Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations

Abstract

We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant M and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon ^{M+1}$. This implies that for such initial data u(0) we control the Sobolev norm of the solution u(t) for time of order $\varepsilon ^{-M}$. Furthermore this property is locally stable: if v(0) is sufficiently close to u(0) (of order $\varepsilon ^{3/2}$) then the solution v(t) is also controled for time of order $\varepsilon ^{-M}$.

Appendices

Appendix A. The case of (NLSP)

As explain in Section 2.2, the main difference between (NLS) and (NLSP) appears when we calculate $Z_4$. Indeed, the resonant normal form procedure used in Section 7.1 leads, in the (NLSP) case, to the following formula (see (19) with ${\hat{V}}_a = a^2$, $a \ne 0$ and ${\hat{V}}_0 = 0$)

$$\begin{aligned} Z_4 = \varphi '(0) \sum _{a\ne b \in {\mathbb {Z}}} \frac{1}{(a - b)^2} I_a I_b. \end{aligned}$$

Thus the frequencies associated with this integrable Hamiltonian are

$$\begin{aligned} \lambda _a (I) = \frac{\partial Z_4}{\partial I_a} = 2\varphi '(0) \sum _{b\ne a \in {\mathbb {Z}}} \frac{1}{(a - b)^2} I_b. \end{aligned}$$

For these frequencies we obtain a much better control of the small denominators that the one obtained for (NLS), in particular, contrary to the (NLS) case (see (21)), the loss of derivative is independent of s.

For ${{\varvec{j}}}= (j_1,\ldots ,j_{2m}) \in {\mathbb {U}}_2\times {\mathbb {Z}}$, if $j_\alpha = (\delta _\alpha ,a_\alpha )$ for $\alpha = 1,\ldots ,2m$, the small denominators in the (NLSP) case are given by

$$\begin{aligned} \omega _{{{\varvec{j}}}}(I)= \sum _{\alpha = 1}^{2m} \delta _\alpha \frac{\partial Z_4}{\partial I_{a_\alpha }} (I) = 2 \varphi '(0) \sum _{\alpha = 1}^{2m} \delta _\alpha \sum _{b \ne a_\alpha \in {\mathbb {Z}}} \frac{1}{(a_\alpha - b)^2} I_b. \end{aligned}$$

Let us remark that $\omega _{{\varvec{j}}}(I)$ has the same structure of the small denominator associated with $Z_6$ used to obtain non resonance estimates, except that $I_b^2$ is replaced by $I_b$ as it can be easily seen by comparing the previous formula with (54). By proceeding as in Section 5, with the crucial use of Lemma 5.11, we obtain the following result whose proof whose proof is left to the reader.

Lemma A.1

Assume that $I_a$, $a\in {\mathbb {Z}}$ are independent random variable with $I_a$ uniformly distributed in $(0,\langle a \rangle ^{-2s - 4})$, then there exists a constant $c>0$ such that for all $\gamma \in (0,1)$ we have

$$\begin{aligned} {\mathbb {P}}\left( \forall {{\varvec{k}}}\in {\mathcal {I}}rr, \ \sharp k \le 2r \Rightarrow |\omega _{{{\varvec{k}}}}( I)| \ge \gamma \left( \prod _{\alpha =1}^{\sharp {{\varvec{k}}}} \langle k_\alpha \rangle ^{-4} \right) \right) \ge 1 - c\gamma . \end{aligned}$$

The major difference with the (NLS) case is that now the small denominator do not depend on s (compare with Lemma 5.10). Hence, the construction can be performed without having to distribute the derivative and we can apply a normal form procedure using only $Z_4$ (and not $Z_4+Z_6$ as in the (NLS) case).

Following the general strategy, for $\varepsilon ,\gamma >0$, $r\ge 1$, $N \ge 1$ and $s\ge 0$, we say that $z\in \ell _s^1$ belongs to the non resonant set ${\mathcal {U}}_{\gamma ,\varepsilon ,r,s}$, if for all ${{\varvec{k}}}\in {\mathcal {I}}rr$ of length $\sharp {{\varvec{k}}}\le 2r$ we have

$$\begin{aligned} |\omega _{{\varvec{k}}}(I)|> \gamma \varepsilon ^2 \left( \prod _{\alpha =1}^{\sharp {{\varvec{k}}}} \langle k_\alpha \rangle ^{-4} \right) ; \end{aligned}$$

and the that $z\in \ell _s^1$ belongs to the truncated non resonant set ${\mathcal {U}}_{\gamma ,\varepsilon ,r,s}^N$, if for all ${{\varvec{k}}}\in {\mathcal {I}}rr$ of length $\sharp {{\varvec{k}}}\le 2r$ such that $\langle \mu _1({{\varvec{k}}})\rangle \le N^2$, we have

$$\begin{aligned} |\omega _{{\varvec{k}}}(I)|> \gamma \varepsilon ^2 N^{-16r}. \end{aligned}$$

(111)

An adapted Proposition 4.4 remains valid, namely: for N large enough depending on $\varepsilon$ and on $\gamma '<\gamma$ we have ${\mathcal {U}}_{\gamma ,\varepsilon ,r,s}\subset {\mathcal {U}}_{\varepsilon ,\gamma ',r,s}^N$. Moreover, by using the previous Lemma, if $z \in \ell _s^1$ depends on random actions $I_a$ independent and uniformly distributed in $(0,\langle a \rangle ^{-2s - 4})$, there exists a constant $c>0$ such that for all $\gamma \in (0,1)$ we have

$$\begin{aligned} \ {\mathbb {P}}\left( \forall \varepsilon >0,\ \varepsilon z\in {\mathcal {U}}_{\gamma ,\varepsilon ,r,s} \right) \ge 1 - c\gamma . \end{aligned}$$

(112)

Note that the difference with Proposition 5.1 is that for one choice of non resonant actions, the non resonance condition holds for all $\varepsilon$. In other words, the phenomenon of resonances between $\varepsilon$ and I cannot occur in the (NLSP) case.

The class of rational Hamiltonians we need is also simpler: we only need to consider ${\mathscr {H}}_\omega$ and ${\mathscr {H}}_\omega ^*$ defined in Section (6), i.e. functionals of the form

$$\begin{aligned} Q_\Gamma [c](z)= \sum _{ \ell \in {\mathbb {Z}}^* } c_\ell (-1)^{n_\ell } \frac{z_{\varvec{\pi }_\ell }}{\displaystyle \prod _{\alpha =1}^{p_{\ell }} \omega _{{{\varvec{k}}}_{\ell ,\alpha }} } . \end{aligned}$$

with the same condition as in the (NLS) case, but without the restrictive condition (vi) on the distribution of derivatives, making the proof of the Poisson bracket estimate considerably much simpler, as can be seen in the next Appendix.

By using the estimate (111), we can prove an equivalent of Lemma (6.3) for this class of functional (with $\alpha _r = 16r$) and the steps of the rational normal form construction can be then followed as in Section 7 under the same condition (99). The optimization process in N and $\gamma$ can then be done in the same way.

In the end, the probability estimate (112) gives Theorem (2.4).

Appendix B. Proof of Lemma 6.6

This section is devoted to the proof of Lemma 6.6. As in the statement of the Lemma, let $W = \omega$ or $W = \Omega$ and let $\Gamma = (\varvec{\pi },{\varvec{{\mathsf {k}}}},{\varvec{{\mathsf {h}}}},n)\in {\mathscr {H}}_{r,W}^*$ and $\Gamma = (\varvec{\pi }',{\varvec{{\mathsf {k}}}}',{\varvec{{\mathsf {h}}}}',n')\in {\mathscr {H}}_{r',W}$.

To compute the poisson bracket between $Q_\Gamma [c]$ and $Q_{\Gamma '}[c']$, we only need to calculate the poisson brackets of the summands (see the expression (59)). Applying the Leibniz’s rule we see that, up to combinatorial factors and finite linear combinations depending on r, four kind of terms appear depending on which part of the Hamiltonians the Poisson bracket applies to:

Type I. The first type of terms we consider are those where the derivatives apply only on the numerators. They are of the form

$$\begin{aligned} \frac{c_\ell c_{\ell '}(-i)^{p_\ell + q_\ell + p_{\ell '} + q_{\ell '}}}{\displaystyle \prod _{\alpha =1}^{n_{\ell }} \omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =n_{\ell }+1}^{p_{\ell }} \Omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =1}^{q_{\ell }} \Omega _{{\varvec{h}}_{\ell ,\alpha }} \prod _{\alpha =1}^{n_{\ell '}} \omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =n_{\ell '}+1}^{p_{\ell '}} \Omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =1}^{q_{\ell '}} \Omega _{ {\varvec{h}}_{\ell ',\alpha }} } \{ z_{\varvec{\pi }_\ell },z_{\varvec{\pi }'_{\ell '}}\} \end{aligned}$$

for some $\ell$ and $\ell '$ in ${\mathbb {Z}}^*$. Let us set ${{\varvec{j}}}= \varvec{\pi }_\ell$ and ${{\varvec{j}}}' = \varvec{\pi }'_{\ell '}$, i.e. $z_{\varvec{\pi }_\ell } = z_{j_1}\cdots z_{j_{2m}}$ and $z_{\varvec{\pi }'_{\ell '}} = z_{j'_1}\cdots z_{j'_{2m'}}$. The product $\{z_{\varvec{\pi }_\ell },z_{\varvec{\pi }_{\ell '}}\}$ is a linear combination of terms of the form $z_{{{\varvec{j}}}''}$ with ${{\varvec{j}}}'' \in {\mathcal {R}}_{m_\ell + m_{\ell '}-1}$ .

Up to a combinatorial factor, linear combinations and renumbering to define the application $\varvec{\pi }''$, we can concentrate on terms $z_{{{\varvec{j}}}''}$ with ${{\varvec{j}}}'' = \varvec{\pi }''_{\ell ''}$ of the form

$$\begin{aligned} z_{j_2}\cdots z_{j_{2m}}z_{j'_2}\cdots z_{j'_{2m'}}, \end{aligned}$$

provided ${\bar{j}}_1 = j'_1$. Clearly, the produced term is of the good form with $r'' = r+ r' - 1$, $n_{\ell '} = n_\ell + n_\ell '$, $q_{\ell '} = q_\ell + q_{\ell '}$ and $p_{\ell ''} = p_{\ell } + p_{\ell '}$. In particular the reality condition is easily verified by considering the terms corresponding to $-\ell$ and $- \ell '$ and imposing $\overline{\pi ''_{\ell ''}} = \pi ''_{-\ell ''}$, and the conditions ${(\mathbf{i })-(\mathbf{v} )}$ of the definition of the class are trivially satisfied. We can also verify that these terms fulfill the conditions defining the subclass ${\mathscr {H}}_{r'',W}$. Indeed, in the case when $W = \omega$, we have $q_{\ell ''} = q_\ell + q_{\ell '} = 0$ and $n_{\ell ''} = n_{\ell } + n_{\ell '} \le 2(r +1) - 5 + 2r' - 6 = 2 (r + r') - 9 \le 2 r'' - 7$. In the case $W = \Omega$, we can set $\alpha _i'' = \alpha _i + \alpha _i'$ for $i = 1,\ldots ,4$ and $\alpha _5'' = \alpha _5 + \alpha _5'+1$, and we can easily check that the relations (67) are satisfied for $\Gamma ''$. Moreover, using (68) and (69), we check that $\alpha _5'' = \alpha _5 + \alpha _5' +1 \le (r + 2) - 4 + r' - 4 +1 \le r''-4$, and similarly that the three conditions in (68) are satisfied.

Fig. 1

Possible configurations arising from the calculation of $\{ z_{{{\varvec{j}}}}, z_{{{\varvec{j}}}'} \}$

It remains to prove the conditions $\mathbf{ (vi) }$ and $\mathbf{(vii) }$ that are the most delicate. We analyze different cases according to which are the largest indices among ${{\varvec{j}}}$, ${{\varvec{j}}}'$ and ${{\varvec{j}}}''$. The three main case are $\langle j_1\rangle \le \langle \mu _3({{\varvec{j}}})\rangle$, $j_1 = \mu _2({{\varvec{j}}})$ and $j_1 = \mu _1({{\varvec{j}}})$, and by symmetry, we are left to the following cases to be studied:

$\langle j_1 \rangle \le \langle \mu _3({{\varvec{j}}}) \rangle$	$\langle j_1' \rangle \le \langle \mu _3({{\varvec{j}}}') \rangle$	$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$	3.3.a
		$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$	3.3.b
	$j_1' = \mu _2({{\varvec{j}}}')$	$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$	$\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	3.2.a
		$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$	3.2.b
		$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$	3.2.c
	$j_1' = \mu _1({{\varvec{j}}}')$	$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$	3.1
$j_1 = \mu _2({{\varvec{j}}})$	$j_1' = \mu _2({{\varvec{j}}}')$	$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$	2.2
	$j_1' = \mu _1({{\varvec{j}}}')$	$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _3({{\varvec{j}}})$	2.1.a
		$\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}}')$	2.1.b
$j_1 = \mu _1({{\varvec{j}}})$	$j_1' = \mu _1({{\varvec{j}}}')$	$\mu _1({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}}')$	1.1.a
		$\mu _1({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$	$\mu _2({{\varvec{j}}}'') = \mu _3({{\varvec{j}}})$	1.1.b

These cases are summarized in Figure 1 where we try to visualize the different configurations.

Cases 3.3. In these cases, we have $\langle j_1\rangle \le \langle \mu _3({{\varvec{j}}})\rangle$ and $\langle j'_1\rangle \le \langle \mu _3({{\varvec{j}}}')\rangle$ and $\mathbf{(vii) }$ is automatically satisfied as $\mu _2({{\varvec{j}}}'')$ is always greater than $\mu _2({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}')$.

To prove $\mathbf{(vi) }$, we must contruct a fontion $\iota ''$ that distributes the derivatives in ${{\varvec{j}}}''$ from the functions $\iota$ and $\iota '$ distributing the derivatives in ${{\varvec{j}}}$ and ${{\varvec{j}}}'$. Note that by induction hypothesis and the definition of the condition (64), the modes $\mu _1({{\varvec{j}}})$, $\mu _{2}({{\varvec{j}}})$, $\mu _1({{\varvec{j}}}')$ and $\mu _2({{\varvec{j}}}')$ are free in the sense that 1 and 2 are not in the image of $\iota$ and $\iota '$.

We see that we can build $\iota ''$ from $\iota$ and $\iota '$ easily if $j_1$ or $j_1'$ do not correspond to some $\mu _{\iota _{\alpha }}({{\varvec{j}}})$ or $\mu _{\iota '_{\alpha }}({{\varvec{j}}}')$, as $j_1$ and $j_1'$ do not appear in ${{\varvec{j}}}''$. We thus see that the issue is to control $\langle j_1 \rangle = \langle j_1' \rangle$ by two free modes and by letting the two highest modes $\mu _1({{\varvec{j}}}'')$ and $\mu _2({{\varvec{j}}}'')$ free. Indeed, in such a case, up to a reconfiguration of $\iota ''$, the relation (64) will hold again for ${{\varvec{j}}}''$, by using the induction hypothesis on ${{\varvec{j}}}$ and ${{\varvec{j}}}'$. By symmetry, we thus are led to distinguish two cases:

Case 3.3.a. $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$. In this case, $\langle j_1 \rangle \le \langle \mu _{2}({{\varvec{j}}})\rangle$, $\langle j_1' \rangle \le \langle \mu _{2}({{\varvec{j}}}')\rangle$ and we can distribute the derivative to the free modes $\mu _2({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}')$ by letting the two highest modes of ${{\varvec{j}}}''$ free.

Case 3.3.b. $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$. In this case, we use the fact that $\langle j_1 \rangle = \langle j_1' \rangle$ to control $\langle j_1 \rangle$ by $\langle \mu _2({{\varvec{j}}}')\rangle$ and $\langle j_1' \rangle$ by $\langle \mu _1({{\varvec{j}}}')\rangle$ which are modes always smaller that $\langle \mu _3({{\varvec{j}}}'')\rangle$.

Cases 3.2. $\langle j_1\rangle \le \langle \mu _3({{\varvec{j}}})\rangle$ and $j_1' = \mu _2({{\varvec{j}}}')$. The main difference with the previous case is that condition (vii) is not automatically satisfied. To prove it, we need a control of $\langle \mu _2({{\varvec{j}}}) \rangle$ and $\langle \mu _2({{\varvec{j}}}') \rangle$ by $\langle \mu _2({{\varvec{j}}}'')\rangle$. But on the other hand, we only need to control $\langle j_1 \rangle$ by one mode, as $j_1'$ was not used in the distribution of the derivative (condition (64)) for ${{\varvec{j}}}'$. As necessarily the first to highest modes of ${{\varvec{j}}}''$ are in the set $\{ \mu _1({{\varvec{j}}}), \mu _2({{\varvec{j}}}),\mu _1({{\varvec{j}}}')\}$ we are thus led to the following three cases:

Case 3.2.a. $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}}')$ and $\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$. In this situation, we can easily control $\langle j_1 \rangle$ by $\langle \mu _2({{\varvec{j}}}) \rangle$ which is free, and fulfill condition (vi). Moreover, we have $\langle \mu _2({{\varvec{j}}}) \rangle \le \langle \mu _1({{\varvec{j}}})\rangle =\langle \mu _2({{\varvec{j}}}'')\rangle$ and $\langle \mu _2({{\varvec{j}}}')\rangle = \langle j_1 \rangle \le \langle \mu _2({{\varvec{j}}})\rangle$ and hence condition (65) for ${{\varvec{j}}}''$ is inherited from condition (vii) for ${{\varvec{j}}}$ and ${{\varvec{j}}}'$.

Case 3.2.b. $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$. Here, we can control $\langle j_1 \rangle = \langle \mu _2({{\varvec{j}}}') \rangle$ by $\langle \mu _1({{\varvec{j}}}')\rangle$ which is free and smaller than $\mu _2({{\varvec{j}}}'')$ which shows $\mathbf (vi)$. Moreover, in this situation, we have $\mu _{2}({{\varvec{j}}}) = \mu _2({{\varvec{j}}}'')$ and $\langle \mu _2({{\varvec{j}}}') \rangle = \langle j_1 \rangle \le \langle \mu _2({{\varvec{j}}})\rangle =\langle \mu _2({{\varvec{j}}}'') \rangle$ so that $\mathbf (vii)$ holds true for ${{\varvec{j}}}''$.

Case 3.2.c. $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$. In this situation $\mathbf (vi)$ can be easily shown as $\langle j_1 \rangle \le \langle \mu _2({{\varvec{j}}})\rangle$ which free and smaller than $\langle \mu _2({{\varvec{j}}}'')\rangle$. To prove $\mathbf (vii)$, we notice that $\mu _2({{\varvec{j}}}) = \mu _2({{\varvec{j}}}'')$ and $\langle \mu _2({{\varvec{j}}}')\rangle = \langle j_1\rangle \le \langle \mu _2({{\varvec{j}}}) \rangle$.

Case 3.1. $\langle j_1\rangle \le \langle \mu _3({{\varvec{j}}})\rangle$ and $j_1' = \mu _1({{\varvec{j}}}')$. In this situation we have $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$. As in the previous case, we only have to distribute derivative in one free mode, i.e. control $\langle j_1 \rangle$ by $\langle \mu _2({{\varvec{j}}}')\rangle$. This is done by using the zero momentum condition: we have $\langle j_1 \rangle = \langle \mu _1({{\varvec{j}}}') \rangle \le C_{r'} \langle \mu _2({{\varvec{j}}}') \rangle$ where $C_{r'}$ depends only on $r'$. This shows $\mathbf (vi)$ and $\mathbf (vii)$ is proved by noticing that $\mu _2({{\varvec{j}}}) = \mu _2({{\varvec{j}}}'')$ and $\langle \mu _2({{\varvec{j}}}') \rangle \le \langle \mu _1({{\varvec{j}}}') \rangle = \langle j_1 \rangle \le \langle \mu _2({{\varvec{j}}}) \rangle$.

Case 2.2. $j_1 = \mu _2({{\varvec{j}}})$ and $j_1' = \mu _2({{\varvec{j}}}')$ and by symmetry we can assume $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'')=\mu _1({{\varvec{j}}}')$. In this case, $\mathbf (vi)$ for ${{\varvec{j}}}''$ is directly inherited from the condition for ${{\varvec{j}}}$ and ${{\varvec{j}}}'$ as $j_1$ and $j_1'$ were not involved in them. To prove $\mathbf (vii)$, we notice that $\langle \mu _2({{\varvec{j}}})\rangle = \langle \mu _2({{\varvec{j}}}')\rangle \le \langle \mu _1({{\varvec{j}}}')\rangle = \langle \mu _2({{\varvec{j}}}'')\rangle$.

Cases 2.1. $j_1 = \mu _2({{\varvec{j}}})$ and $j_1' = \mu _1({{\varvec{j}}}')$. In this necessarily, we have $\mu _1({{\varvec{j}}}'') = \mu _1({{\varvec{j}}})$. As in the previous case, $\mathbf (vi)$ is easily obtained. To prove $\mathbf (vii)$ we have to distinguish two cases:

Case 2.1.a. $\mu _2({{\varvec{j}}}'') = \mu _3({{\varvec{j}}})$, which means in particular that $\langle \mu _2({{\varvec{j}}}')\rangle \le \langle \mu _3({{\varvec{j}}})\rangle = \langle \mu _2({{\varvec{j}}}'')\rangle$. Moreover, by using the zero-momentum condition, we have $\langle \mu _2({{\varvec{j}}}) \rangle = \langle \mu _1({{\varvec{j}}}') \le C_{r'} \langle \mu _2({{\varvec{j}}}')\rangle \le C_{r'} \langle \mu _2({{\varvec{j}}}'')\rangle$ and this shows the result.

Case 2.1.a. $\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}}')$. In this situation we just need to prove that $\langle \mu _2({{\varvec{j}}})\rangle$ is controlled by $\langle \mu _2({{\varvec{j}}}'')\rangle$ which is ensured by the fact that $\langle \mu _2({{\varvec{j}}})\rangle = \langle \mu _1({{\varvec{j}}}')\rangle \le C_{r'} \langle \mu _2({{\varvec{j}}}')\rangle$ by using the zero momentum condition.

Cases 1.1. $j_1 = \mu _1({{\varvec{j}}})$ and $j_1' = \mu _1({{\varvec{j}}}')$. As before, $\mathbf (vi)$ is easily obtained. To verify $\mathbf (vii)$, by symmetry, we have only two cases to examine:

Case 1.1.a. $\mu _1({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _2({{\varvec{j}}}')$. In this situation, we have by using the zero momentum condition $\langle \mu _2({{\varvec{j}}})\rangle =\langle \mu _1({{\varvec{j}}}'')\rangle \le C_{r''}\langle \mu _2({{\varvec{j}}}'')\rangle$ which shows $\mathbf (vii)$.

Case 1.1.b. $\mu _1({{\varvec{j}}}'') = \mu _2({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}'') = \mu _3({{\varvec{j}}})$. In this case we have necessarily $\langle \mu _2({{\varvec{j}}}')\rangle \le \langle \mu _3({{\varvec{j}}})\rangle \le \langle \mu _2({{\varvec{j}}})\rangle = \langle \mu _1({{\varvec{j}}}'') \rangle$ and we conclude by using the zero momentum condition as in the previous case.

To conclude the analysis of this type, we just observe that (80) is a consequence of the fact that in all the previous cases, we have $\langle \mu _1({{\varvec{j}}}'') \rangle \le \max (\langle \mu _1({{\varvec{j}}}'') \rangle , \langle \mu _1({{\varvec{j}}}'') \rangle )$ and the definition (66) of ${\mathcal {N}}_{\Gamma }(c)$.

Type II. The second type of terms we consider are those where one $\omega _{{{\varvec{k}}}_{\ell ,\alpha }}$ appears in the Poisson bracket. They are of the form

$$\begin{aligned} \frac{c_\ell c_{\ell '}(-i)^{p_\ell + q_\ell + p_{\ell '} + q_{\ell '}}z_{\varvec{\pi }_\ell }}{\displaystyle \prod _{\alpha =1}^{n_{\ell } - 1} \omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =n_{\ell }+1}^{p_{\ell }} \Omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =1}^{q_{\ell }} \Omega _{{\varvec{h}}_{\ell ,\alpha }} \prod _{\alpha =1}^{n_{\ell '}} \omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =n_{\ell '}+1}^{p_{\ell '}} \Omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =1}^{q_{\ell '}} \Omega _{ {\varvec{h}}_{\ell ',\alpha }} } \{ \frac{1}{\omega _{{{\varvec{k}}}_{\ell ,n_\ell }}},z_{\varvec{\pi }'_{\ell '}}\} \end{aligned}$$

Let us set ${{\varvec{j}}}^* ={{\varvec{k}}}_{\ell ,n_\ell } = (j^*_1,\ldots ,j_{\sharp {{\varvec{k}}}_{\ell ,n_\ell }}^*)$. The Poisson bracket above is in general zero, except if one of the index of ${{\varvec{j}}}^*$ is conjugated to one of the index of ${{\varvec{j}}}' = \varvec{\pi }'_{\ell '}$. We can assume here that ${\bar{j}}^*_1 = j'_1$. In this case, we have

$$\begin{aligned} \left\{ \frac{1}{\omega _{{{\varvec{j}}}^*}} ,z_{{{\varvec{j}}}'}\right\} = \pm i \frac{1}{\omega _{{{\varvec{j}}}^*}^2} z_{{{\varvec{j}}}'} \end{aligned}$$

(113)

So the new term is of the good form with ${{\varvec{j}}}'' = {{\varvec{j}}}\cup {{\varvec{j}}}'$ and up to a combinatorial factor, linear combinations and renumbering we can define the application $\varvec{\pi }''$ in such a way that ${{\varvec{j}}}'' = \varvec{\pi }''_{\ell ''}$. The term in the denominator has one more factor repeating the index ${{\varvec{k}}}_{\ell ,n_\ell }$. Hence we have $m_{\ell ''} = m_\ell + m_{\ell '}$, $n_{\ell ''} = n_\ell + n_{\ell '} + 1$, $p_{\ell ''} = p_\ell + p_{\ell '} + 1$, $q_{\ell ''} = q_\ell + q_{\ell '}$ and $r'' = r + r' - 1$. As in the Type I case, we can fulfill the reality condition by considering the terms corresponding to $-\ell$ and $-\ell '$ and imposing $\overline{\pi ''_{\ell ''}} = \pi ''_{-\ell ''}$, and the conditions ${(\mathbf{i })-(\mathbf{v} )}$ of the definition of the class are hence satisfied. Moreover, we can verify that these terms fulfill the conditions associated with the subclass ${\mathscr {H}}_{r'',W}$. In the case when $W = \omega$, we have $q_{\ell ''} = q_\ell + q_{\ell '} = 0$ and $n_{\ell ''} = n_{\ell } + n_{\ell '} + 1 \le 2(r +1) - 5 + 2r' - 5 = 2 (r + r') - 8 \le 2 r'' - 6$. Moreover, in the case $W = \Omega$, we can set $\alpha _i'' = \alpha _i + \alpha _i'$ for $i \in \{ 1,3,4\}$ and $\alpha _i'' = \alpha _i + \alpha _i' + 1$ for $i \in \{ 2,5\}$ and check that the relations (67) and (68) are satisfied for $\Gamma ''$.

In this case $\mu _2({{\varvec{j}}}'')$ is necessarily greater than $\mu _2({{\varvec{j}}})$ and $\mu _2({{\varvec{j}}}')$, so that $\mathbf (vii)$ is easily proved.

To prove $\mathbf (vi)$, we observe that the functions $\iota$ and $\iota '$ distribute the indices ${{\varvec{k}}}_{\ell ,\alpha }$ and ${{\varvec{k}}}'_{\ell ',\alpha }$ to some indices in ${{\varvec{j}}}$ and ${{\varvec{j}}}'$ respectively that are always lower than the third ones. Hence we have four free indices, and two new derivatives to distribute coming from the presence of the new term $\omega _{{{\varvec{j}}}^*}$. We can distinguish two cases:

• $\langle j_1'\rangle \le \langle \mu _2({{\varvec{j}}}')\rangle$. In this situation, we use $\mathbf (vi)$ saying that $\langle \mu _{\min }({{\varvec{j}}}^*) \rangle \le C \langle \mu _2({{\varvec{j}}})\rangle$. Hence as $\langle \mu _{\min }({{\varvec{j}}}^*)\rangle \le \langle j_1^*\rangle = \langle j_1' \rangle \le \langle \mu _2({{\varvec{j}}}')\rangle$, we can construct $\iota ''$ from $\iota$ and $\iota '$ and by making ${{\varvec{j}}}^*$ correspond to the third and fourth largest indices amongst $\mu _1({{\varvec{j}}}),\mu _2({{\varvec{j}}}), \mu _1({{\varvec{j}}}')$ and $\mu _2({{\varvec{j}}}')$.

• $j_1' = \mu _1({{\varvec{j}}}')$. We still have by $\mathbf (vi)$ that $\langle \mu _{\min }({{\varvec{j}}}^*) \rangle \le C \langle \mu _2({{\varvec{j}}})\rangle$. Moreover by zero momentum condition, we have $\langle \mu _{\min }({{\varvec{j}}}^*)\rangle \le \langle j_1^*\rangle = \langle j_1' \rangle = \langle \mu _1({{\varvec{j}}}')\rangle \le C_{r'}\langle \mu _2({{\varvec{j}}}')\rangle$ and we are back the the previous case.

Type III. Now we consider terms where one $\Omega _{{{\varvec{k}}}_{\ell ,\alpha }}$ appears in the Poisson bracket. They are of the form

$$\begin{aligned} \frac{c_\ell c_{\ell '}(-i)^{p_\ell + q_\ell + p_{\ell '} + q_{\ell '}}z_{\varvec{\pi }_\ell }}{\displaystyle \prod _{\alpha =1}^{n_{\ell }} \omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =n_{\ell }+1}^{p_{\ell } - 1} \Omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =1}^{q_{\ell }} \Omega _{{\varvec{h}}_{\ell ,\alpha }} \prod _{\alpha =1}^{n_{\ell '}} \omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =n_{\ell '}+1}^{p_{\ell '}} \Omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =1}^{q_{\ell '}} \Omega _{ {\varvec{h}}_{\ell ',\alpha }} } \left\{ \frac{1}{\Omega _{{{\varvec{k}}}_{\ell ,p_\ell }}},z_{\varvec{\pi }'_{\ell '}}\right\} \end{aligned}$$

Let us set ${{\varvec{j}}}^* ={{\varvec{k}}}_{\ell ,p_\ell } = (j^*_1,\ldots ,j_{\sharp {{\varvec{k}}}_{\ell ,p_\ell }}^*)$, ${{\varvec{j}}}= \varvec{\pi }_\ell$ and ${{\varvec{j}}}' = \varvec{\pi }'_{\ell '}$ as before. To compute the Poisson bracket there two case to examine.

• First, if $\overline{{{\varvec{j}}}^*} \cap {{\varvec{j}}}'= \emptyset$ then

  $$\begin{aligned} \left\{ \frac{1}{\Omega _{{{\varvec{j}}}^*}} ,z_{{{\varvec{j}}}'}\right\} = \pm i P(I)\frac{z_{{{\varvec{j}}}'}}{\Omega _{{{\varvec{j}}}^*}^2}, \end{aligned}$$

  where, in view of the form $Z_6$ (see (37)) , P is a polynomial of degree 1 with real coefficients. Up to a combinatorial factor, linear combinations and renumbering we can define the application $\varvec{\pi }''$ satisfying the reality condition, and we can set $m_{\ell ''} = m_{\ell } + m_{\ell '} + 1$, $n_{\ell ''} = n_\ell + n_{\ell '}$, $p_{\ell ''} = p_\ell + p_{\ell '}$ and $q_{\ell ''} = q_\ell + q_{\ell '} + 1$. The conditions ${(\mathbf{i })-(\mathbf{v} )}$ of the definition of the class are hence satisfied. Moreover, we can set $\alpha _i'' = \alpha _i + \alpha _i'$ for $i \in \{ 1,2,3\}$, $\alpha _i'' = \alpha _i + \alpha _i'+1$ for $i \in \{ 4,5\}$ and check that the relations (67) and (68) are satisfied for $\Gamma ''$. Moreover, $\mathbf (vii)$ is satisfied as $\langle \mu _2({{\varvec{j}}})\rangle \le \langle \mu _2({{\varvec{j}}}'')\rangle$ and $\langle \mu _2({{\varvec{j}}}')\rangle \le \langle \mu _2({{\varvec{j}}}'')\rangle$, and $\mathbf (vi)$ is also satisfied as there is no new derivative to distribute.

• If one of the index of ${{\varvec{j}}}^*$ is conjugate of one of the index of ${{\varvec{j}}}'$, then we get

  $$\begin{aligned} \left\{ \frac{1}{\Omega _{{{\varvec{j}}}^*}} ,z_{{{\varvec{j}}}'}\right\} = \pm i \frac{z_{{{\varvec{j}}}'}}{\Omega _{{{\varvec{j}}}^*}^2} + \pm i P(I)\frac{z_{{{\varvec{j}}}'}}{\Omega _{{{\varvec{j}}}^*}^2}, \end{aligned}$$

  (114)

  where P is a polynomial of degree 1 with real coefficients. We thus treat the second term as previously. To treat the first term, we use the same analysis than the one in type II with $n_{\ell '} = n_{\ell } + n_{\ell '}$, $p_{\ell ''} = p_{\ell } + p_{\ell '} + 1$ $q_{\ell ''} =q_{\ell } + q_{\ell '}$. The only difference is that we set $\alpha _i'' = \alpha _i + \alpha _i'$ for $i \in \{ 1,2,4\}$ and $\alpha _i'' = \alpha _i + \alpha _i' + 1$ for $i \in \{ 3,5\}$ but the distribution of derivatives is achieved in a similar way.

Type IV. Finally we consider terms where one $\Omega _{{\varvec{h}}_{\ell ,\alpha }}$ appears in the Poisson bracket. They are of the form

$$\begin{aligned} \frac{c_\ell c_{\ell '}(-i)^{p_\ell + q_\ell + p_{\ell '} + q_{\ell '}}z_{\varvec{\pi }_\ell }}{\displaystyle \prod _{\alpha =1}^{n_{\ell }} \omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =n_{\ell }+1}^{p_{\ell }} \Omega _{{{\varvec{k}}}_{\ell ,\alpha }} \prod _{\alpha =1}^{q_{\ell } - 1} \Omega _{{\varvec{h}}_{\ell ,\alpha }} \prod _{\alpha =1}^{n_{\ell '}} \omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =n_{\ell '}+1}^{p_{\ell '}} \Omega _{{{\varvec{k}}}'_{\ell ',\alpha }} \prod _{\alpha =1}^{q_{\ell '}} \Omega _{ {\varvec{h}}_{\ell ',\alpha }} } \left\{ \frac{1}{\Omega _{{\varvec{h}}_{\ell ,q_\ell }}},z_{\varvec{\pi }'_{\ell '}}\right\} \end{aligned}$$

It is almost the same as type III except that to deal with the first term in the right-hand side of (114) we count one $\Omega _{{{\varvec{j}}}^*}$ in the denominator as $\Omega _{{\varvec{h}}_{\ell '',q_{\ell ''}}}$ with $q_{\ell ''} = q_{\ell } + q_{\ell '}+1$ and the other is counted as $\Omega _{{{\varvec{k}}}_{\ell '',p_{\ell ''}}}$ with $p_{\ell ''} = p_{\ell } + p_{\ell '}$. The analysis is then the same as in Type II for the distribution of derivatives.