2.15.30 phi equation \(\phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0\)

problem number 139

Added December 27, 2018.

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

phi equation. Solve for \(phi(x,t)\)

\[ \phi _{tt} - \phi _{xx} - \phi + \phi ^3 = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[phi[x, t], t, t] - D[phi[x, t], x, x] - phi[x, t] + phi[x, t]^3 == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, phi[x, t], {x, t}], 60*10]];
 

\begin{align*}& \left \{\phi (x,t)\to -\tanh \left (c_2 t-\sqrt {\frac {1}{2}+c_2{}^2} x+c_3\right )\right \}\\& \left \{\phi (x,t)\to \tanh \left (c_2 t-\sqrt {\frac {1}{2}+c_2{}^2} x+c_3\right )\right \}\\& \left \{\phi (x,t)\to -\tanh \left (c_2 t+\sqrt {\frac {1}{2}+c_2{}^2} x+c_3\right )\right \}\\& \left \{\phi (x,t)\to \tanh \left (c_2 t+\sqrt {\frac {1}{2}+c_2{}^2} x+c_3\right )\right \}\\\end{align*}

Maple ✓

restart; 
pde := diff(phi(x,t),t$2)-diff(phi(x,t),x$2) - phi(x,t) + phi(x,t)^3=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',PDEtools:-TWSolutions(pde,phi(x,t))),output='realtime'));
 

\begin{align*}& \{\phi \left (x , t\right ) = -1\}\\& \{\phi \left (x , t\right ) = 1\}\\& \left \{\phi \left (x , t\right ) = -\tanh \left (-\frac {\sqrt {4 c_2^{2}-2}\, t}{2}+c_2 x +c_1 \right )\right \}\\& \left \{\phi \left (x , t\right ) = \tanh \left (\frac {\sqrt {4 c_2^{2}-2}\, t}{2}+c_2 x +c_1 \right )\right \}\\& \left \{\phi \left (x , t\right ) = \tanh \left (-\frac {\sqrt {4 c_2^{2}-2}\, t}{2}+c_2 x +c_1 \right )\right \}\\& \left \{\phi \left (x , t\right ) = -\tanh \left (\frac {\sqrt {4 c_2^{2}-2}\, t}{2}+c_2 x +c_1 \right )\right \}\\ \end{align*}