2.15.18 Klein Gordon \(u_{xx}+u_{yy}+ u^2=0\)

problem number 127

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'));
 

\[u \left (x , y\right ) = -6 \left (c_{1}^{2}+c_{2}^{2}\right ) \WeierstrassP \left (c_{1} x +c_{2} y +2 c_{3}, 0, c_{4}\right )\]

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