Since series (2.5)–(2.7) may not converge, we can still assume that \((\Phi ,\Pi ,A,\delta )\) are all close to the AdS solution (0, 0, 1, 0) and expand \((\Phi ,\Pi ,A,\delta )\) using (2.8)–(2.10). First, we solve at the linear level and obtain the periodic expressions

$$\begin \Phi _(\tau ,x):= \cos (\tau ) e^_(x), \\ \Pi _(\tau ,x) := - \omega _ \sin (\tau ) e_(x), \\ A_(\tau ,x) := \cos ^2(\tau ) \Gamma _(x)+ \sin ^2(\tau ) \Gamma _(x),\\ \delta _(\tau ,x):= -\cos ^2(\tau ) \Gamma _(x) - \sin ^2(\tau ) \Gamma _(x), \end\right. } \end$$

where

$$\begin&\Gamma _(x):=\frac\int _^ \left( e^_(y)\right) ^2 \tan ^2(y) \textrmy, \\&\Gamma _(x):= \omega _^2 \frac\int _^ \left( e_(y)\right) ^2 \tan ^2(y) \textrmy, \\&\Gamma _(x):= \int _^ \left( e^_(y)\right) ^2 \sin (y)\cos (y) \textrmy, \\&\Gamma _(x):=\omega _^2 \int _^ \left( e_(y)\right) ^2 \sin (y)\cos (y) \textrmy. \end$$

For simplicity, we have fixed \(\gamma = 0\) and use Lemma 2.2 to compute

$$\begin&\Gamma _(x) = \frac \Big ( 12 x -3\sin (2x)-3\sin (4x)+\sin (6x) \Big ), \\&\Gamma _(x)= - \frac \Big ( -12 x -3\sin (2x)+3\sin (4x)+\sin (6x) \Big ), \\&\Gamma _(x) = \frac \Big ( 25 +20\cos (x) +3\cos (4x) \Big ), \\&\Gamma _(x) = - \frac \Big ( -93 +56 \cos (2x) + 28 \cos (4x) +8 \cos (6x) +\cos (8x) \Big ). \end$$

For future reference, observe that

$$\begin \left\| \Gamma _ \right\| _\left[ 0,\frac \right] } \lesssim 1, \quad \forall a \in \. \end$$

(4.1)

Second, we get the following nonlinear system for the error terms \((\Psi ,\Sigma ,B,\Theta )\),

$$\begin&(\omega _ + \epsilon ^ \theta _+\epsilon ^ \eta _)\partial _\Psi \\&\qquad = ( \theta _+\epsilon ^ \eta _) \sin (\tau ) e_^(x) \\&\qquad \quad + \partial _ \Bigg ( \Sigma +\omega _\sin (\tau )e_(x)(A_+\delta _) \Bigg ) \\&\qquad \quad + \epsilon ^ \partial _ \Bigg ( -\Sigma (A_+\delta _) -\omega _ \sin (\tau ) e_(x) ( -B + \Theta + A_\delta _ ) \Bigg ) \\&\qquad \quad + \epsilon ^ \partial _ \Bigg ( \Sigma (-B+\Theta +A_ \delta _) -\omega _\sin (\tau )e_ (-\Theta A_ +B \delta _ ) \Bigg ) \\&\qquad \quad + \epsilon ^ \partial _ \Bigg ( \Sigma (-\Theta A_ +B \delta _) + \omega _\sin (\tau )e_ B \Theta \Bigg ) \\&\qquad \quad - \epsilon ^ \partial _ \Big ( \Sigma \Theta B \Big ), \end$$

$$\begin&(\omega _ + \epsilon ^ \theta _+\epsilon ^ \eta _)\partial _\Sigma \\&\qquad = \omega _ (\theta _+\epsilon ^ \eta _) \cos (\tau ) e_ \\&\qquad \quad + \Bigg ( \frac \left( \Psi - \cos (\tau ) e_^(A_+\delta _) \right) -\cos (\tau ) \left( e_^ (A_+\delta _) \right) ^ + \partial _\Psi \Bigg ) \\&\qquad \quad + \epsilon ^ \Bigg ( \frac \left( -\Psi (A_+\delta _) +\cos (\tau ) e_^ (-B+\Theta + A_ \delta _) \right) \\&\qquad \quad -\left( \Psi (A_+\delta _) \right) ^+ \cos (\tau ) \left( e_^ (-B+\Theta +A_\delta _) \right) ^ \Bigg ) \\&\qquad \quad + \epsilon ^ \Bigg ( \frac \left( \Psi (-B+\Theta +A_\delta _) + \cos (\tau ) e_^ (-A_\Theta + \delta _B) \right) \\&\qquad \quad + \cos (\tau ) \left( e_^(\delta _B-A_\Theta ) \right) ^ + \left( \Psi (-B+\Theta +A_\delta _) \right) ^ \Bigg ) \\&\quad \qquad + \epsilon ^ \Bigg ( \frac \left( \Psi (-\Theta A_ + \delta _B) - \cos (\tau ) e_^ B \Theta \right) \\&\qquad \quad - \cos (\tau ) \left( e_^ B \Theta \right) ^ + \left( \delta _ \Psi B \right) ^ - \left( A_ \Psi \Theta \right) ^ \Bigg ) \\&\qquad \quad - \epsilon ^ \Bigg ( \frac B \Theta \Psi + \left( B \Theta \Psi \right) ^ \Bigg ), \end$$

$$\begin&\partial _ B + \frac B \\&\qquad = \frac \Bigg ( 2 \cos (\tau ) e_^ \Psi - 2 \omega _ \sin (\tau ) e_ \Sigma \\&\qquad \quad d - A_ \cos ^2(\tau ) (e_^)^2 - \omega _^2 A_ \sin ^2(\tau ) e_^2 \Bigg ) \\&\qquad \quad +\frac \epsilon ^2 \Bigg ( \Psi ^2 + \Sigma ^2 - 2 A_ \cos (\tau ) e_^ \Psi + 2 \omega _ A_ \sin (\tau )e_ \Sigma \\&\qquad \quad - \cos ^2(\tau ) (e_^)^2 B - \omega _^2 \sin ^2(\tau ) (e_ )^2 B \Bigg ) \\&\qquad \quad +\frac \epsilon ^4 \Bigg ( -A_ \Psi ^2 -A_ \Sigma ^2 -2 \cos (\tau ) e_^ B \Psi + 2 \omega _ \sin (\tau ) e_ B \Sigma \Bigg ) \\&\qquad \quad -\frac \epsilon ^6 \Big ( B \left( \Psi ^2 + \Sigma ^2 \right) \Big ), \end$$

$$\begin \partial _\Theta&= \frac \Bigg ( 2 \cos (\tau ) e_^ \Psi - 2 \omega _\sin (\tau ) e_ \Sigma - \delta _ \cos ^(\tau ) (e_^)^2 - \omega _^2 \delta _ \sin ^ (\tau ) e_^2 \Bigg ) \\&\quad + \frac \epsilon ^ \Bigg ( \Psi ^2+\Sigma ^2 - 2\delta _ \cos (\tau ) e_^ \Psi + 2 \omega _ \sin (\tau ) e_ \delta _ \Sigma \\&\quad + \cos ^2(\tau ) (e_^)^2 \Theta + \omega _^2 \sin ^2(\tau ) e_^2 \Bigg ) \\&\quad + \frac \epsilon ^ \Bigg ( -\delta _ \Psi ^2 - \delta _ \Sigma ^2 + 2 \cos (\tau ) e_^ \Theta \Psi - 2 \omega _ \sin (\tau ) e_ \Theta \Sigma \Bigg ) \\&\quad +\frac \epsilon ^ \Big ( \Theta \left( \Psi ^2 + \Sigma ^2 \right) \Big ). \end$$

4.1 Definition of the Fourier Constants

As before, we expand the error terms \((\Psi , \Sigma , B, \Theta )\) in terms of the eigenfunctions of the linearized operator as follows

$$\begin \Psi (\tau ,x)&= \sum _^ \psi _(\tau ) \frac^(x)}}, \quad \Sigma (\tau ,x) = \sum _^ \sigma _(\tau ) e_(x), \\ B(\tau ,x)&= \sum _^ b _(\tau ) e_(x),\quad \Theta (\tau ,x) = \sum _^ \xi _(\tau ) e_(x). \end$$

After substituting these expressions into the equations above, we take the inner product \((\cdot | e_^)\) (for the equation for \(\Psi \)) and \((\cdot | e_)\) (for the equations for \(\Sigma ,B\) and \(\Theta \)) in both sides. A long but straightforward computation yields that all the interactions with respect the space variable x are included in the following Fourier constants:

$$\begin \mathbb _&:= \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) e_(x) e_(x) \tan ^2(x) \textrmx, \\ \mathbb _&:= \int _^} \Gamma _(x) \Gamma _(x) e_(x) e_(x) \tan ^2(x) \textrmx, \\ \mathbb _&:= \int _^} \left( \Gamma _(x)\Gamma _(x) - \Gamma _(x)\Gamma _(x) \right) e_(x) e_(x) \tan ^2(x) \textrmx \\ \mathbb _&:= \int _^} e_(x) e_(x) e_(x) \tan ^2(x) \textrmx, \\ \mathbb _&:= \int _^} \Gamma _(x) e_(x) e_(x) e_(x) \tan ^2(x) \textrmx, \\ \mathbb _&:= \int _^} e_(x) e_(x) e_(x) e_(x) \tan ^2(x) \textrmx, \end$$

for the equation for \(\Psi \),

$$\begin \overline}_&:= \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx, \\ \overline}_&:= \int _^} \Gamma _(x) \Gamma _(x) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx ,\\ \overline}_&:= \int _^} \left( \Gamma _(x)\Gamma _(x) - \Gamma _(x)\Gamma _(x) \right) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx ,\\ \overline}_&:= \int _^} e_(x) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx ,\\ \overline}_&:= \int _^} \Gamma _(x) e_(x) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx ,\\ \overline}_&:= \int _^} e_(x) e_(x) \frac^(x)}} \frac^(x)}} \tan ^2(x) \textrmx, \end$$

for the equation for \(\Sigma \),

$$\begin \mathbb _&:=\int _^} \frac^(x)}} \frac^(x)}} \frac^(x)}} \frac \textrmx ,\\ \mathbb _&:=\int _^} e_(x) e_(x) \frac^(x)}} \frac \textrmx ,\\ \mathbb _&:=\int _^} \Gamma _(x) \frac^(x)}} \frac^(x)}} \frac^(x)}} \frac \textrmx ,\\ \mathbb _&:=\int _^} \Gamma _(x) e_(x) e_(x) \frac^(x)}} \frac \textrmx ,\\ \mathbb _&:=\int _^} e_(x) \frac^(x)}} \frac^(x)}} \frac^(x)}} \frac \textrmx ,\\ \mathbb _&:=\int _^} e_(x) e_(x) e_(x) \frac^(x)}} \frac \textrmx, \end$$

for the equation for \(\Theta \), and finally

$$\begin \overline}_&:=\int _^} \frac^(x)}} \frac^(x)}} \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx ,\\ \overline}_&:=\int _^} e_(x) e_(x) \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx ,\\ \overline}_&:=\int _^} \Gamma _ (x) \frac^(x)}} \frac^(x)}} \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx ,\\ \overline}_&:=\int _^} \Gamma _ (x) e_(x) e_(x) \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx ,\\ \overline}_&:=\int _^} e_(x) \frac^ (x)}} \frac^ (x)}} \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx ,\\ \overline}_&:=\int _^} e_(x) e_ (x) e_ (x) \left( \int _^} e_(y) \sin (y) \cos (y) \textrmy \right) \tan ^2(x) \textrmx, \end$$

for the equation for B. In addition, the nonlinear system for the error terms boils down to

$$\begin \frac \psi _(\tau ) - \frac}+\theta _ \epsilon ^2 +\eta _ \epsilon ^4} \sigma _(\tau )&= \frac+\theta _ \epsilon ^2+\eta _ \epsilon ^4} \frac_(\tau ) }}\\&+ \frac+\theta _ \epsilon ^2++\eta _ \epsilon ^4} \frac}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau ))}}, \\ \frac \sigma _(\tau ) + \frac}+\theta _ \epsilon ^2+\eta _ \epsilon ^4} \psi _(\tau )&= \frac+\theta _ \epsilon ^2+\eta _ \epsilon ^4} ^_(\tau ) \\&+ \frac+\theta _ \epsilon ^2+\eta _ \epsilon ^4}^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )), \end$$

subject to the constraints

$$\begin b_ (\tau )&= ^_(\tau ) + ^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) ,\\ \xi _ (\tau )&= ^_(\tau ) + ^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )). \end$$

Here, the source terms are given explicitly in terms of the Fourier constants by

$$\begin \frac_(\tau )}}&= \omega _ \Big ( \frac^2\theta _\delta _}^2} \sin (\tau ) + \omega _ \mathbb _ \sin (\tau ) \cos ^2(\tau ) + \omega _ \mathbb _ \sin ^3(\tau ) \Big ) \\&\quad + \omega _ \epsilon ^2 \Big ( \frac^2\eta _\delta _}^2} \sin (\tau )+ \omega _ \mathbb _\sin (\tau ) \cos ^(\tau ) + \omega _ \mathbb _ \sin ^ (\tau ) \\&\quad + \omega _ \mathbb _ \sin ^3(\tau ) \cos ^2(\tau ) \Big ),\\ ^_(\tau )&= \omega _ \omega _\Big ( \frac\delta _}} \cos (\tau ) + \overline}_ \cos ^3(\tau ) + \overline}_ \cos (\tau ) \sin ^2(\tau ) \Big ) \\&\quad +\omega _ \omega _ \epsilon ^2 \Big ( \frac\delta _}} \cos (\tau )+\overline}_\cos ^(\tau ) + \overline}_ \cos ^(\tau ) \sin ^ (\tau ) \\&\quad + \overline}_ \cos ^3(\tau ) \sin ^2(\tau ) \Big ),\\ ^_(\tau )&= \omega _^2 \Big ( - \cos ^4(\tau ) \overline}_- \sin ^2(\tau ) \cos ^2(\tau ) \overline}_\\&\quad - \sin ^2(\tau ) \cos ^2(\tau ) \overline}_ - \sin ^4(\tau ) \overline}_ \Big ), \\ ^_(\tau )&= \frac^2}} \Big ( \cos ^4(\tau ) \mathbb _ + \cos ^2(\tau ) \sin ^2(\tau ) \mathbb _ \\&\quad + \sin ^4(\tau ) \mathbb _ + \cos ^2(\tau ) \sin ^2(\tau ) \mathbb _ \Big ), \end$$

where \(\delta _\) stands for the Kronecker’s delta, whereas the linear and nonlinear terms are given by

$$\begin&^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) = \epsilon ^2 \Bigg ( -\cos ^2(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau ) \\&\quad -\sin ^2(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau ) \\&\quad -\omega _ \sin (\tau ) \sum _^ \omega _^2 \mathbb _ ( \xi _(\tau )-b_(\tau )) \Bigg ) \\&\quad + \epsilon ^4 \Bigg ( -\cos ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau ) - \cos ^(\tau ) \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )\\&\quad + \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )+ \omega _ \sin (\tau ) \cos ^(\tau ) \sum _^ \omega _^2 \mathbb _ \xi _(\tau ) \\&\quad + \omega _ \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ \xi _(\tau ) + \omega _ \sin (\tau ) \cos ^(\tau ) \sum _^ \omega _^2 \mathbb _ b_(\tau ) \\&\quad + \omega _ \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ b_(\tau ) + \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )(-b_(\tau )+\xi _(\tau )) \Bigg ) \\&\quad + \epsilon ^6 \Bigg ( \omega _ \sin (\tau ) \sum _^ \omega _^2 \mathbb _ b_(\tau ) \xi _(\tau )\\&\quad - \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ \xi _(\tau )\sigma _(\tau ) - \sin ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )b_(\tau ) \\&\quad - \cos ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )\xi _(\tau ) - \cos ^(\tau ) \sum _^ \omega _^2 \mathbb _ \sigma _(\tau )b_(\tau ) \Bigg ) \\&\quad + \epsilon ^8 \Bigg ( - \sum _^ \omega _^2 \mathbb _ \xi _(\tau ) \sigma _(\tau ) b_(\tau ) \Bigg ), \end$$

$$\begin&^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) = \epsilon ^2 \Bigg ( \cos ^2(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) \\&\quad +\sin ^2(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau )- \cos (\tau ) \sum _^ \omega _ \omega _ \overline}_ (\xi _(\tau )-b_(\tau )) \Bigg ) \\&\quad + \epsilon ^4 \Bigg ( \cos ^(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) + \cos ^(\tau ) \sin ^(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) \\&\quad + \sin ^(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) + \sin ^(\tau ) \cos (\tau ) \sum _^ \omega _ \omega _\overline}_ \xi _(\tau ) \\&\quad + \cos ^(\tau ) \sum _^ \omega _ \omega _\overline}_ \xi _(\tau ) \\&\quad + \cos ^(\tau ) \sum _^ \omega _ \omega _ \overline}_ b_(\tau ) \\&\quad + \cos (\tau ) \sin ^(\tau ) \sum _^\omega _ \omega _ \overline}_ b_(\tau ) + \sum _^ \omega _ \overline}_ \psi _(\tau )(b_(\tau )-\xi _(\tau )) \Bigg ) \\&\quad + \epsilon ^6 \Bigg ( \cos (\tau ) \sum _^ \omega _ \omega _ \overline}_ b_(\tau ) \xi _(\tau ) + \cos ^(\tau ) \sum _^ \omega _\overline}_ \xi _(\tau ) \psi _(\tau ) \\&\quad + \cos ^(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) b_(\tau ) \\&\quad + \sin ^(\tau ) \sum _^ \omega _\overline}_ \psi _(\tau ) \xi _(\tau ) + \sin ^(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) b_(\tau ) \Bigg ) \\&\quad + \epsilon ^8 \Bigg ( \sum _^ \omega _ \overline}_ \xi _(\tau ) \psi _(\tau ) b_(\tau ) \Bigg ), \end$$

$$\begin&^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) = \Bigg ( 2 \cos (\tau ) \sum _^ \omega _\overline}_ \psi _(\tau ) - 2 \omega _ \sin (\tau ) \sum _^ \overline}_ \sigma _(\tau ) \Bigg ) \\&\quad + \epsilon ^2 \Bigg ( - 2 \cos ^3(\tau ) \sum _^ \omega _\overline}_ \psi _(\tau ) - 2 \cos (\tau ) \sin ^2(\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) \\&\quad + 2 \omega _ \cos ^2(\tau ) \sin (\tau ) \sum _^ \overline}_ \sigma _(\tau ) + 2 \omega _ \sin ^3(\tau ) \sum _^ \overline}_ \sigma _(\tau )\\&\quad - \cos ^2(\tau ) \sum _^ \omega _^2 \overline}_ b_(\tau ) \\&\quad - \sin ^2(\tau ) \sum _^ \omega _^2 \overline}_ b_(\tau ) + \sum _^ \overline}_ \psi _(\tau ) \psi _(\tau ) + \sum _^ \overline}_ \sigma _(\tau )\sigma _(\tau ) \Bigg ) \\&\quad + \epsilon ^4 \Bigg ( -\cos ^2(\tau ) \sum _^ \overline}_ \psi _(\tau ) \psi _(\tau ) -\sin ^2(\tau ) \sum _^ \overline}_ \psi _(\tau ) \psi _(\tau ) \\&\quad - 2 \cos (\tau ) \sum _^ \omega _ \overline}_ \psi _(\tau ) b_(\tau ) -\cos ^2(\tau ) \sum _^ \overline}_ \sigma _(\tau ) \sigma _(\tau ) \\&\quad -\sin ^2(\tau ) \sum _^ \overline}_ \sigma _(\tau ) \sigma _(\tau ) +2 \sin (\tau ) \sum _^\omega _ \overline}_ \sigma _(\tau ) b_(\tau ) \Bigg ) \\&\quad + \epsilon ^6 \Bigg ( - \sum _^ \overline}_ \psi _(\tau ) \psi _(\tau ) b_ (\tau ) - \sum _^ \overline}_ \sigma _(\tau ) \sigma _(\tau ) b_ (\tau ) \Bigg ), \end$$

$$\begin&^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) = \Bigg ( 2 \cos (\tau ) \sum _^ \omega _ \frac_}} \psi _(\tau ) - 2 \omega _ \sin (\tau ) \sum _^ \frac_}} \sigma _(\tau ) \Bigg ) \\&\quad + \epsilon ^2 \Bigg ( 2 \cos ^3(\tau ) \sum _^ \omega _ \frac_}} \psi _(\tau ) + 2 \cos (\tau ) \sin ^2(\tau ) \sum _^ \omega _ \frac_}} \psi _(\tau ) \\&\quad -2 \omega _ \sin ^3(\tau ) \sum _^ \frac_}} \sigma _(\tau ) \\&\quad - 2 \omega _ \cos ^2(\tau )\sin (\tau ) \sum _^ \frac_}} \sigma _(\tau ) + \cos ^2(\tau ) \sum _^ \omega _^2 \frac_}} \xi _(\tau ) \\&\quad + \omega _^2 \sin ^2(\tau ) \sum _^ \omega _\frac_}^2} \xi _(\tau ) \\&\quad + \sum _^ \frac_}} \psi _(\tau ) \psi _(\tau ) + \sum _^ \frac_}} \sigma _(\tau )\sigma _(\tau ) \Bigg ) \\&\quad + \epsilon ^4 \Bigg ( \cos ^2(\tau ) \sum _^ \frac_}} \psi _(\tau ) \psi _(\tau ) + \sin ^2(\tau ) \sum _^ \frac_}} \psi _(\tau ) \psi _(\tau ) \\&\quad + 2 \cos (\tau ) \sum _^ \omega _ \frac_}} \psi _(\tau ) \xi _(\tau ) +\cos ^2(\tau ) \sum _^ \frac_}} \sigma _(\tau ) \sigma _(\tau ) \\&\quad + \sin ^2(\tau ) \sum _^ \frac_}} \sigma _(\tau ) \sigma _(\tau ) - 2 \omega _ \sin (\tau ) \sum _^ \frac_}} \sigma _(\tau ) \xi (\tau ) \Bigg ) \\&\quad + \epsilon ^6 \Bigg ( - \sum _^ \frac_}} \psi _(\tau )\psi _(\tau ) \xi _ (\tau ) + \sum _^ \frac_}} \sigma _(\tau ) \sigma _(\tau ) \xi _ (\tau ) \Bigg ). \end$$

4.2 Approximate Periodic Solution and Small Divisors

Similarly to the first approach with the infinite sum, the linear and homogeneous part of the ordinary differential equation for \((\psi _,\sigma _)\) is simply the equation for the harmonic oscillator and hence we can use the variational constants formula to solve it. We find the fixed-point formulation

$$\begin \psi _(\tau ) =&^_(\tau ) + \int _^ \Bigg ( \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \frac}_(\psi (s),\sigma (s),b(s),\xi (s))}} \\&+ \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} ^}_(\psi (s),\sigma (s),b(s),\xi (s)) \Bigg ) ds, \\ \sigma _(\tau ) =&^_(\tau ) + \int _^ \Bigg ( \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \frac}_(\psi (s),\sigma (s),b(s),\xi (s))}} \\&+ \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} ^}_(\psi (s),\sigma (s),b(s),\xi (s)) \Bigg ) ds, \end$$

subject to the constrain equations

$$\begin b_ (\tau )&= ^_(\tau ) + ^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) ,\\ \xi _ (\tau )&= ^_(\tau ) + ^}_(\psi (\tau ),\sigma (\tau ),b(\tau ),\xi (\tau )) \end$$

where

$$\begin&^_(\tau ) = \cos \left( \frac \tau } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) \psi _(0) + \sin \left( \frac \tau } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) \sigma _(0) \\&\quad + \int _^ \Bigg ( \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \frac_(s)}} + \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} ^_(s) \Bigg ) ds, \\&^_(\tau ) = -\sin \left( \frac \tau } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) \psi _(0) + \cos \left( \frac \tau } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) \sigma _(0) \\&\quad + \int _^ \Bigg ( \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} \frac_(s)}} + \frac (\tau -s)} +\theta _\epsilon ^2+ \eta _\epsilon ^4} \right) } +\theta _\epsilon ^2+ \eta _\epsilon ^4} ^_(s) \Bigg ) ds. \end$$

Furthermore, one can use various trigonometric identities to write

$$\begin ^_(s)&= K_(\epsilon ^) \sin (s) + \Lambda _(\epsilon ^) \sin (3s) + M_(\epsilon ^) \sin (5s), \\ ^_(s)&= N_(\epsilon ^) \cos (s) + \Xi _(\epsilon ^) \cos (3s) + T_(\epsilon ^) \cos (5s) \end$$

where

$$\begin K_(\epsilon ^)&:= \omega _ \omega _^2 \Bigg (\frac \omega _ \theta _}^2} + \frac \mathbb _ + \frac\mathbb _ \\&\quad + \epsilon ^2 \left( \frac \omega _ \eta _}^2}+ \frac \mathbb _ + \frac \mathbb _ + \frac \mathbb _ \right) \Bigg ), \\ \Lambda _(\epsilon ^)&:= \omega _\omega _^2 \Bigg ( \frac \mathbb _ -\frac \mathbb _ + \epsilon ^2 \left( \frac \mathbb _ -\frac \mathbb _ +\frac \mathbb _ \right) \Bigg ), \\ M_(\epsilon ^)&:= \omega _\omega _^2 \epsilon ^2 \Bigg ( \frac \mathbb _ +\frac \mathbb _ - \frac \mathbb _ \Bigg ), \\ N_(\epsilon ^)&:= \omega _ \omega _ \left( \frac \theta _}} + \frac \overline}_ + \frac \overline}_\right. \\&\quad \left. + \epsilon ^2 \left( \frac \eta _}}+ \frac \overline}_ + \frac \overline}_ + \frac \overline}_ \right) \right) , \\ \Xi _(\epsilon ^)&:= \omega _ \omega _ \left( \frac \overline}_ -\frac \overline}_ + \epsilon ^2 \left( \frac \overline}_ + \frac \overline}_ +\frac \overline}_ \right) \right) , \\ T_(\epsilon ^)&:=\omega _ \omega _ \epsilon ^2 \Big ( \frac \overline}_ +\frac \overline}_ - \frac \overline}_ \Big ). \end$$

Now, computing the integrals above we obtain

$$\begin ^_(\tau )&= ^}_(\epsilon ^) \cos (\tau ) + ^}_(\epsilon ^) \cos (3 \tau ) + ^}_(\epsilon ^) \cos (5\tau ) \\&\quad + \Big ( \psi _(0) + ^}_(\epsilon ^) \Big ) \cos \left( \frac \tau } +\theta _\epsilon ^2+\eta _\epsilon ^4} \right) \\&\quad + \sigma _(0) \sin \left( \frac \tau } +\theta _\epsilon ^2+\eta _\epsilon ^4} \right) , \\ ^_(\tau )&= ^}_(\epsilon ^) \sin (\tau ) + ^}_(\epsilon ^) \sin (3 \tau ) + ^}_(\epsilon ^) \sin (5\tau ) \\&\quad + \sigma _(0) \cos \left( \frac \tau } +\theta _\epsilon ^2+\eta _\epsilon ^4} \right) \\&\quad - \Big ( \psi _(0) - ^}_(\epsilon ^) \Big ) \sin \left( \frac \tau } +\theta _\epsilon ^2+\eta _\epsilon ^4} \right) , \end$$

where

$$\begin ^}_(\epsilon ^)&= \frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) K_(\epsilon ^2) + \omega _^2 N_(\epsilon ^2) }-\omega _-\epsilon ^2 \theta _ -\epsilon ^4 \eta _) (\omega _+\omega _+\epsilon ^2 \theta _+\epsilon ^4 \eta _ ) }, \\ ^}_(\epsilon ^)&= \frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) \Lambda _(\epsilon ^2) + \omega _^2 \Xi _(\epsilon ^2) }-3\omega _-3\epsilon ^2 \theta _-3\epsilon ^4 \eta _ ) (\omega _+3\omega _+3\epsilon ^2 \theta _ +3\epsilon ^4 \eta _) }, \\ ^}_(\epsilon ^)&= \frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) M_(\epsilon ^2) + \omega _^2 T_(\epsilon ^2) }-5\omega _-5\epsilon ^2 \theta _ -5\epsilon ^4 \eta _) (\omega _+5\omega _+5\epsilon ^2 \theta _ +5\epsilon ^4 \eta _) }, \\ ^}_(\epsilon ^)&= -\frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) K_(\epsilon ^2) + \omega _^2 N_(\epsilon ^2) }-\omega _-\epsilon ^2 \theta _ -\epsilon ^4 \eta _) (\omega _+\omega _+\epsilon ^2 \theta _ +\epsilon ^4 \eta _) } \\&\quad -\frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) \Lambda _(\epsilon ^2) + \omega _^2 \Xi _(\epsilon ^2) }-3\omega _-3\epsilon ^2 \theta _-3\epsilon ^4 \eta _ ) (\omega _+3\omega _+3\epsilon ^2 \theta _+3\epsilon ^4 \eta _ ) } \\&\quad -\frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) M_(\epsilon ^2) + \omega _^2 T_(\epsilon ^2) }-5\omega _-5\epsilon ^2 \theta _-5\epsilon ^4 \eta _ ) (\omega _+5\omega _+5\epsilon ^2 \theta _+5\epsilon ^4 \eta _ ) } \end$$

and

$$\begin ^}_(\epsilon ^)&= - \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) N_(\epsilon ^2) + K_(\epsilon ^2) }-\omega _-\epsilon ^2 \theta _ -\epsilon ^4 \eta _) (\omega _+\omega _+\epsilon ^2 \theta _ +\epsilon ^4 \eta _) }, \\ ^}_(\epsilon ^)&= - \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) \Xi _(\epsilon ^2) + \Lambda _(\epsilon ^2) }-3\omega _-3\epsilon ^2 \theta _ -3\epsilon ^4 \eta _) (\omega _+3\omega _+3\epsilon ^2 \theta _+3\epsilon ^4 \eta _ ) }, \\ ^}_(\epsilon ^)&= - \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) T_(\epsilon ^2) + M_(\epsilon ^2) }-5\omega _-5\epsilon ^2 \theta _ -5\epsilon ^4 \eta _) (\omega _+5\omega _+5\epsilon ^2 \theta _+5\epsilon ^4 \eta _ ) }, \\ ^}_(\epsilon ^)&= \frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) K_(\epsilon ^2) + \omega _^2 N_(\epsilon ^2) }-\omega _-\epsilon ^2 \theta _ -\epsilon ^4 \eta _) (\omega _+\omega _+\epsilon ^2 \theta _+\epsilon ^4 \eta _ ) } \\&\quad + \frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) \Lambda _(\epsilon ^2) + \omega _^2 \Xi _(\epsilon ^2) }-3\omega _-3\epsilon ^2 \theta _-3\epsilon ^4 \eta _ ) (\omega _+3\omega _+3\epsilon ^2 \theta _ +3\epsilon ^4 \eta _) } \\&\quad +\frac} \frac+\epsilon ^2 \theta _+\epsilon ^4 \eta _) M_(\epsilon ^2) + \omega _^2 T_(\epsilon ^2) }-5\omega _-5\epsilon ^2 \theta _-5\epsilon ^4 \eta _ ) (\omega _+5\omega _+5\epsilon ^2 \theta _ +5\epsilon ^4 \eta _) }. \end$$

Notice that conditions like

$$\begin&\omega _\pm \omega _ \pm \epsilon ^2 \theta _ \pm \epsilon ^4 \eta _ \ne 0, \quad \omega _\pm 3\omega _ \pm 3\epsilon ^2 \theta _ \pm 3\epsilon ^4 \eta _ \ne 0,\\&\omega _\pm 5\omega _ \pm 5\epsilon ^2 \theta _ \pm 5\epsilon ^4 \eta _ \ne 0, \end$$

are closely related to small divisors and play an important role in KAM theory [3,4,5,6,7,8]. Observe that the identity

$$\begin ^}_(\epsilon ^) +^}_(\epsilon ^) = 0 \end$$

holds true for all \(i=0,1,\dots \) and all \(\epsilon >0\). Here, all the constants involved

$$\begin ^}_(\epsilon ^),^}_(\epsilon ^),^}_(\epsilon ^),^}_(\epsilon ^), ^}_(\epsilon ^),^}_(\epsilon ^),^}_(\epsilon ^),^}_(\epsilon ^) \end$$

depend explicitly on the Fourier constants defined above and most importantly depend only on one index. Due to this fact, we can compute them (see Lemma B.1 in Appendix B). The fact that these constants are given in closed forms has a numerous advantages. First, we get the asymptotic behaviour for large i and fixed \(\epsilon >0\),

$$\begin&^}_(\epsilon ^) \simeq \frac^5},^}_(\epsilon ^) \simeq \frac^5},^}_(\epsilon ^)\simeq \frac^5},^}_(\epsilon ^)\simeq \frac^5}, \text i \longrightarrow \infty \\&^}_(\epsilon ^) \simeq \frac^6},^}_(\epsilon ^) \simeq \frac^6},^}_(\epsilon ^)\simeq \frac^6},^}_(\epsilon ^)\simeq \frac^5}, \text i \longrightarrow \infty . \end$$

Second, we get their asymptotic behaviour for sufficiently small \(\epsilon \) and fixed \(i=0,1,\dots \),

$$\begin ^}_(\epsilon ^)&= \left\ \left( -3+\frac} \right) \epsilon ^ + (...)1+(...)\epsilon ^2, & \text i = 0 \\ (...)1+(...)\epsilon ^2, & \text i \ne 0 \end\right. , \\ ^}_(\epsilon ^)&= \left\ \frac}} \left( \omega _ \mathbb _-\omega _ \mathbb _+3 \omega _ \overline}_-3 \omega _ \overline}_ \right) \epsilon ^\\ + (...)1+(...)\epsilon ^2, & \text i = 3 \\ (...)1+(...)\epsilon ^2, & \text i \ne 3 \end\right. ,\\ ^}_(\epsilon ^)&= \left\ (...)1+(...)\epsilon ^2, & \text i = 6 \\ (...)\epsilon ^2, & \text i \ne 6 \end\right. , \\ ^}_(\epsilon ^)&= \left\ \left( -3+\frac} \right) \epsilon ^ + (...)1+(...)\epsilon ^2, & \text i = 0 \\ \frac}} \left( \omega _ \mathbb _-\omega _ \mathbb _+3 \omega _ \overline}_-3 \omega _ \overline}_ \right) \epsilon ^\\ +(...)1+(...)\epsilon ^2, & \text i = 3 \\ (...)1+(...)\epsilon ^2, & \text i = 6 \\ (...)1+(...)\epsilon ^2, & \text i \ne 0,3,6 \end\right. \end$$

$$\begin ^}_(\epsilon ^)&= \left\ \left( -3+\frac} \right) \epsilon ^ + (...)1+(...)\epsilon ^2, & \text i = 0 \\ (...)1+(...)\epsilon ^2, & \text i \ne 0 \end\right. , \\ ^}_(\epsilon ^)&= \left\ \frac}} \left( \omega _ \mathbb _-\omega _ \mathbb _+3 \omega _ \overline}_-3 \omega _ \overline}_ \right) \epsilon ^\\ + (...)1+(...)\epsilon ^2, & \text i = 3 \\ (...)1+(...)\epsilon ^2, & \text i \ne 3 \end\right. ,\\ ^}_(\epsilon ^)&= \left\ (...)1+(...)\epsilon ^2, & \text i = 6 \\ (...)\epsilon ^2, & \text i \ne 6 \end\right. , \\ ^}_(\epsilon ^)&= \left\ \left( -3+\frac} \right) \epsilon ^ + (...)1+(...)\epsilon ^2, & \text i = 0 \\ \frac}} \left( \omega _ \mathbb _-\omega _ \mathbb _+3 \omega _ \overline}_-3 \omega _ \overline}_ \right) \\ \epsilon ^ +(...)1+(...)\epsilon ^2, & \text i = 3 \\ (...)1+(...)\epsilon ^2, & \text i = 6 \\ (...)1+(...)\epsilon ^2, & \text i \ne 0,3,6 \end\right. \end$$

We compute

$$\begin&\mathbb _ = - \frac\pi }, ~\mathbb _ = - \frac\pi },\\&\overline}_ = - \frac\pi }, ~\overline}_ = - \frac\pi } \end$$

and hence

$$\begin \omega _ \mathbb _-\omega _ \mathbb _+3 \omega _ \overline}_-3 \omega _ \overline}_ = 0. \end$$

Consequently, by the structure of the equations, \(^}_,^}_,^}_\) and \(^}_\) cannot blow up as \(\epsilon \) does to zero. However, we choose

$$\begin \theta _:= \frac \end$$

to ensure that every component of the periodic parts \(^_(\tau )\) and \(^_(\tau )\) of \(\psi _\) and \(\sigma _\), respectively, is bounded as \(\epsilon \) goes to zero. This choice coincides with the choice of \(\theta _\) from the first approach as well as with the numerical computations of Rostworowski-Maliborski [30]. Similarly, using various trigonometric identities, we get

$$\begin ^_(\tau )&= ^}_ + ^}_ \cos (2 \tau )+ ^}_\cos (4 \tau ) , \\ ^_(\tau )&= ^}_ + ^}_ \cos (2 \tau )+ ^}_\cos (4 \tau ), \end$$

where

$$\begin ^}_&=- \frac^2} \Big ( 3 \overline}_ + \overline}_ + \overline}_ + 3 \overline}_ \Big ),\\ ^}_&= \frac^2 } \Big ( -\overline}_ + \overline}_ \Big ),\\ ^}_&= \frac^2} \Big (- \overline}_ + \overline}_ + \overline}_ - \overline}_\Big ),\\ ^}_&= \frac^2}^2} \Big ( 3 \mathbb _ + \mathbb _ + \mathbb _ + 3 \mathbb _ \Big ),\\ ^}_&=\frac^2}^2} \Big ( \mathbb _ -\mathbb _ \Big ),\\ ^}_&= \frac^2}^2} \Big ( \mathbb _ - \mathbb _ - \mathbb _ + \mathbb _ \Big ). \end$$

As before, all the constants involved

$$\begin ^}_,^}_,^}_, ^}_,^}_,^}_ \end$$

depend explicitly on the Fourier constants defined above and most importantly depend only on one index. Due to this fact, we can compute them (see Lemma B.1 in Appendix B). We immediately get their asymptotic behaviour for large i,

$$\begin&^}_\simeq \frac^},^}_\simeq \frac^},^}_ \simeq \frac^}, \text i \longrightarrow \infty \\&^}_ \simeq \frac^},^}_\simeq \frac^},^}_ \simeq \frac^}, \text i \longrightarrow \infty . \end$$

4.3 Choice of the Initial Data

We choose

$$\begin \psi _(0):= - ^}_(\epsilon ^) = ^}_(\epsilon ^), \quad \sigma _(0):=0 \end$$

so that

$$\begin ^_(\tau )&= ^}_(\epsilon ^) \cos (\tau ) + ^}_(\epsilon ^) \cos (3 \tau ) + ^}_(\epsilon ^) \cos (5\tau ), \\ ^_(\tau )&= ^}_(\epsilon ^) \sin (\tau ) + ^}_(\epsilon ^) \sin (3 \tau ) + ^}_(\epsilon ^) \sin (5\tau ). \end$$

This choice is motivated by the fact that the source terms \(^_(\tau )\) and \(^_(\tau )\) of the solutions \(\psi _(\tau )\) and \(\sigma _(\tau )\) would give rise to a periodic term. Indeed,

and similarly for \(\Sigma , B\) and \(\Theta \).

4.4 Growth and Decay of the Fourier Constants

We are interested in the asymptotic behaviour of all the Fourier constants which appear using this approach. To begin with, we split them into five groups as follows

$$\begin \mathcal _&:= \Bigg \\mathbb _, \omega _\mathbb _, \omega _\mathbb _,\omega _\mathbb _, \omega _\mathbb _, \omega _\mathbb }_,\\&\quad \omega _\mathbb }_, \omega _\mathbb }_, \omega _\mathbb }_, \omega _\mathbb }_, \omega _\mathbb _, \omega _\mathbb _,\omega _\mathbb _, \omega _\mathbb _,\omega _\mathbb _,\\&\quad \omega _\mathbb _,\omega _\mathbb _, \omega _\mathbb _,\omega _\mathbb _, \frac \mathbb _ }}, \frac \mathbb _ }}, \frac \mathbb _ }},\\&\quad \frac \mathbb _}}, \frac \mathbb _}},\omega _ \mathbb }_, \omega _ \mathbb }_, \omega _ \mathbb }_,\omega _ \mathbb }_, \omega _ \mathbb }_ \Bigg \}, \\ \mathcal _&:= \Bigg \ \mathbb _,\omega _ \mathbb _,\omega _ \mathbb _,\omega _ \mathbb _,\frac \mathbb _}},\\&\quad \omega _ \mathbb }_,\omega _ \mathbb }_,\omega _ \mathbb }_,\omega _ \mathbb }_,\omega _ \mathbb }_ \Bigg \}, \\ \mathcal _&:= \Bigg \_}},\frac_}},\frac_}},\frac_}},\frac_}}, \frac^2 \mathbb _}}, \frac_}}, \omega _ \omega _\mathbb }_,\omega _ \omega _\mathbb }_,\\&\quad \omega _ \omega _\mathbb }_,\omega _ \omega _\mathbb }_, \\&\quad \omega _ \omega _\mathbb }_, \mathbb }_,\mathbb }_,\mathbb }_,\mathbb }_,\mathbb }_,\omega _^2\mathbb }_,\mathbb }_ \Bigg \}, \\ \mathcal _&:= \Bigg \_}},\frac_}}, \frac_}},\frac_}},\frac_}},\frac_}}, \frac_}},\frac_}},\frac_}},\frac_}}, \frac_}} \\&\quad \overline}_, \overline}_,\overline}_,\overline}_,\overline}_, \overline}_,\\&\quad \overline}_,\overline}_, \overline}_, \overline}_, \overline}_, \omega _ \omega _\overline}_ \Bigg \}, \\ \mathcal &:= \Bigg \ \mathbb _, \omega _\mathbb _,\frac_}},\frac_} }, \omega _\mathbb }_, \omega _\mathbb _, \omega _\mathbb }_, \mathbb }_, \overline }_ \Bigg \}. \end$$

As before, we shall use the notation

$$\begin \sum _ f(a\pm b \pm c) = f(a + b + c) + f(a + b - c) + f(a - b + c) + f(a - b - c), \end$$

that is summation with respect to all possible combinations of plus and minus and expressions like \(\omega _ \pm \omega _ \pm \omega _\) stand not only for \(\omega _ + \omega _ + \omega _\) and \(\omega _ - \omega _ - \omega _\) but also for \(\omega _ + \omega _ - \omega _\) and \(\omega _ - \omega _ + \omega _\), that is considering all possible combinations of plus and minus. We will use the leading-order terms (Remark 2.3) together with the asymptotic behaviour of the oscillatory integrals (Lemma 2.6), the orthogonality properties (Lemma 2.2), the \(L^-\)bounds for quantities related to the eigenfunctions (Lemma 2.4) and the \(L^-\)bounds of the weights \(\Gamma _\) (estimate (4.1)).

4.4.1 Fourier Constants in \(\mathcal _\), \(\mathcal _\), \(\mathcal _\) and \(\mathcal _\)

First, we focus on the elements of \(\mathcal _\).

Proposition 4.1

(Fourier constants in \(\mathcal _\)). The following growth and decay estimates hold.

Growth and decay estimates for the Fourier constants in \(\mathcal _\) as \(i,j \longrightarrow + \infty \)

Constant

F

1st derivative \(\ne 0\)

\( \omega _ - \omega _ \longrightarrow \infty \)

  

\(\ne 0\)

  

\(\displaystyle \omega _ \mathbb _, \omega _\mathbb }_ \)

\(\displaystyle \Gamma _-\Gamma _ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _)^4} \right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_\)

\(\displaystyle \Gamma _-\Gamma _ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _)^4}\right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_\)

\(\displaystyle \Gamma _ \Gamma _ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _)^4} \right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_\)

\(\displaystyle \Gamma _ \Gamma _ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _)^4}\right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_\)

\(\displaystyle \Gamma _ \Gamma _+ \Gamma _ \Gamma _ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _)^4}\right) \)

\(\displaystyle \omega _\mathbb _\)

\(\displaystyle e_ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _ \right) \)

\(\displaystyle \sum _\mathcal \left( \frac } \pm \omega _)^4}\right) \)

\(\displaystyle \omega _\mathbb _\)

\(\displaystyle \Gamma _ e_ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _\mathcal \left( \frac } \pm \omega _)^} \right) \)

\(\displaystyle \omega _\mathbb _\)

\(\displaystyle \Gamma _ e_ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _\mathcal \left( \frac } \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb _\)

\(\displaystyle \Gamma _ e_ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _\mathcal \left( \frac } \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb _\)

\(\displaystyle \Gamma _ e_ \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _\right) \)

\(\displaystyle \sum _\mathcal \left( \frac } \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_, \frac\mathbb _}} \)

\(\displaystyle \Gamma _ e_^\sin \cos \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _^ \right) \)

\(\displaystyle \frac} \sum _\mathcal \left( \frac \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_,\frac\mathbb _}} \)

\(\displaystyle \Gamma _ e_^\sin \cos \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _^ \right) \)

\(\displaystyle \frac} \sum _\mathcal \left( \frac \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_,\frac\mathbb _}} \)

\(\displaystyle \Gamma _ e_^\sin \cos \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _^ \right) \)

\(\displaystyle \frac} \sum _\mathcal \left( \frac \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_,\frac\mathbb _}} \)

\(\displaystyle \Gamma _ e_^\sin \cos \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _^ \right) \)

\(\displaystyle \frac} \sum _\mathcal \left( \frac \pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_, \frac\mathbb _}}\)

\(\displaystyle e_^\sin \cos \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _^ \right) \)

\(\displaystyle \frac} \sum _\mathcal \left( \frac \pm \omega _)^} \right) \)

Proof

All these estimates follow directly from Lemma 2.9 and in particular from

$$\begin \int _^} F(x) \cos ( 2 b x ) \textrmx&= \sum _^ \frac}} \left( (-1)^ F^ \left( \frac \right) -F^ \left( 0 \right) \right) \\&\quad + \mathcal \left( \frac} \right) , \end$$

as \(b \longrightarrow \infty \). However, we illustrate the proof only for the first constant, namely \(\omega _\mathbb _\). For large values of i, j and in the case where both \(\omega _ \pm \omega _ \longrightarrow \infty \) (equivalently when \(\omega _ - \omega _ \longrightarrow \infty \)),

$$\begin \mathbb _&:= \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) e_(x)e_(x)\tan ^2(x) \textrmx\\&\simeq \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) \sin (\omega _x) \sin (\omega _x) \textrmx \\&= \frac \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) \cos ((\omega _-\omega _)x) \textrmx \\&\quad - \frac \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) \cos ((\omega _+\omega _)x) \textrmx. \end$$

Observe that both \(\omega _ + \omega _\) and \(\omega _ - \omega _\) are even,

$$\begin \omega _ + \omega _ = 2(3 +i+j), \quad \omega _ - \omega _ = 2(i-j). \end$$

We define \(F(x):=\Gamma _(x) - \Gamma _(x)\). If \(\omega _ - \omega _ \longrightarrow \infty \), then Lemma 2.9 applies and since

$$\begin F^ \left( \frac\right) = F^ \left( 0 \right) =0,~~~F^ \left( \frac \right) = -54 \ne 0 \end$$

we infer

$$\begin \mathbb _=\mathcal \left( \frac - \omega _)^4} + \frac+ \omega _)^4} \right) , \end$$

as \(i,j \longrightarrow \infty \). On the other hand, if , then we see that

$$\begin \left| \mathbb _ \right|&:=\left| \int _^} \left( \Gamma _(x) - \Gamma _(x) \right) e_(x)e_(x)\tan ^2(x) \textrmx \right| \\&\le \left\| \Gamma _ - \Gamma _ \right\| _\left[ 0,\frac\right] } \left\| e_\tan \right\| _\left[ 0,\frac\right] } \left\| e_\tan \right\| _\left[ 0,\frac\right] } \lesssim 1. \end$$

In conclusion,

as \(i,j \longrightarrow \infty \). \(\square \)

Second, we focus on the elements of \(\mathcal _\).

Proposition 4.2

(Fourier constants in \(\mathcal _\)). Let \(N\in \mathbb \). The following growth and decay estimates hold.

Growth and decay estimates for the Fourier constants in \(\mathcal _\) as \(i,j,k \longrightarrow + \infty \)

Constant

F

1st derivative \(\ne 0\)

\(\forall ~\omega _ \pm \omega _ \pm \omega _ \longrightarrow \infty \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_ \)

\(\displaystyle \frac} \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _ \omega _ \right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _\pm \omega _)^} \right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_ \)

\(\displaystyle \frac} \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _ \omega _ \right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _\pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_ \)

\(\displaystyle \frac} \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _ \omega _ \right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _\pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb _, \omega _\mathbb }_ \)

\(\displaystyle \frac} \)

\(\displaystyle F^ \left( \frac \right) \ne 0 \)

\(\mathcal \left( \omega _ \omega _ \right) \)

\(\displaystyle \sum _ \mathcal \left( \frac } \pm \omega _\pm \omega _)^}\right) \)

\(\displaystyle \omega _\mathbb }_, \frac\mathbb _}} \)

\(\displaystyle e_^ \cos ^2 \)

None