I'm looking at this example regarding dispersive relations from modified PDE equations:
I'm working with $a=1$. I derived the modified equation, but I don't understand how the author makes the conclusions about the group velocity. After taking Fourier transforms of the modified equation and solving, I have $$\hat{v}(\xi, t) = [i\xi - \gamma i\xi^3]\hat{v}(\xi, t),$$ for $\gamma = \frac{2h^2 - 3kh + k^2}{6}.$ So, I think the group velocity should be $1 - \gamma \xi^2.$ But this can't be right, because then solving the inequality $1 - \gamma \xi^2 < a = 1$ does not result in the condition $1 < k/h < 2,$ which is what the author has. How does he get these conditions?