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'));
\[u \left ( x,t \right ) ={\frac {1}{{\it \_C1}} \left ( \left ( \RootOf \left ( \int ^{-{\frac {-{{\it \_Z}}^{2}+{\it \_C2}}{{\it \_C1}}}}\!{\frac {\sqrt {{\it \_C1}\,{\it \_f}+{\it \_C2}}}{\sqrt {-{{\it \_C1}}^{3}{\it \_C4}\,{\it \_f}-{{\it \_C1}}^{2}{\it \_C2}\,{\it \_C4}+{\it \_C1}\,{{\it \_f}}^{3}+2\,{\it \_C1}\,{{\it \_f}}^{2}k-{\it \_C3}\,{{\it \_C1}}^{2}+{\it \_C2}\,{{\it \_f}}^{2}}}}{d{\it \_f}}{\it \_C1}+{\it \_C1}\,{\it \_C5}-{\it \_C1}\,x-t{\it \_C2}-{\it \_C3} \right ) \right ) ^{2}-{\it \_C2} \right ) }\] Answer in terms of RootOf.
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