Bretherton equation

Bretherton equation is a nonlinear partial differential equation introduced by Chris Bretherton：[1]

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

Analytic solution

{u(x, t) = -(1/30)*(_C5^2+_C4^2)*\sqrt(30)/(\sqrt(\alpha)*_C4^2)-2*\sqrt(30)*_C4^2*WeierstrassP(_C3+_C4*x+_C5*t, -(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)*_C4^2*WeierstrassP(_C3+_C4*x+_C5*t, -(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-1*I+(18.208764436791314548+0.*I)*JacobiDN(1.22+1.3*x+(15.095721921379631782+24.271454992000839308*I)*t, 1.1984050731412980386-.42555954136146448708*I)^1.5
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
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
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
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
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

Traveling wave plot

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

See also

Reference

  1. ↑ Li Zhibing, Traveling wave solutions of nonlinear mathematical physics equations, P 152 SCIENCEP, 2008(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}
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