= Opposite propagation directions in the annular displacement equation
{title2=$\omega=\sigma k^3/(\beta\alpha_1),\quad A=-3\beta\alpha_1c\,\mathrm{sech}^2(k_s\xi)/\alpha_2$}
With no forcing, the linear <dispersion relation> has positive <phase velocity> $\sigma k^2/(\beta\alpha_1)$ and <group velocity> three times this. A decaying permanent wave with $\xi=x+ct$ instead obeys $\sigma(A')^2=A^2(\beta\alpha_1c+\alpha_2A/3)$. For positive coefficients a nonzero homoclinic wave requires $c>0$ and is negative, travelling upstream. Its inverse width is $k_s=\frac12\sqrt{\beta\alpha_1c/\sigma}$. The opposite directions reflect the nonlinear amplitude-dependent propagation speed.
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