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In this article the authors wrote 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, $$\tau=\omega(\varepsilon) t$$ where they argued that $\omega$ is analytic near the threshold value, $\omega = 1$. Later they argue that $\omega_{1} = 0$ because they are looking for bounded solutions. They argue this is equivalent to a lack of the resonance term $\omega_{1} \ddot{\phi}_{1}$ in their equation (11).

Can anyone provide a justification for these two assumptions and why these assumptions are valid?

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Related question which also links to arXiv:0802.3525: physics.stackexchange.com/q/52590/2451 and links therein. –  Qmechanic Oct 9 at 18:48

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