28 Korteweg-deVries PDE

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

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

problem number 165

From Mathematica symbolic PDE document.

Solve for \(u(x,t)\) \[ \frac{\partial ^3 u}{\partial x^3} + \frac{\partial u}{\partial t}- 6 u(x,t) \frac{\partial u}{\partial x} = 0 \]

Reference https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation

Mathematica

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

Maple

\[ u \left ( x,t \right ) =2\,{{\it \_C2}}^{2} \left ( \tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}-1/6\,{\frac{8\,{{\it \_C2}}^{3}-{\it \_C3}}{{\it \_C2}}} \]