2.15.20 Korteweg de Vries (KdV) \(u_t + (u_x)^3+ 6 u u_x = 0\)

problem number 129

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Korteweg de Vries (KdV). Solve for \(u(x,t)\) \[ u_t + u_{xxx}+ 6 u u_x = 0 \]

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


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

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