= Bogomolny classification of a rescaled phi-six kink
{c}
{title2=$\phi_{\sigma,\eta}(x)=\sigma b[1+e^{-2\eta b^2(x-x_0)}]^{-1/2}$}
For $U=\phi^2(\phi^2-b^2)^2/2$ and $b>0$, the three <scalar-field vacua> are $-b,0,b$. The static first integral is $\phi_x^2=2U$. In either adjacent interval, $y=\phi^2/b^2$ obeys $y_x=\pm2b^2y(1-y)$, giving the displayed four oriented <kink> profiles with $\sigma,\eta\in\{1,-1\}$. Each has rest <energy> $b^4/4$. The <intermediate-vacuum obstruction to a kink> excludes a direct static connection from $-b$ to $b$: reaching zero would give zero derivative, and uniqueness of the smooth second-order field equation forces the zero solution. A <Lorentz boost> replaces $x-x_0$ by $(x-vt-x_0)/\sqrt{1-v^2}$.
// Destination: relativistic-quantum-field.bigb
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