2.6.1 \(u_{xxx} + u_t -6 u u_x = 0\)

problem number 91

From Mathematica symbolic PDE document.

Korteweg-deVries (waves on shallow water surfaces) with no initial conditions

Solve for \(u(x,t)\) \[ u_{xxx} + u_t -6 u u_x = 0 \] Reference https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation

Mathematica


\[\left \{\left \{u(x,t)\to \frac {12 c_1{}^3 \tanh ^2(c_2 t+c_1 x+c_3)-8 c_1{}^3+c_2}{6 c_1}\right \}\right \}\]

Maple


\[u \left (x , t\right ) = 2 c_{2}^{2} \left (\tanh ^{2}\left (c_{3} t +c_{2} x +c_{1}\right )\right )+\frac {-8 c_{2}^{3}+c_{3}}{6 c_{2}}\]