2.15.26 Sine Gordon \(\phi _{tt} - \phi _{xx} + \sin \phi = 0\)

problem number 135

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Sine Gordon. Solve for \(u(x,t)\) \[ \phi _{tt} - \phi _{xx} + \sin \phi = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[phi[x, t], {t, 2}] - D[phi[x, t], {x, 2}] + Sin[phi[x, t]] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, phi[x, t], {x, t}], 60*10]];
 

Failed

Maple ✗

restart; 
pde := diff(phi(x,t),t$2)-diff(phi(x,t),x$2)+sin(phi(x,t))=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,phi(x,t))),output='realtime'));
 

sol=()

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