Added December 27, 2018.
Special case Klein Gordon (nonlinear). Solve for \(u(x,y)\) \[ u_{xx}+u_{yy}+ u^2=0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = Laplacian[u[x, y], {x, y}] + u[x, y]^2 == 0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
Failed
Maple ✓
restart; pde := diff(u(x,y),x$2)+diff(u(x,y),y$2)+u(x,y)^2=0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y))),output='realtime'));
\[u \left ( x,y \right ) =-6\,{\it WeierstrassP} \left ( {\it \_C1}\,x+{\it \_C2}\,y+2\,{\it \_C3},0,{\it \_C4} \right ) \left ( {{\it \_C1}}^{2}+{{\it \_C2}}^{2} \right ) \]
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