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


\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


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