Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Boussinesq. Solve for \(u(x,t)\) \[ u_{tt}-u_{xx}-u_{xxxx} - 3 (u^2)_{xx} = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[u[x, t], {t, 2}] - D[u[x, t], {x, 2}] - D[u[x, t], {x, 4}] - 3*D[u[x, t]^2, {x, 2}] == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
\[\left \{\left \{u(x,t)\to \frac {1}{6} \left (-12 c_1{}^2 \tanh ^2(c_2 t+c_1 x+c_3)+8 c_1{}^2+\frac {c_2{}^2}{c_1{}^2}-1\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(u(x,t),t$2)-diff(u(x,t),x$2)-diff(u(x,t),x$4)- 3 * diff( u(x,t)^2, x$2)=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[u \left ( x,t \right ) =-2\,{{\it \_C2}}^{2} \left ( \tanh \left ( {\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) \right ) ^{2}+1/6\,{\frac {8\,{{\it \_C2}}^{4}-{{\it \_C2}}^{2}+{{\it \_C3}}^{2}}{{{\it \_C2}}^{2}}}\]
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