2.15.16 Kadomtsev Petviashvili \( \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \)

problem number 125

Added December 27, 2018.

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

Kadomtsev Petviashvili. Solve for \(u(x,y,t)\) \[ \left ( u_t + u u_x + \epsilon ^2 u_{xxx} \right )_x + \lambda u_{yy} = 0 \]

Mathematica


\[\left \{\left \{u(x,y,t)\to -\frac {12 c_3 c_1{}^3 \text {eps}^2 \tanh ^2(c_3 t+c_1 x+c_2 y+c_4)+c_3 \left (c_3-8 c_1{}^3 \text {eps}^2\right )+c_2{}^2 \lambda }{c_1 c_3}\right \}\right \}\]

Maple


\[u \left (x , y , t\right ) = \frac {-12 c_{2}^{4} \epsilon ^{2} \left (\tanh ^{2}\left (c_{4} t +c_{2} x +c_{3} y +c_{1}\right )\right )+8 c_{2}^{4} \epsilon ^{2}-c_{3}^{2} \lambda -c_{2} c_{4}}{c_{2}^{2}}\]

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