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 ✓
ClearAll["Global`*"]; pde = D[u[x, t], t] + D[u[x, t], {x,3}] + 6*u[x, t]*D[u[x, t], x] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to -\frac {12 c_1{}^3 \tanh ^2(c_2 t+c_1 x+c_3)-8 c_1{}^3+c_2}{6 c_1}\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,t),t)+ diff( u(x,t),x$3 ) + 6 * u(x,t)* diff(u(x,t),x) = 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+{\frac {8\,{{\it \_C2}}^{3}-{\it \_C3}}{6\,{\it \_C2}}}\]
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