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 ✓
ClearAll["Global`*"]; pde = D[D[u[x, y, t], t] + u[x, y, t]*D[u[x, y, t], x] + eps^2*D[u[x, y, t], {x, 3}], t] + lambda*D[u[x, y, t], {y, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];
\[\left \{\left \{u(x,y,t)\to -\frac {12 c_3 c_1{}^3 \text {eps}^2 \tanh ^2(c_3 t+c_1 x+c_2 y+c_4)+c_3 \left (c_3-8 c_1{}^3 \text {eps}^2\right )+c_2{}^2 \lambda }{c_1 c_3}\right \}\right \}\]
Maple ✓
restart; pde := diff( diff(u(x,y,t),t)+u(x,y,t)*diff(u(x,y,t),x)+epsilon^2* diff(u(x,y,t),x$3),x)+ lambda*diff(u(x,y,t),y$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));
\[u \left (x , y , t\right ) = \frac {-12 c_{2}^{4} \epsilon ^{2} \left (\tanh ^{2}\left (c_{4} t +c_{2} x +c_{3} y +c_{1}\right )\right )+8 c_{2}^{4} \epsilon ^{2}-c_{3}^{2} \lambda -c_{2} c_{4}}{c_{2}^{2}}\]
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