323671954306I)JacobiNC(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
:p[7] := -5.1805015655073574818+2.2007257608074972052I+(21.217747598613725681-27.288660748456732356I)JacobiSN(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
:p[8] := -5.7120768471542134149-.77489365858670457610I+(21.217747598613725681-27.288660748456732356I)JacobiCN(1.22+1.3x+(15.095721921379631782+24.271454992000839308I)t, 1.1984050731412980386-.42555954136146448708*I)^1.5
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行波图
参考文献

*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

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李志斌编著 《非线性数学物理方程的行波解》 科学出版社

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