Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Hunter Saxton. Solve for \(u(x,t)\) \[ \left ( u_t + u u_x) \right )_x = \frac {1}{2} (u_x)^2 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[D[u[x, t], t] + u[x, t]*D[u[x, t], x], x] == (1*D[u[x, t], x]^2)/2; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
Failed
Maple ✓
restart; pde := diff( (diff(u(x,t),t)+ u(x,t)* diff(u(x,t),x)) , x) = 1/2* (diff(u(x,t),x))^2; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t),'build')),output='realtime'));
\[u \left (x , t\right ) = \frac {2 \RootOf \left (-c_{2} \mathit {\_c}_{1}^{3}-x \mathit {\_c}_{1}^{3}+2 c_{1}^{2} \ln \left (\sqrt {\mathit {\_Z}}\, \mathit {\_c}_{1}+c_{1}\right )+\mathit {\_Z} \mathit {\_c}_{1}^{2}-2 c_{1} \sqrt {\mathit {\_Z}}\, \mathit {\_c}_{1}\right )}{t \mathit {\_c}_{1}+2 c_{3}}\] with RootOf
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