Added December 27, 2018.
Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations
Camassa Holm. Solve for \(u(x,t)\) \[ u_t + 2 k u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx}+ u u_{xxx} \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[u[x, t], t] + 2*k*D[u[x, t], x] - D[D[u[x, t], {x, 2}], t] + 3*u[x, t]*D[u[x, t], x] == 2*D[u[x, t], x]*D[u[x, t], {x, 2}] + u[x, t]*D[u[x, t], {x, 3}]; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
Failed
Maple ✓
restart; pde := diff(u(x,t),t)+2*k*diff(u(x,t),x)- diff(u(x,t),x,x,t)+3*u(x,t)*diff(u(x,t),x)=2*diff(u(x,t),x)*diff(u(x,t),x$2)+u(x,t)*diff(u(x,t),x$3); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
\[u \left (x , t\right ) = \frac {\RootOf \left (c_{1} c_{5}-c_{1} x +c_{1} \left (\int _{}^{-\frac {-\mathit {\_Z}^{2}+c_{2}}{c_{1}}}\frac {\sqrt {c_{1} \mathit {\_f} +c_{2}}}{\sqrt {-c_{4} c_{1}^{3} \mathit {\_f} +c_{1} \mathit {\_f}^{3}+2 c_{1} \mathit {\_f}^{2} k -c_{1}^{2} c_{2} c_{4}+c_{2} \mathit {\_f}^{2}-c_{3} c_{1}^{2}}}d\mathit {\_f} \right )-c_{2} t -c_{3}\right )^{2}-c_{2}}{c_{1}}\] Answer in terms of RootOf.
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