2.14.18 Klein Gordon $u_{xx}+u_{yy}+u^2=0$

problem number 120

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'));
 

$$u(x,y)=-6(c_1^2+c_2^2)\mathrm{WeierstrassP}(c_{1}x+c_{2}y+2c_3,0,c_4)$$

____________________________________________________________________________________