I'm trying to understand the derivation of the solvability condition for the Duffing equation.
Using the method of multiple scales we can write the first order problem for the Duffing equation as:
$$\partial^2_t Y_1 + Y_1 = - Y_0^3 -2 \partial_{t,t_1} Y_0$$
Where $t_1 = \varepsilon t$ and $Y_0(t,t_1)=A(t_1)e^{it} + C.C.$ (C.C. stands for complex conjugate). To get the solvability equation from here we express the previous equation as (we look for solutions in $L^2$):
$$ \mathcal{L}(Y_1) = b$$
And then if we consider $u \in$ Ker$(\mathcal{L}^\dagger)$ we would have:
$$<u|b>=<u|\mathcal{L}(Y_1)>=<\mathcal{L}^\dagger(u)|Y_1>=0$$
I have found that $\mathcal{L}=\mathcal{L}^\dagger$ so $u=Ce^{it}+De^{-it}$. Taking now the scalar product is clear that to get 0 we need to impose the condition
$$-3A^2A^* -2i \frac{dA}{dt_1} = 0$$
but there is also another couple of terms of the form $-A^3 (De^{2it}+Ce^{4it})$. Why are these terms not relevant? I don't see that they go to 0 when you integrate. I know that they're bounded, but I don't know now how to see that they are exactly 0.
Why don't we consider these terms for the solvability condition?