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IN this article the author written a nonlinear equation as $$-\frac{\partial^2 \phi}{\partial t^2}+ \nabla \phi= \phi+ \sum_{k=2}^\infty g_k \phi^k$$

They have scaled the time as, $$\tau=\omega(\varepsilon) t$$ they have said in equation (8) is $\omega=1$ because of threshold and in the equation (14) they have said that $\omega_1 =0$ because ofbounded solution and absence of bounded solution no resonance occur.

Can you explain the two conditions please that why is this?

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