\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 07, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/07\hfil Global periodic solutions for NLS-KdV systems]
{Global well-posedness of NLS-KdV systems for periodic functions}

\author[C. Matheus\hfil EJDE-2006/07\hfilneg]
{Carlos Matheus} 

\address{Carlos Matheus \newline
%Instituto Nacional de Matem\'atica Pura e Aplicada
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320,
Brazil}
\email{matheus@impa.br}

\thanks{Submitted November 13, 2006. Published January 2, 2007.}
\subjclass[2000]{35Q55} 
\keywords{Global well-posedness; Schr\"odinger-Korteweg-de Vries system;
 \hfill\break\indent I-method}

\begin{abstract}
 We prove that the Cauchy problem of the
 Schr\"odinger-Korteweg-deVries (NLS-KdV) system for
 periodic functions is globally well-posed for initial
 data in the energy space $H^1\times H^1$. More precisely, we
 show that the non-resonant NLS-KdV system is globally well-posed for
 initial data in $H^s(\mathbb{T})\times H^s(\mathbb{T})$
 with $s>11/13$ and the resonant NLS-KdV system is globally well-posed
 with $s>8/9$.
 The strategy is to apply the I-method used by Colliander, Keel,
 Staffilani, Takaoka and Tao. By doing this, we improve the
 results by Arbieto, Corcho and Matheus concerning the global
 well-posedness of  NLS-KdV systems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}


\section{Introduction}\label{s.intro}

We consider the Cauchy problem of the Schr\"odinger-Korteweg-deVries
(NLS-KdV) system
\begin{equation}\label{e.nls-kdv}
\begin{gathered}
i\partial_tu + \partial_x^2u = \alpha uv + \beta |u|^2u,\\
\partial_tv + \partial_x^3v + \tfrac{1}{2}\partial_x(v^2) =
\gamma \partial_x(|u|^2),\\
u(x,0)=u_0(x),\quad v(x,0)=v_0(x), \quad  t\in\mathbb{R}.
\end{gathered}
\end{equation}
This system   appears naturally
in fluid mechanics and plasma physics as a model of interaction between
a short-wave $u=u(x,t)$ and a long-wave $v=v(x,t)$.

In this paper we are interested in global solutions of the NLS-KdV system for
rough initial data. Before stating our main results, let us recall some of the
recent theorems of local and global well-posedness theory of the Cauchy
problem \eqref{e.nls-kdv}.

For continuous spatial variable (i.e., $x\in\mathbb{R}$), Corcho and
Linares \cite{Corcho}
recently proved that the NLS-KdV system is locally well-posed for initial data
$(u_0,v_0)\in H^k(\mathbb{R})\times H^s(\mathbb{R})$ with $k\geq 0$, $s>-3/4$
and
\begin{gather*}
 k-1\leq s\leq 2k-1/2 \quad\text{if }k\leq 1/2 ,\\
 k-1\leq s<k+1/2 \quad\text{if } k>1/2.
\end{gather*}
Furthermore, they  prove the global well-posedness of the NLS-KdV
system in the energy $H^1\times H^1$ using three conserved quantities
discovered by  Tsutsumi \cite{MTsutsumi}, whenever $\alpha\gamma>0$.

Also, Pecher \cite{Pecher} improved this global well-posedness
result by an application of the I-method of Colliander, Keel, Stafillani,
Takaoka and Tao (see for instance \cite{CKSTT1}) combined with some refined
bilinear estimates. In particular, Pecher proved that, if $\alpha\gamma>0$,
the NLS-KdV system is globally well-posed for initial data
$(u_0,v_0)\in H^s\times H^s$ with $s>3/5$ in the resonant case $\beta=0$ and
$s>2/3$ in the non-resonant case $\beta\neq 0$.

On the other hand, in the periodic setting (i.e., $x$ isn the space of
periodic functions $\mathbb{T}$), Arbieto,
Corcho and Matheus \cite{ACM} proved the local well-posedness of the NLS-KdV
system for initial data $(u_0,v_0)\in H^k\times H^s$ with $0\leq s\leq 4k-1$ and
$-1/2\leq k-s\leq 3/2$. Also, using the same three conserved quantities
discovered by  Tsutsumi, one obtains the global well-posedness of NLS-KdV on
$\mathbb{T}$ in the energy space $H^1\times H^1$ whenever $\alpha\gamma>0$.

Motivated by this scenario, we combine the new bilinear estimates of Arbieto,
Corcho and Matheus \cite{ACM} with the I-method of Tao and his collaborators to
prove the following result.

\begin{theorem}\label{t.A}
The NLS-KdV system \eqref{e.nls-kdv} on $\mathbb{T}$
is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times
H^s(\mathbb{T})$ with $s>11/13$ in the non-resonant case $\beta\neq 0$ and
$s>8/9$ in the resonant case $\beta=0$, whenever $\alpha\gamma>0$.
\end{theorem}

The paper is organized as follows. In the section \ref{s.preliminaries}, we
discuss the preliminaries for the proof of the theorem \ref{t.A}:
Bourgain spaces and its properties, linear estimates, standard estimates for
the non-linear terms $|u|^2 u$ and $\partial_x (v^2)$, the bilinear estimates of
Arbieto, Corcho and Matheus \cite{ACM} for the coupling terms $uv$ and
$\partial_x(|u|^2)$, the I-operator and its properties. In the section \ref{s.local},
we apply the results of the section \ref{s.preliminaries} to get a variant of
the local well-posedness result of \cite{ACM}. In the
section \ref{s.conservation}, we recall some conserved quantities
of \eqref{e.nls-kdv} and its modification by the introduction of the I-operator;
moreover, we prove that two of these modified energies are almost
conserved. Finally, in the section \ref{s.global}, we combine the almost
conservation results in section \ref{s.conservation} with the local
well-posedness result in section \ref{s.local} to conclude the proof of the
theorem \ref{t.A}.

\section{Preliminaries}\label{s.preliminaries}

A successful procedure to solve some dispersive equations (such as the
nonlinear Schr\"odinger and KdV equations) is to use the Picard's fixed point
method in the following spaces:
\begin{gather*}
\|f\|_{X^{k,b}}:= \Big(\int\sum_{n\in\mathbb{Z}} \langle n\rangle^{2k}
\langle\tau+n^2\rangle^{2b}|\widehat{f}(n,\tau)| d\tau\Big)^{1/2}
= \|U(-t) f\|_{H_t^b(\mathbb{R},H_x^k)},\\
\|g\|_{Y^{s,b}}:= \Big(\int\sum_{n\in\mathbb{Z}} \langle n\rangle^{2s}
\langle\tau-n^3\rangle^{2b}|\widehat{g}(n,\tau)| d\tau\Big)^{1/2}
= \|V(-t) f\|_{H_t^b(\mathbb{R},H_x^s)},
\end{gather*}
where $\langle\cdot\rangle:= 1+ |\cdot|$, $U(t)=e^{it\partial_x^2}$ and
$V(t)=e^{-t\partial_x^3}$.
These spaces are called Bourgain spaces. Also,
we introduce the restriction in time norms
\begin{equation*}
\|f\|_{X^{k,b}(I)}:=\inf_{\widetilde{f}|_I=f} \|\widetilde{f}\|_{X^{k,b}}
\quad \text{and} \quad
\|g\|_{Y^{s,b}(I)}:=\inf_{\widetilde{g}|_I=g} \|\widetilde{g}\|_{Y^{s,b}}
\end{equation*}
where $I$ is a time interval.

The interaction of the Picard method has been based around the spaces
$Y^{s,1/2}$. Because we are interested in the continuity of the flow associated
to \eqref{e.nls-kdv} and the $Y^{s,1/2}$ norm do not control the
$L_t^{\infty}H_x^s$ norm, we modify the Bourgain spaces as follows:
\begin{gather*}
\|u\|_{X^k}:= \|u\|_{X^{k,1/2}} + \|\langle n\rangle^k\widehat{u}(n,\tau)\|_{L_n^2
L_{\tau}^1} ,\\
\|v\|_{Y^{s}}:=\|v\|_{Y^{s,1/2}} + \|\langle n\rangle^s\widehat{v}(n,\tau)\|_{L_n^2
L_{\tau}^1}
\end{gather*}
and, given a time interval $I$, we consider the restriction in time of the
$X^k$ and $Y^s$ norms
\begin{equation*}
\|u\|_{X^{k}(I)}:=\inf_{\widetilde{u}|_I=u} \|\widetilde{u}\|_{X^{k}}
\quad \text{and} \quad
\|v\|_{Y^{s}(I)}:=\inf_{\widetilde{v}|_I=v} \|\widetilde{v}\|_{Y^{s}}
\end{equation*}

Furthermore, the mapping properties of $U(t)$ and $V(t)$ naturally leads one to
consider the companion spaces
\begin{gather*}
\|u\|_{Z^k}:=\|u\|_{X^{k,-1/2}}+
\Big\|\frac{\langle n\rangle^k\widehat{u}(n,\tau)}{\langle\tau+n^2\rangle}\Big\|_{L_n^2
L_{\tau}^1} , \\
\|v\|_{W^s}:=\|v\|_{Y^{s,-1/2}}+
\Big\|\frac{\langle n\rangle^s\widehat{v}(n,\tau)}{\langle\tau-n^3\rangle}\Big\|_{L_n^2
L_{\tau}^1}
\end{gather*}
In the sequel, $\psi$ denotes a non-negative smooth bump function supported on
$[-2,2]$ with $\psi=1$ on $[-1,1]$ and $\psi_{\delta}(t):=\psi(t/\delta)$ for
any $\delta>0$.

\noindent\textbf{Notation}. Fix $(k,s)$ a pair of indices such that
the local well-posedness of the periodic NLS-KdV system holds. Given
two non-negative real numbers $A$ and $B$, we write $A\lesssim B$
whenever $A\leq C\cdot B$, where $C=C(k,s)$ is a constant which may
depend only on $(k,s)$. Also, we write $A\gtrsim B$ if $A\geq c\cdot
B$, where $c=c(k,s)$ is sufficiently small (depending only on
$(k,s)$), and $A\sim B$ if $A\lesssim B\lesssim A$. Furthermore, we
use $A\ll B$ to mean $A\leq c\dot B$ where $c=c(k,s)$ is a small
constant (depending only on $(k,s)$), and $A\gg B$ to denote $A\geq
C\cdot B$ with $C=C(k,s)$ a large constant. Finally, given, for
instance, a function $\psi$ and a number $b$, we put also
$A\lesssim_{\psi,b} B$ to mean $A\leq C\cdot B$ where
$C=C(k,s,\psi,b)$ is a constant depending also on the specified
function $\psi$ and number $b$ (besides $(k,s)$).

\bigskip

Next, we recall some properties of the Bourgain spaces:

\begin{lemma}\label{l.Strichartz}
$X^{0,3/8}([0,1]), Y^{0,1/3}([0,1])\subset
L^4(\mathbb{T}\times [0,1])$. More precisely,
\begin{equation*}
\|\psi(t) f\|_{L_{xt}^4}\lesssim \|f\|_{X^{0,3/8}} \quad \text{and} \quad
\|\psi(t) g\|_{L_{xt}^4}\lesssim \|g\|_{Y^{0,1/3}}.
\end{equation*}
\end{lemma}

For the proof of the above lemma see \cite{Bourgain}.
Another basic property of these spaces are their stability under time
localization:

\begin{lemma}\label{l.time-localization}
Let $X^{s,b}_{\tau = h(\xi)}:= \{f :
\langle\tau-h(\xi)\rangle^b\langle\xi\rangle^s |\widehat{f}(\tau,\xi)|\in L^2\}$. Then
\begin{equation*}
\|\psi(t) f\|_{X^{s,b}_{\tau=h(\xi)}}\lesssim_{\psi,b}
\|f\|_{X^{s,b}_{\tau=h(\xi)}}
\end{equation*}
for any $s,b\in\mathbb{R}$. Moreover, if $-1/2<b'\leq b<1/2$, then for any
$0<T<1$, we have
\begin{equation*}
\|\psi_T(t) f\|_{X_{\tau=h(\xi)}^{s,b'}}\lesssim_{\psi,b',b} T^{b-b'}
\|f\|_{X^{s,b}_{\tau=h(\xi)}}.
\end{equation*}
\end{lemma}

\begin{proof}
First of all, note that $\langle\tau-\tau_0-h(\xi)\rangle^{b}\lesssim_b
\langle\tau_0\rangle^{|b|}\langle\tau-h(\xi)\rangle^{b}$,
from which we obtain
$$
\|e^{it\tau_0}f\|_{X_{\tau=h(\xi)}^{s,b}}\lesssim_b \langle\tau_0\rangle^{|b|}
\|f\|_{X_{\tau=h(\xi)}^{s,b}}.
$$
Using that $\psi(t)=\int\widehat{\psi}(\tau_0) e^{it\tau_0}d\tau_0$, we
conclude
$$
\|\psi(t)f\|_{X_{\tau=h(\xi)}^{s,b}}\lesssim_b
\Big(\int|\widehat{\psi}(\tau_0)| \langle\tau_0\rangle^{|b|}\Big)
\|f\|_{X_{\tau=h(\xi)}^{s,b}}.
$$
Since $\psi$ is smooth with compact support, the first estimate follows.

Next we prove the second estimate. By conjugation we may assume $s=0$ and,
by composition it suffices to treat the cases $0\leq b'\leq b$ or
$\leq b'\leq b\leq 0$. By duality, we may take $0\leq b'\leq b$.
Finally, by interpolation with the trivial case $b'=b$, we may consider
$b'=0$. This reduces matters to show that
$$
\|\psi_T(t)f\|_{L^2}\lesssim_{\psi,b} T^b\|f\|_{X_{\tau=h(\xi)}^{0,b}}
$$
for $0<b<1/2$. Partitioning the frequency spaces into the cases
$\langle\tau-h(\xi)\rangle\geq 1/T$ and $\langle\tau-h(\xi)\leq 1/T$, we see that in the
former case we'll have
$$
\|f\|_{X_{\tau=h(\xi)}^{0,0}}\leq T^b\|f\|_{X_{\tau=h(\xi)}^{0,b}}
$$
and the desired estimate follows because the multiplication by $\psi$ is a
bounded operation in Bourgain's spaces. In the latter case, by Plancherel and
Cauchy-Schwarz
\begin{align*}
\|f(t)\|_{L_x^2}&\lesssim \|\widehat{f(t)}(\xi)\|_{L_{\xi}^2}\\
& \lesssim
\big\|\int_{\langle\tau-h(\xi)\rangle\leq 1/T}|\widehat{f}(\tau,\xi)|d\tau)
\big\|_{L_{\xi}^2} \\
&\lesssim_b T^{b-1/2}
\big\|\int\langle\tau-h(\xi)\rangle^{2b} |\widehat{f}(\tau,\xi)|^2
d\tau)^{1/2}\big\|_{L_{\xi}^2} \\
&= T^{b-1/2}\|f\|_{X_{\tau=h(\xi)}^{s,b}}.
\end{align*}
Integrating this against $\psi_T$ concludes the proof of the lemma.
\end{proof}

Also, we have the following duality relationship between $X^k$ (resp., $Y^s$)
and $Z^k$ (resp., $W^s$):

\begin{lemma}\label{l.duality}
We have
\begin{gather*}
\big|\int \chi_{[0,1]}(t) f(x,t) g(x,t)\,dx\,dt\big|
\lesssim \|f\|_{X^s} \|g\|_{Z^{-s}},\\
\big|\int \chi_{[0,1]}(t) f(x,t) g(x,t)\,dx\,dt\big|\lesssim \|f\|_{Y^s}
\|g\|_{W^{-s}}
\end{gather*}
for any $s$ and any $f, g$ on $\mathbb{T}\times\mathbb{R}$.
\end{lemma}

\begin{proof} See \cite[p. 182--183]{CKSTT} (note that, although this result is
stated only for the spaces $Y^s$ and $W^s$, the same proof adapts for the spaces
$X^k$ and $Z^k$).
\end{proof}

Now, we recall some linear estimates related to the semigroups $U(t)$ and
$V(t)$:

\begin{lemma}[Linear estimates]\label{l.linear}
It holds
\begin{gather*}
\|\psi(t)U(t)u_0\|_{Z^k}\lesssim \|u_0\|_{H^k},\\
\|\psi(t)V(t)v_0\|_{W^s}\lesssim \|v_0\|_{H^s};
\\
\|\psi_{T}(t)\int_0^t U(t-t') F(t') dt'\|_{X^k}\lesssim \|F\|_{Z^k}, \\
\|\psi_{T}(t)\int_0^t V(t-t') G(t') dt'\|_{Y^s}\lesssim \|G\|_{W^s}.
\end{gather*}
\end{lemma}

For a proof of the above lemma, see \cite{CKSTT1}, \cite{CKSTT} or \cite{ACM}.
Furthermore, we have the following well-known multiinear estimates for the cubic
term $|u|^2 u$ of the nonlinear Schr\"odinger equation and the nonlinear term
$\partial_x(v^2)$ of the KdV equation:

\begin{lemma}\label{l.u2u}
$\|uv\overline{w}\|_{Z^k}\lesssim \|u\|_{X^{k,\frac{3}{8}}}\|v\|_{X^{k,\frac{3}{8}}}
\|w\|_{X^{k,\frac{3}{8}}}$ for any $k\geq 0$.
\end{lemma}

For the proof of the above lemma, see See \cite{Bourgain} and \cite{ACM}.

\begin{lemma}\label{l.dv2}
$\|\partial_x(v_1 v_2)\|_{W^s}\lesssim
\|v_1\|_{Y^{s,\frac{1}{3}}}\|v_2\|_{Y^{s,\frac{1}{2}}}+
\|v_1\|_{Y^{s,\frac{1}{2}}}\|v_2\|_{Y^{s,\frac{1}{3}}}$ for any $s\geq -1/2$, if
$v_1=v_1(x,t)$ and $v_2=v_2(x,t)$ are $x$-periodic functions having zero
$x$-mean for all $t$.
\end{lemma}

The proof of the above lemma can be found in \cite{Bourgain}, \cite{CKSTT1}
and \cite{ACM}.
Next, we revisit the bilinear estimates of mixed Schr\"odinger-Airy type of
Arbieto, Corcho and Matheus \cite{ACM} for the coupling terms $uv$
and $\partial_x(|u|^2)$ of the NLS-KdV system.

\begin{lemma}\label{l.uv}
$\|uv\|_{Z^k}\lesssim \|u\|_{X^{k,\frac{3}{8}}}\|v\|_{Y^{s,\frac{1}{2}}} +
\|u\|_{X^{k,\frac{1}{2}}}\|v\|_{Y^{s,\frac{1}{3}}}$ whenever $s\geq 0$ and
$k-s\leq 3/2$.
\end{lemma}

\begin{lemma}\label{l.du2}
$\|\partial_x(u_1\overline{u_2})\|_{W^s}\lesssim \|u_1\|_{X^{k,3/8}}\|u_2\|_{X^{k,1/2}} +
\|u_1\|_{X^{k,1/2}}\|u_2\|_{X^{k,3/8}}$ whenever $1+s\leq 4k$ and $k-s\geq
-1/2$.
\end{lemma}

\begin{remark} \label{rmk2.1} \rm
Although the lemmas \ref{l.uv} and \ref{l.du2} are not stated as
above in \cite{ACM}, it is not hard to obtain them from the
calculations of Arbieto, Corcho and Matheus.
\end{remark}

Finally, we introduce the I-operator: let $m(\xi)$ be a smooth non-negative
symbol on $\mathbb{R}$ which equals $1$ for $|\xi|\leq 1$ and equals
$|\xi|^{-1}$ for $|\xi|\geq 2$. For any $N\geq 1$ and $\alpha\in\mathbb{R}$,
denote by $I_N^{\alpha}$ the spatial Fourier multiplier
\begin{equation*}
\widehat{I_N^{\alpha}f}(\xi) = m\big(\frac{\xi}{N}\big)^{\alpha}
\widehat{f}(\xi).
\end{equation*}

For latter use, we recall the following general interpolation lemma.

\begin{lemma}[{\cite[Lemma 12.1]{CKSTT}}]\label{l.interpolation}
Let $\alpha_0>0$ and $n\geq 1$. Suppose $Z, X_1, \dots, X_n$
are translation-invariant Banach spaces and $T$ is a translation
invariant $n$-linear operator such that
\begin{equation*}
\|I_1^{\alpha} T(u_1,\dots, u_n)\|_{Z}\lesssim \prod_{j=1}^n
\|I_1^{\alpha}u_j\|_{X_j},
\end{equation*}
for all $u_1,\dots,u_n$ and $0\leq\alpha\leq\alpha_0$. Then
\begin{equation*}
\|I_N^{\alpha} T(u_1,\dots, u_n)\|_{Z}\lesssim \prod_{j=1}^n
\|I_N^{\alpha}u_j\|_{X_j}
\end{equation*}
for all $u_1,\dots,u_n$, $0\leq\alpha\leq\alpha_0$ and $N\geq 1$. Here the
implicit constant is independent of $N$.
\end{lemma}

After these preliminaries, we can proceed to the next section where
a variant of the local well-posedness of Arbieto, Corcho and
Matheus is obtained.
In the sequel we take $N\gg 1$ a large integer and denote by $I$ the operator
$I=I_N^{1-s}$ for a given $s\in\mathbb{R}$.

\section{A variant local well-posedness result}\label{s.local}

This section is devoted to the proof of the following proposition.

\begin{proposition}\label{p.local}
For any $(u_0,v_0)\in H^s(\mathbb{T})\times
H^s(\mathbb{T})$ with $\int_{\mathbb{T}} v_0 = 0$ and $s\geq 1/3$, the periodic
NLS-KdV system \eqref{e.nls-kdv}
has a unique local-in-time solution on the time interval $[0,\delta]$ for some
$\delta\leq 1$ and
\begin{equation}\label{e.local}
\delta\sim
\begin{cases}
(\|Iu_0\|_{X^1}+\|Iv_0\|_{Y^1})^{-\frac{16}{3}-}, &\text{if } \beta\neq 0, \\
(\|Iu_0\|_{X^1}+\|Iv_0\|_{Y^1})^{-8-}, &\text{if } \beta = 0.
\end{cases}
\end{equation}
Moreover, we have $\|Iu\|_{X^1}+\|Iv\|_{Y^1}\lesssim
\|Iu_0\|_{X^1}+\|Iv_0\|_{Y^1}$.
\end{proposition}

\begin{proof}
We apply the I-operator to the NLS-KdV system \eqref{e.nls-kdv} so
that
\begin{gather*}
i Iu_t + Iu_{xx} = \alpha I(uv) + \beta I(|u|^2 u), \\
Iv_t + Iv_{xxx} + I(v v_x) = \gamma I(|u|^2)_x, \\
Iu(0) = Iu_0, \quad Iv(0) = Iv_0.
\end{gather*}
To solve this equation, we seek for some fixed point of the integral maps
\begin{gather*}
\Phi_1(Iu, Iv):= U(t) Iu_0 -i\int_0^t U(t-t') \{\alpha I(u(t')v(t')) + \beta
I(|u(t')|^2 u(t'))\} dt', \\
\Phi_2(Iu, Iv):= V(t)Iv_0 -\int_0^t V(t-t')\{I(v(t')v_x(t')) - \gamma
I(|u(t')|^2)_x\} dt'.
\end{gather*}
The interpolation lemma \ref{l.interpolation} applied to the linear and
multilinear estimates in the lemmas \ref{l.linear}, \ref{l.u2u}, \ref{l.dv2},
 \ref{l.uv} and \ref{l.du2} yields, in view of the
lemma \ref{l.time-localization},
\begin{gather*}
\|\Phi_1(Iu, Iv)\|_{X^1}\lesssim \|Iu_0\|_{H^1} + \alpha\delta^{\frac{1}{8}-}
\|Iu\|_{X^1}\|Iv\|_{Y^1} + \beta\delta^{\frac{3}{8}-}\|Iu\|_{X^1}^3, \\
\|\Phi_2(Iu, Iv)\|_{Y^1}\lesssim \|Iv_0\|_{H^1} +
\delta^{\frac{1}{6}-}\|Iv\|_{Y^1}^2 +
\gamma\delta^{\frac{1}{8}-}\|Iu\|_{X^1}^2.
\end{gather*}
In particular, these integrals maps are contractions provided that
$\beta\delta^{\frac{3}{8}-}(\|Iu_0\|_{H^1}+\|Iv_0\|_{H^1})^2 \ll 1$ and
$\delta^{\frac{1}{8}-}(\|Iu_0\|_{H^1}+\|Iv_0\|_{H^1})\ll 1$. This completes the
proof.
\end{proof}

\section{Modified energies}\label{s.conservation}

We define the following three quantities:
\begin{gather}\label{e.mass}
M(u):=\|u\|_{L^2}, \\
\label{e.momentum}
L(u,v):= \alpha \|v\|_{L^2}^2 + 2\gamma\int \Im (u\overline{u_x}) dx, \\
\label{e.energy}
E(u,v):= \alpha\gamma\int v |u|^2 dx + \gamma\|u_x\|_{L^2}^2 + \frac{\alpha}{2}
\|v_x\|_{L^2}^2 -\frac{\alpha}{6}\int v^3 dx + \frac{\beta\gamma}{2}\int |u|^4
dx.
\end{gather}
In the sequel, we suppose $\alpha\gamma>0$. Note that
\begin{gather}\label{e.L1}
|L(u,v)|\lesssim \|v\|_{L^2}^2 + M \|u_x\|_{L^2} ,\\
\label{e.L2}
\|v\|_{L^2}^2\lesssim |L| + M \|u_x\|_{L^2}.
\end{gather}
Also, the Gagliardo-Nirenberg and Young inequalities implies
\begin{gather}\label{e.E1}
\|u_x\|_{L^2}^2+\|v_x\|_{L^2}^2\lesssim |E| + |L|^{\frac{5}{3}} + M^8 + 1,\\
\label{e.E2}
|E|\lesssim \|u_x\|_{L^2}^2 + \|v_x\|_{L^2}^2 + |L|^{\frac{5}{3}} + M^8 + 1
\end{gather}
In particular, combining the bounds (\ref{e.L1}) and (\ref{e.E2}),
\begin{equation}\label{e.E3}
|E|\lesssim \|u_x\|_{L^2}^2 + \|v_x\|_{L^2}^2 + \|v\|_{L^2}^{\frac{10}{3}} +
M^{10} + 1.
\end{equation}
Moreover, from the bounds (\ref{e.L2}) and (\ref{e.E1}),
\begin{equation}\label{e.E4}
\|v\|_{L^2}^2\lesssim |L| + M |E|^{1/2} + M^6 + 1
\end{equation}
and hence
\begin{equation}\label{e.E5}
\|u\|_{H^1}^2 + \|v\|_{H^1}^2\lesssim |E| + |L|^{5/3} + M^8 + 1
\end{equation}
\begin{equation}\label{e.al}
\begin{aligned}
&\frac{d}{dt} L(Iu, Iv) \\
&= 2\alpha\int Iv (Iv Iv_x - I(v v_x)) dx +
2\alpha\gamma\int Iv (I(|u|^2)-|Iu|^2)_x dx \\
&\quad + 4\alpha\gamma\Re\int I\overline{u}_x (Iu Iv - I(uv)) dx
+ 4\beta\gamma\Re\int ((Iu)^2 I\overline{u} - I(u^2\overline{u})) I\overline{u}_x dx \\
&=: \sum_{j=1}^4 L_j.
\end{aligned}
\end{equation}
and
\begin{equation}\label{e.ae}
\begin{aligned}
&\frac{d}{dt} E(Iu, Iv) \\
&= \alpha\int (I(vv_x)-IvIv_x)Iv_{xx} dx +
\frac{\alpha}{2}\int (Iv)^2 (I(vv_x)-IvIv_x) dx + \\
&\quad+ 2\beta\gamma\Im\int (I(|u|^2 u)_x - ((Iu)^2 I\overline{u})_x) I\overline{u}_x dx \\
&\quad+ \alpha\gamma\int |Iu|^2 (Iv Iv_x - I(v v_x)) dx + \alpha\gamma\int (|Iu|^2 -
I(|u|^2))Iv Iv_x dx \\
&\quad+ \alpha\gamma\int Iv_{xx} (|Iu|^2-I(|u|^2))_x dx -2\alpha\gamma\Im\int Iu_x
(I(\overline{u}v)-I\overline{u} Iv)_x dx \\
&\quad+ \alpha\gamma^2 \int (I(|u|^2) - |Iu|^2)_x |Iu|^2 dx + 2\alpha^2\gamma\Im\int
Iv Iu (I(\overline{u}v - I\overline{u} Iv)) dx\\
&\quad + 2\beta^2\gamma\Im\int Iu(I\overline{u})^2 (I(|u|^2 u) - (Iu)^2 I\overline{u}) dx \\
&\quad -2\alpha\beta\gamma\Im\int Iv Iu (I(|u|^2 \overline{u}) - Iu (I\overline{u}))^2 dx
-2\alpha\beta\gamma\Im\int (Iu)^2 I\overline{u} (I(\overline{u}v) - I\overline{u} Iv) dx \\
&=: \sum_{j=1}^{12} E_j
\end{aligned}
\end{equation}

\subsection{Estimates for the modified L-functional}\label{s.l}

\begin{proposition}\label{p.al}
Let $(u,v)$ be a solution of \eqref{e.nls-kdv} on
the time interval $[0,\delta]$. Then, for any $N\geq 1$ and $s>1/2$,
\begin{equation}\label{e.aL}
\begin{aligned}
&|L(Iu(\delta), Iv(\delta)) - L(Iu(0), Iv(0))|\\
&\lesssim N^{-1+}\delta^{\frac{19}{24}-} (\|Iu\|_{X^{1,1/2}}+\|Iv\|_{Y^{1,1/2}})^3 +
N^{-2+}\delta^{\frac{1}{2}-}\|Iu\|_{X^{1,1/2}}^4.
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
Integrating (\ref{e.al}) with respect to $t\in [0,\delta]$, it
follows that we have to bound the (integral over $[0,\delta]$ of the) four
terms on the right hand side. To simplify the computations, we assume that the
Fourier transform of the functions are non-negative and we ignore the appearance
of complex conjugates (since they are irrelevant in our subsequent arguments).
Also, we make a dyadic decomposition of the frequencies $|n_i|\sim N_j$ in many
places. In particular, it will be important to get extra factors $N_j^{0-}$
everywhere in order to sum the dyadic blocks.

We begin with the estimate of $\int_{0}^{\delta} L_1$. It is sufficient to show
that
\begin{equation}\label{e.aL1}
\begin{aligned}
&\int_{0}^{\delta}\sum_{n_1+n_2+n_3=0}
\Big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\Big|
\widehat{v_1}(n_1,t) |n_2|\widehat{v_2}(n_2,t) \widehat{v_3}(n_3,t)\\
&\lesssim N^{-1}\delta^{\frac{5}{6}-}\prod_{j=1}^3 \|v_j\|_{Y^{1,1/2}}
\end{aligned}
\end{equation}

\noindent $\bullet$ $|n_1|\ll |n_2|\sim |n_3|$, $|n_2|\gtrsim N$. In this case, note that
\begin{gather*}
\big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\big|\lesssim
|\frac{\nabla m(n_2)\cdot n_1}{m(n_2)}|\lesssim \frac{N_1}{N_2},
\text{if } |n_1|\leq N, \\
\big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\big|\lesssim
\big(\frac{N_1}{N}\big)^{1/2}, \text{if } |n_1|\geq N.
\end{gather*}
Hence, using the lemmas \ref{l.Strichartz} and \ref{l.time-localization}, we
obtain
\begin{equation*}
\big|\int_{0}^{\delta} L_1\big|\lesssim \frac{N_1}{N_2} \|v_1\|_{L^4} \|(v_2)_x\|_{L^4}
\|v_3\|_{L^2}\lesssim N^{-2+}\delta^{\frac{5}{6}-} N_{\rm max}^{0-}
\prod_{j=1}^3\|v_i\|_{Y^{1,1/2}}
\end{equation*}
if $|n_1|\leq N$, and
\begin{equation*}
|\int_{0}^{\delta} L_1|\lesssim \big(\frac{N_1}{N}\big)^{1/2} \frac{1}{N_1
N_3} \delta^{\frac{5}{6}-}\prod_{j=1}^3\|v_i\|_{Y^{1,1/2}}\lesssim
N^{-2+}\delta^{\frac{5}{6}-} N_{\rm max}^{0-}
\prod_{j=1}^3\|v_i\|_{Y^{1,1/2}}.
\end{equation*}

\noindent $\bullet$
$|n_2|\ll |n_1|\sim |n_3|$, $|n_1|\gtrsim N$.
This case is similar to the previous one.

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$. The multiplier is bounded by
\begin{equation*}
\big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\big|\lesssim
\big(\frac{N_1}{N}\big)^{1-}.
\end{equation*}
In particular, using the lemmas \ref{l.Strichartz}
and \ref{l.time-localization},
\begin{equation*}
|\int_{0}^{\delta} L_1|\lesssim \big(\frac{N_1}{N}\big)^{1-} \|v_1\|_{L^2}
\|(v_2)_x\|_{L^4} \|v_3\|_{L^4}\lesssim N^{-1+}\delta^{\frac{5}{6}-}
N_{\rm max}^{0-}\prod_{j=1}^3 \|v_i\|_{Y^{1,1/2}}.
\end{equation*}


Now, we estimate $\int_{0}^{\delta} L_2$. Our task is to prove that
\begin{equation}\label{e.aL2}
\begin{aligned}
&\int_0^{\delta}\sum_{n_1 + n_2 + n_3 = 0}
\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1) m(n_2)}\Big| |n_1+n_2|
\widehat{u_1}(n_1,t) \widehat{u_2}(n_2,t) \widehat{v_3}(n_3,t) \\
&\lesssim N^{-1+}\delta^{\frac{19}{24}-}\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}
\|v_3\|_{Y^{1,1/2}}
\end{aligned}
\end{equation}


\noindent$\bullet$
$|n_2|\ll |n_1|\sim |n_3|\gtrsim N$. We estimate the multiplier by
\begin{equation*}
\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1) m(n_2)}\Big|\lesssim
\langle(\frac{N_2}{N})^{1/2}\rangle.
\end{equation*}
Thus, using $L^2_{xt}L^4_{xt}L^4_{xt}$ H\"older inequality and the
lemmas \ref{l.Strichartz} and \ref{l.time-localization}
\begin{align*}
\int_0^{\delta}L_2 &\lesssim \langle\big(\frac{N_2}{N}\big)^{1/2}\rangle
\frac{1}{\langle N_2\rangle N_3}\delta^{\frac{19}{24}-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}
\|v_3\|_{Y^{1,1/2}} \\
&\lesssim N^{-1+}\delta^{\frac{19}{24}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}\|v_3\|_{Y^{1,1/2}}.
\end{align*}

\noindent$\bullet$  $|n_1|\ll |n_2|\sim |n_3|$. This case is similar to the
previous one.

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$. Estimating the
multiplier by
\begin{equation*}
\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1) m(n_2)}\Big|\lesssim
\big(\frac{N_2}{N}\big)^{1-}
\end{equation*}
we conclude
\begin{equation*}
\begin{aligned}
\int_0^{\delta} L_2&\lesssim \big(\frac{N_2}{N}\big)^{1-} \frac{1}{N_1 N_2}
\delta^{\frac{19}{24}-}\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}\|v_3\|_{Y^{1,1/2}}
\\
&\lesssim N^{-2+}\delta^{\frac{19}{24}-}N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}\|v_3\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation*}

Next, let us compute $\int_0^{\delta} L_3$. We claim that
\begin{equation}\label{e.aL3}
\begin{aligned}
&\int_0^{\delta}\sum_{n_1+n_2+n_3=0}
\Big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\Big| \widehat{u_1}(n_1,t)
\widehat{v_2}(n_2,t) |n_3|\widehat{u_3}(n_3,t) \\
&\lesssim N^{-2+}\delta^{\frac{19}{24}-}
\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}\|u_3\|_{X^{1,1/2}}
\end{aligned}
\end{equation}


\noindent$\bullet$ $|n_2|\ll |n_1|\sim |n_3|$, $|n_1|\gtrsim N$. The multiplier
is bounded by
\[
\big|\frac{m(n_1+n_2) - m(n_1) m(n_2)}{m(n_1) m(n_2)}\big|\lesssim
\begin{cases}
|\frac{\nabla m(n_1)\cdot n_2}{m(n_1)}|\lesssim \frac{N_2}{N_1},
&\text{if } |n_2|\leq N, \\
\big(\frac{N_2}{N}\big)^{1/2}, &\text{if } |n_2|\geq N.
\end{cases}
\]
So, it is not hard to see that
\begin{equation*}
\int_0^{\delta} L_3 \lesssim N^{-2+}\delta^{\frac{19}{24}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}\|u_3\|_{X^{1,1/2}}
\end{equation*}

\noindent$\bullet$ $|n_1|\ll |n_2|\sim |n_3|$, $|n_2|\gtrsim N$.
This case is completely similar to the previous one.

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$. Since the multiplier
is bounded by $N_2 / N$, we get
\begin{equation*}
\int_0^{\delta} L_3 \lesssim
N^{-2+}\delta^{\frac{19}{24}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}\|u_3\|_{X^{1,1/2}}.
\end{equation*}


Finally, it remains to estimate the contribution of $\int_0^{\delta} L_4$. It
suffices to see that
\begin{equation}\label{e.aL4}
\begin{aligned}
&\int_0^{\delta}\sum_{n_1 + n_2 + n_3 + n_4= 0}
\Big|\frac{m(n_1+n_2+n_3)-m(n_1)m(n_2)m(n_3)}{m(n_1) m(n_2) m(n_3)}
\Big| |n_4|
\prod_{j=1}^4 \widehat{u_j}(n_j,t) \\
&\lesssim N^{-2+}\delta^{\frac{1}{2}-}
\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}
\end{aligned}
\end{equation}

\noindent$\bullet$  $N_1, N_2, N_3\gtrsim N$. Since the multiplier verifies
\begin{equation*}
\Big|\frac{m(n_1+n_2+n_3)-m(n_1)m(n_2)m(n_3)}{m(n_1) m(n_2) m(n_3)}\Big|
\lesssim \left(\frac{N_1}{N}\frac{N_2}{N}\frac{N_3}{N}\right)^{1/2},
\end{equation*}
appliying $L^4_{xt}L^4_{xt}L^4_{xt}L^4_{xt}$ H\"older inequality and
the lemmas \ref{l.Strichartz}, \ref{l.time-localization}, we have
\begin{align*}
\int_0^{\delta} L_4&\lesssim
\left(\frac{N_1}{N}\frac{N_2}{N}\frac{N_3}{N}\right)^{1/2}
\frac{\delta^{\frac{1}{2}-}}{N_1 N_2 N_3} \prod_{j=1}^4
\|u_j\|_{X^{1,1/2}}\\
&\lesssim N^{-3+}\delta^{\frac{1}{2}-} N_{\rm max}^{0-}
\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{align*}

\noindent$\bullet$ $N_1\sim N_2\gtrsim N$ and $N_3, N_4\ll N_1, N_2$. Here the multiplier is
bounded by $\left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2} \langle
\big(\frac{N_3}{N}\big)^{1/2}\rangle$. Hence,
\begin{align*}
\int_0^{\delta} L_4
&\lesssim \left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2} \langle
\big(\frac{N_3}{N}\big)^{1/2}\rangle
\frac{\delta^{\frac{1}{2}-}}{N_1 N_2 \langle N_3\rangle}
\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}\\
&\lesssim N^{-2+}\delta^{\frac{1}{2}-} N_{\rm max}^{0-}
\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{align*}

\noindent$\bullet$ $N_1\sim N_4\gtrsim N$ and $N_2, N_3\ll N_1, N_4$. In this case we have
the following estimates for the multiplier
\begin{align*}
&\Big|\frac{m(n_1+n_2+n_3)-m(n_1)m(n_2)m(n_3)}{m(n_1) m(n_2) m(n_3)}
\Big| \\
&\lesssim
\begin{cases}
\big|\frac{\nabla m(n_1) (n_2+n_3)}{m(n_1)}\big|\lesssim \frac{N_2+N_3}{N_1},
&\text{if } N_2, N_3\leq N\\
\left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2}
\langle (\frac{N_3}{N})^{1/2} \rangle, &\text{if } N_2\geq N,\\
\left(\frac{N_1}{N}\frac{N_3}{N}\right)^{1/2}
\langle (\frac{N_2}{N})^{1/2} \rangle, &\text{if } N_3\geq N.
\end{cases}
\end{align*}
Therefore, it is not hard to see that, in any of the situations
$N_2, N_3\leq N$, $N_2\geq N$ or $N_3\geq N$, we have
\begin{equation*}
\int_0^{\delta} L_4\lesssim N^{-2+}\delta^{\frac{1}{2}-} N_{\rm max}^{0-}
\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{equation*}

\noindent$\bullet$ $N_1\sim N_2\sim N_4\gtrsim N$ and $N_3\ll N_1, N_2, N_4$. Here we have
the following bound
\begin{equation*}
\int_0^{\delta} L_4 \lesssim
\left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2}
\langle\big(\frac{N_3}{N}\big)^{1/2}\rangle \frac{\delta^{\frac{1}{2}-}}{N_1
N_2 N_3} \prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{equation*}

At this point, clearly the bounds (\ref{e.aL1}), (\ref{e.aL2}), (\ref{e.aL3})
and (\ref{e.aL4}) concludes the proof of the proposition \ref{p.al}.
\end{proof}


\subsection{Estimates for the modified E-functional}\label{s.e}

\begin{proposition}\label{p.ae}
Let $(u,v)$ be a solution of \eqref{e.nls-kdv} on
the time interval $[0,\delta]$ such that $\int_{\mathbb{T}}v =0$. Then, for any
$N\geq 1$, $s>1/2$,
\begin{equation}\label{e.aE}
\begin{aligned}
&|E(Iu(\delta), Iv(\delta))-E(Iu(0), Iv(0))|\\
&\lesssim \left(N^{-1+}\delta^{\frac{1}{6}-}+N^{-\frac{2}{3}+}\delta^{\frac{3}{8}-} +
N^{-\frac{3}{2}+}\delta^{\frac{1}{8}-} \right) (\|Iu\|_{X^1}+ \|Iv\|_{Y^1})^3\\
&\quad +N^{-1+}\delta^{\frac{1}{2}-} (\|Iu\|_{X^1}+\|Iv\|_{Y^1})^4 +
N^{-2+}\delta^{\frac{1}{2}-}\|Iu\|_{X^1}^4 (\|Iu\|_{X^1}^2+\|Iv\|_{Y^1}).
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
Again we integrate (\ref{e.ae}) with respect to $t\in [0,\delta]$,
decompose the frequencies into dyadic blocks, etc., so that our objective is to
bound the (integral over $[0,\delta]$ of the) $E_j$ for each $j=1,\dots, 12$.

For the expression $\int_0^{\delta} E_1$, apply the lemma \ref{l.duality}. We
obtain
\begin{equation*}
|\int_0^{\delta} E_1|\lesssim \|Iv_{xx}\|_{Y^{-1}} \|Iv Iv_x - I(v v_x)\|_{W^1}
\lesssim \|Iv\|_{Y^1} \|Iv Iv_x - I(v v_x)\|_{W^1}
\end{equation*}
Writing the definition of the norm $W^1$, it suffices to prove the bound
\begin{equation}
\begin{aligned}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle^{1/2}}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3,\tau_3}} \\
&+ \Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L^1_{\tau_3}}\\
& \lesssim  N^{-1+}\delta^{\frac{1}{6}-}\|v_1\|_{Y^{1,1/2}}\|v_2\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation}
Recall that the dispersion relation
$\sum_{j=1}^3 \tau_j-n_j^3 = -3n_1 n_2 n_3$ implies that,
since $n_1 n_2 n_3\neq 0$, if we put $L_j:=|\tau_j - n_j^3|$ and
$L_{\rm max} = \max\{L_j;j=1,2,3\}$, then
$L_{\rm max}\gtrsim \langle n_1\rangle \langle n_2\rangle \langle n_3\rangle$.

\noindent$\bullet$ $|n_2|\sim |n_3|\gtrsim N$, $|n_1|\ll |n_2|$. The multiplier is bounded by
\begin{equation*}
\Big|\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|\lesssim
\begin{cases}
\frac{N_1}{N_2}, &\text{if } |n_1|\leq N, \\
\big(\frac{N_1}{N}\big)^{1/2}, &\text{if } |n_1|\geq N.
\end{cases}
\end{equation*}
Thus, if $|\tau_3- n_3^3| = L_{\rm max}$, we have
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle^{1/2}}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3,\tau_3}} \\
&\lesssim
\begin{cases}
\frac{N_1}{N_2}\frac{N_3}{(N_1 N_2 N_3)^{1/2}}
\|v_1\|_{L_{xt}^4}\|(v_2)_x\|_{L^4_{xt}}\\
\lesssim N^{-1+}
\delta^{\frac{1}{3}-} N_{\rm max}^{0-} \|v_1\|_{Y^{1,1/2}} \|v_2\|_{Y^{1,1/2}},
&\text{if } |n_1|\leq N, \\[3pt]
\big(\frac{N_1}{N_2}\big)^{1/2}\frac{N_3}{N_1}
\frac{1}{(N_1 N_2 N_3)^{1/2}}
\|v_1\|_{L_{xt}^4}\|(v_2)_x\|_{L^4_{xt}}\\
\lesssim N^{-\frac{3}{2}+}
\delta^{\frac{1}{3}-} N_{\rm max}^{0-} \|v_1\|_{Y^{1,\frac{1}{2}}}
\|v_2\|_{Y^{1,\frac{1}{2}}},
&\text{if } |n_1|\geq N.
\end{cases}
\end{align*}
and
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L^1_{\tau_3}} \\
&\lesssim
\begin{cases}
\frac{N_1}{N_2}\frac{N_3}{(N_1 N_2 N_3)^{\frac{1}{2}-}}\|v_1\|_{L_{xt}^4}
\|(v_2)_x\|_{L^4_{xt}}\\
\lesssim N^{-1+}\delta^{\frac{1}{3}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,1/2}} \|v_2\|_{Y^{1,1/2}},
&\text{if } |n_1|\leq N, \\
\big(\frac{N_1}{N_2}\big)^{1/2} \frac{N_3}{N_1}
\frac{\delta^{\frac{1}{3}-}}{(N_1 N_2
N_3)^{\frac{1}{2}-}} \|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}}\\
\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{3}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}},
&\text{if } |n_1|\geq N.
\end{cases}
\end{align*}
If either $|\tau_1-n_1^3|=L_{\rm max}$ or $|\tau_2-n_2^3|=L_{\rm max}$, we have
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle^{1/2}}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3,\tau_3}} \\
&\lesssim
\begin{cases}
\frac{N_1}{N_2}\frac{N_3}{(N_1 N_2 N_3)^{1/2}}
\frac{\delta^{\frac{1}{6}-}}{N_1}
\|v_1\|_{Y^{1,\frac{1}{2}}}\|v_2\|_{Y^{1,\frac{1}{2}}}\\
\lesssim N^{-1+}
\delta^{\frac{1}{6}-} N_{\rm max}^{0-} \|v_1\|_{Y^{1,1/2}} \|v_2\|_{Y^{1,1/2}},
&\text{if } |n_1|\leq N, \\
\big(\frac{N_1}{N_2}\big)^{1/2}\frac{N_3}{N_1}
\frac{1}{(N_1 N_2 N_3)^{1/2}}
\delta^{\frac{1}{6}-}\|v_1\|_{Y^{1,\frac{1}{2}}}
\|v_2\|_{Y^{1,\frac{1}{2}}}\\
\lesssim N^{-\frac{3}{2}+}
\delta^{\frac{1}{3}-} N_{\rm max}^{0-} \|v_1\|_{Y^{1,\frac{1}{2}}}
\|v_2\|_{Y^{1,\frac{1}{2}}},
&\text{if } |n_1|\geq N.
\end{cases}
\end{align*}
and
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L^1_{\tau_3}} \\
&\lesssim
\begin{cases}
\frac{N_1}{N_2}\frac{N_3}{(N_1 N_2
N_3)^{\frac{1}{2}-}}\frac{\delta^{\frac{1}{6}-}}{N_1}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}}\\
\lesssim N^{-1+}\delta^{\frac{1}{6}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,1/2}} \|v_2\|_{Y^{1,1/2}},
&\text{if } |n_1|\leq N, \\
\big(\frac{N_1}{N_2}\big)^{1/2} \frac{N_3}{N_1}
\frac{\delta^{\frac{1}{6}-}}{(N_1 N_2
N_3)^{\frac{1}{2}-}} \|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}}\\
\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{6}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}},
&\text{if } |n_1|\geq N.
\end{cases}
\end{align*}

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$. Estimating the multiplier by
\begin{equation*}
\Big|\frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|\lesssim \big(\frac{N_1}{N}\big)^{1-},
\end{equation*}
we have that, if $|\tau_3-n_3^3|=L_{\rm max}$,
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle^{1/2}}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3,\tau_3}} \\
&+\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L^1_{\tau_3}} \\
&\lesssim \big\{\big(\frac{N_1}{N}\big)^{1-}
\frac{N_3}{(N_1 N_2 N_3)^{1/2}}
\frac{\delta^{\frac{1}{3}-}}{N_1} +
\big(\frac{N_1}{N}\big)^{1-}\frac{N_3}{(N_1 N_2 N_3)^{\frac{1}{2}-}}
\frac{\delta^{\frac{1}{3}-}}{N_1}\big\}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}} \\
&\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{3}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}}
\end{align*}
and, if either $|\tau_1-n_1^3|=L_{\rm max}$ or $|\tau_2-n_2^3|=L_{\rm max}$,
\begin{align*}
&\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle^{1/2}}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3,\tau_3}} \\
&+\Big\|\frac{\langle n_3 \rangle}{\langle\tau_3- n_3^3\rangle}
\int \sum \frac{m(n_1+n_2) -
m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{v_1}(n_1,\tau_1) \ n_2 \
\widehat{v_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L^1_{\tau_3}} \\
&\lesssim \big\{\big(\frac{N_1}{N}\big)^{1-}
\frac{N_3}{(N_1 N_2 N_3)^{1/2}}\frac{\delta^{\frac{1}{6}-}}{N_1}
+ \big(\frac{N_1}{N}\big)^{1-}
\frac{N_3}{(N_1 N_2 N_3)^{\frac{1}{2}-}}\frac{\delta^{\frac{1}{6}-}}{N_1}
\big\}\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}} \\
&\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{6}-}N_{\rm max}^{0-}
\|v_1\|_{Y^{1,\frac{1}{2}}} \|v_2\|_{Y^{1,\frac{1}{2}}}.
\end{align*}

For the expression $\int_0^{\delta}E_2$, it suffices to prove that
\begin{equation}\label{e.aE2}
\begin{aligned}
&\big|\int_0^{\delta}\sum \frac{m(n_3+n_4)-m(n_3)m(n_4)}{m(n_3)m(n_4)}
\widehat{v_1}(n_1,t)\widehat{v_2}(n_2,t)\widehat{v_3}(n_3,t) \ n_4 \
\widehat{v_4}(n_4,t)\big|\\
& \lesssim  N^{-2+}\delta^{\frac{2}{3}-}\prod_{j=1}^4\|v_j\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation}
Since at least two of the $N_i$ are bigger than $N/3$, we can assume
that $N_1\geq N_2\geq N_3$ and $N_1\gtrsim N$. Hence,
\begin{equation*}
\int_0^{\delta} E_2\lesssim
\begin{cases}
\big(\frac{N_1}{N}\big)^{1-}\frac{\delta^{\frac{2}{3}-}}{N_1 N_2 N_3}
\prod_{j=1}^4 \|v_j\|_{Y^{1,1/2}}\lesssim N^{-2+}\delta^{\frac{2}{3}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|v_j\|_{Y^{1,1/2}}, \\
\quad \text{if } |n_3|\sim |n_4|\gtrsim N, \\[3pt]
\frac{N_3}{N_4}\frac{\delta^{\frac{2}{3}-}}{N_1 N_2 N_3}
\prod_{j=1}^4 \|v_j\|_{Y^{1,1/2}}\lesssim N^{-2+}\delta^{\frac{2}{3}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|v_j\|_{Y^{1,1/2}}, \\
\quad \text{if } |n_3|\ll |n_4|, |n_3|\leq N |n_4|\gtrsim N, \\[3pt]
\big(\frac{N_3}{N}\big)^{1/2}
\frac{\delta^{\frac{2}{3}-}}{N_1 N_2 N_3}
\prod_{j=1}^4 \|v_j\|_{Y^{1,\frac{1}{2}}}
\lesssim N^{-2+}\delta^{\frac{2}{3}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|v_j\|_{Y^{1,\frac{1}{2}}},\\
\quad \text{if }
|n_3|\ll |n_4|, |n_3|\geq N, |n_4|\gtrsim N.
\end{cases}
\end{equation*}
Next, we estimate the contribution of $\int_0^{\delta} E_3$. We claim that
\begin{equation}\label{l.aE3}
\begin{aligned}
&\int_{0}^{\delta}\sum\frac{m(n_1n_2n_3) -
m(n_1)m(n_2)m(n_3)}{m(n_1)m(n_2)m(n_3)} \widehat{u_1}(n_1,t)
\widehat{u_2}(n_2,t) \widehat{u_3}(n_3,t)\ |n_4|^2 \
\widehat{u_4}(n_4,t)\\
&\lesssim  N^{-1+}\delta^{\frac{1}{2}-}\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{aligned}
\end{equation}

\noindent$\bullet$ $|n_1|\sim |n_2|\sim |n_3|\sim |n_4|\gtrsim N$.
Since the multiplier satisfies
\begin{equation*}
\frac{m(n_1n_2n_3) - m(n_1)m(n_2)m(n_3)}{m(n_1)m(n_2)m(n_3)}
\lesssim \big(\frac{N_1}{N}\big)^{\frac{3}{2}}
\end{equation*}
we obtain
\begin{equation*}
\int_0^{\delta} E_3\lesssim \big(\frac{N_1}{N}\big)^{\frac{3}{2}}
\frac{N_4}{N_1N_2N_3}\delta^{\frac{1}{2}-}
\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}\lesssim N^{-2+}\delta^{\frac{1}{2}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{equation*}

\noindent$\bullet$ Exactly two frequencies are bigger than $N/3$. We
consider the most difficult case $|n_4|\gtrsim N$, $|n_1|\sim |n_4|$
and $|n_2|, |n_3|\ll |n_1|, |n_4|$. The multiplier is estimated by
\begin{equation*}
\frac{m(n_1n_2n_3) -
m(n_1)m(n_2)m(n_3)}{m(n_1)m(n_2)m(n_3)}\lesssim
\begin{cases}
\langle\big(\frac{N_3}{N}\big)^{1/2}\rangle
\big(\frac{N_2}{N}\big)^{1/2}, &\text{if } |n_2|\geq N,\\
\langle\big(\frac{N_2}{N}\big)^{1/2}\rangle
\big(\frac{N_3}{N}\big)^{1/2}, &\text{if } |n_3|\geq N,\\
\frac{N_2+N_3}{N_1}, &\text{if } |n_2|, |n_3|\leq N.
\end{cases}
\end{equation*}
Thus,
\begin{equation*}
\int_0^{\delta}E_3\lesssim N^{-1+}\delta^{\frac{1}{2}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{equation*}

\noindent$\bullet$ Exactly three frequencies are bigger than $N/3$.
The most difficult case is $|n_1|\sim |n_2|\sim |n_4|\gtrsim N$ and
$|n_3|\ll |n_1|, |n_2|, |n_4|$. Here the multiplier is bounded by
\begin{equation*}
\frac{m(n_1n_2n_3) -
m(n_1)m(n_2)m(n_3)}{m(n_1)m(n_2)m(n_3)}\lesssim
\left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2}
\langle\big(\frac{N_3}{N}\big)^{1/2}\rangle.
\end{equation*}
Hence,
\begin{align*}
&\int_0^{\delta} E_3\lesssim
\left(\frac{N_1}{N}\frac{N_2}{N}\right)^{1/2}
\langle\big(\frac{N_3}{N}\big)^{1/2}\rangle \frac{N_4}{N_1 N_2 N_3}
\delta^{\frac{1}{2}-}\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}\\
&\lesssim N^{-1+}\delta^{\frac{1}{2}-}
N_{\rm max}^{0-}\prod_{j=1}^4 \|u_j\|_{X^{1,1/2}}.
\end{align*}

The contribution of $\int_0^{\delta} E_4$ is controlled if we are
able to show that
\begin{equation}\label{e.aE4}
\begin{aligned}
&\int_0^{\delta}\sum \frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)}
\widehat{v_1}(n_1,t)\ |n_2| \ \widehat{v_2}(n_2,t) \widehat{u_3}(n_3,t)
\widehat{u_4}(n_4,t)\lesssim \\
& N^{-1+}\delta^{\frac{7}{12}-}\prod_{j=1}^2
\|u_j\|_{X^{1,1/2}}\|v_j\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation}
We crudely bound the multiplier by
\begin{equation*}
|\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)}|\lesssim
\big(\frac{N_{\rm max}}{N}\big)^{1-}.
\end{equation*}
The most difficult case is $|n_2|\geq N$. We have two possibilities:

\noindent$\bullet$ Exactly two frequencies are bigger than $N/3$. We
can assume $N_3\ll N_2$. In particular,
\begin{align*}
\int_0^{\delta}E_4&\lesssim \big(\frac{N_{\rm max}}{N}\big)^{1-}
\frac{\delta^{\frac{7}{12}}}{N_1 N_3 N_4}\prod_{j=1}^2
\|u_j\|_{X^{1,\frac{1}{2}}}\|v_j\|_{Y^{1,\frac{1}{2}}}\\
&\lesssim N^{-1+}\delta^{\frac{7}{12}-}N_{\rm max}^{0-}\prod_{j=1}^2
\|u_j\|_{X^{1,\frac{1}{2}}}\|v_j\|_{Y^{1,\frac{1}{2}}}.
\end{align*}

\noindent$\bullet$ At least three frequencies are bigger than $N/3$.
In this case,
\begin{equation*}
\int_0^{\delta}E_4\lesssim
N^{-2+}\delta^{\frac{7}{12}-}N_{\rm max}^{0-}\prod_{j=1}^2
\|u_j\|_{X^{1,\frac{1}{2}}}\|v_j\|_{Y^{1,\frac{1}{2}}}.
\end{equation*}

The expression $\int_0^{\delta}E_5$ is controlled if we are able to prove
\begin{equation}\label{e.aE5}
\begin{aligned}
&\int_0^{\delta}\sum \frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)}
\widehat{u_1}(n_1,t) \widehat{u_2}(n_2,t) \widehat{v_3}(n_3,t)
\ |n_4| \ \widehat{v_4}(n_4,t)\\
&\lesssim  N^{-1+}\delta^{\frac{7}{12}-}\prod_{j=1}^2
\|u_j\|_{X^{1,1/2}}\|v_j\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation}
This follows directly from the previous analysis for (\ref{e.aE4}).

For the term $\int_0^{\delta} E_6$, we apply the lemma \ref{l.duality}
to obtain
\begin{equation*}
\int_{0}^{\delta}E_6\lesssim \|(Iv)_{xx}\|_{Y^{-1}}\|(|Iu|^2 -
I(|u|^2))_x\|_{W^1}\lesssim \|Iv\|_{Y^1}\|(|Iu|^2 -
I(|u|^2))_x\|_{W^1}.
\end{equation*}
So, the definition of the $W^1$ norm means that we have to prove
\begin{equation}\label{e.aE6}
\begin{aligned}
&\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L_{n_3,\tau_3}^2} \\
&+\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L_{\tau_3}^1}\\
&\lesssim \left\{N^{-\frac{3}{2}+}\delta^{\frac{1}{8}-} +
N^{-\frac{2}{3}}\delta^{\frac{3}{8}-}\right\}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}.
\end{aligned}
\end{equation}
Note that $\sum\tau_j = 0$ and $\sum n_j = 0$. In particular, we obtain the
dispersion relation
\begin{equation*}
\tau_3 - n_3^3 + \tau_2 + n_2^2 + \tau_1 + n_1^2 = -n_3^3+n_1^2+n_2^2.
\end{equation*}

\noindent$\bullet$ $|n_1|\gtrsim N$, $|n_2|\ll |n_1|$. Denoting by $L_1:=|\tau_1+n_1^2|$,
$L_2:=|\tau_2+n_2^2|$ and $L_3:=|\tau_3-n_3^3|$, the dispersion relation says
that in the present situation $L_{\rm max}:=\max\{L_j\}\gtrsim N_3^3$. Since the
multiplier is bounded by
\begin{equation*}
\Big|\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|\lesssim
\begin{cases}
\frac{\nabla m(n_1) n_2}{m(n_1)}\lesssim \frac{N_2}{N_1}, &\text{if }
|n_2|\leq N, \\
\big(\frac{N_2}{N}\big)^{1/2}, &\text{if } |n_2|\geq N,
\end{cases}
\end{equation*}
we deduce that
\begin{align*}
&\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L_{n_3,\tau_3}^2} \\
&+\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L_{\tau_3}^1}\\
&\lesssim \frac{N_3^2}{N_3^{\frac{3}{2}-}}\frac{\delta^{\frac{1}{8}-}}{N N_1}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}} \\
&\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{8}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}.
\end{align*}

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$, $|n_3|^3\gg |n_2|^2$. In the present
case the
multiplier is bounded by $\big(\frac{N_1}{N}\big)^{1-}$ and the dispersion
relation says that $L_{\rm max}\gtrsim N_3^3$. Thus,
\begin{align*}
&\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L_{n_3,\tau_3}^2}  \\
&+\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L_{\tau_3}^1}\\
&\lesssim  \frac{N_3^2}{N_3^{\frac{3}{2}-}}\big(\frac{N_1}{N}\big)^{1-}
\frac{\delta^{\frac{1}{8}-}}{N_1 N_2}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}} \\
&\lesssim N^{-\frac{3}{2}+}\delta^{\frac{1}{8}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}.
\end{align*}

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$ and $|n_3|^3\lesssim |n_2|^2$. Here the
dispersion relation does not give useful information about $L_{\rm max}$.
Since the multiplier is estimated by $\big(\frac{N_2}{N}\big)^{1/2}$,
we obtain the crude bound
\begin{align*}
&\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L_{n_3,\tau_3}^2}  \\
&+\Big\|\frac{\langle n_3\rangle}{\langle\tau_3-n_3^3\rangle^{1/2}} |n_3| \int\sum
\frac{m(n_1+n_2) - m(n_1)m(n_2)}{m(n_1)m(n_2)} \widehat{u_1}(n_1,\tau_1)
\widehat{u_2}(n_2,\tau_2)\Big\|_{L^2_{n_3}L_{\tau_3}^1}\\
&\lesssim N_3^2\big(\frac{N_2}{N}\big)^{1/2}
\frac{\delta^{\frac{3}{8}-}}{N_1 N_2}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}} \\
&\lesssim N^{-\frac{2}{3}+}\delta^{\frac{3}{8}-} N_{\rm max}^{0-}
\|u_1\|_{X^{1,1/2}}\|u_2\|_{X^{1,1/2}}.
\end{align*}

Next, the desired bound related to $\int_0^{\delta} E_7$ follows from
\begin{equation}\label{e.aE7}
\begin{aligned}
&\int_0^{\delta}\sum\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|
|n_1+n_2|\widehat{u_1}(n_1,t)\widehat{v_2}(n_2,t) |n_3|\widehat{u_3}(n_3,t)
\\&\lesssim  N^{-1+}\delta^{\frac{19}{24}-}\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}
\|u_3\|_{X^{1,1/2}}
\end{aligned}
\end{equation}

\noindent$\bullet$ $|n_1|\ll |n_2|\gtrsim N$. The multiplier is
 $\lesssim (|n_2|/N)^{1/2}$ so that
\begin{align*}
\int_0^{\delta}E_7&\lesssim \frac{1}{N^{1/2}} \int_0^{\delta}\sum |n_1+n_2|
\widehat{u_1}(n_1,t) |n_2|^{1/2} \widehat{v_2}(n_2,t) |n_3|\widehat{u_3}(n_3,t)
\\
&\lesssim  N^{-1}\delta^{\frac{19}{24}-}\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}
\|u_3\|_{X^{1,1/2}}.
\end{align*}

\noindent$\bullet$ $|n_1|\sim |n_2|\gtrsim N$. The multiplier is $\lesssim |n_2|/N$. Hence,
\begin{equation*}
\int_0^{\delta}E_7\lesssim N^{-1}\delta^{\frac{19}{24}-}
\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}\|u_3\|_{X^{1,1/2}}.
\end{equation*}
\item $|n_1|\gtrsim N$, $|n_2|\leq N$. The multiplier is again $\lesssim N_2/N$,
so that it can be estimated as above.


Now we turn to the term $\int_0^{\delta}E_8$. The objective is to show that
\begin{equation}\label{e.aE8}
\int_0^{\delta}\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|
|n_1+n_2|\prod_{j=1}^4\widehat{u_j}(n_j,t)\lesssim
N^{-1+}\delta^{\frac{1}{2}-}\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}
\end{equation}

\noindent$\bullet$ At least three frequencies are bigger than $N/3$.
We can assume $|n_1|\geq |n_2|$. The multiplier is bounded by
$N_{\rm max}/N$ so that
\begin{equation*}
\int_0^{\delta}E_8\lesssim \frac{N_{\rm max}}{N}
\frac{\delta^{\frac{1}{2}-}}{N_2 N_3 N_4}
\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}\lesssim
N^{-2+}\delta^{\frac{1}{2}-}N_{\rm max}^{0-}
\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}.
\end{equation*}

\noindent$\bullet$ Exactly two frequencies are bigger than $N/3$.
Without loss of generality, we suppose $|n_1|\sim |n_2|\gtrsim N$
and $|n_3|, |n_4|\ll N$. Since the multiplier satisfies
\begin{equation*}
\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|\lesssim
\big(\frac{N_{\rm max}}{N}\big)^{1-},
\end{equation*}
we get the bound
\begin{equation*}
\int_0^{\delta}E_8\lesssim \big(\frac{N_{\rm max}}{N}\big)^{1-}
\frac{\delta^{\frac{1}{2}-}}{N_2 N_3 N_4}
\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}\lesssim
N^{-1+}\delta^{\frac{1}{2}-}N_{\rm max}^{0-}
\prod_{j=1}^4\|u_j\|_{X^{1,1/2}}.
\end{equation*}


The contribution of $\int_0^{\delta}E_9$ is estimated if we prove that
\begin{equation}\label{e.aE9}
\begin{aligned}
&\int_0^{\delta}\Big|\frac{m(n_1+n_2)-m(n_1)m(n_2)}{m(n_1)m(n_2)}\Big|
\widehat{u_1}(n_1,t)\widehat{v_2}(n_2,t)\widehat{u_3}(n_3,t)\widehat{v_4}(n_4,t)
\\
&\lesssim N^{-2+}\delta^{\frac{7}{12}-}\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}
\|u_3\|_{X^{1,1/2}}\|v_4\|_{Y^{1,1/2}}.
\end{aligned}
\end{equation}
This follows since at least two frequencies are bigger than $N/3$
and the multiplier is always bounded by $(N_{\rm max}/N)^{1-}$, so
that
\begin{align*}
\int_0^{\delta}E_9&\lesssim \big(\frac{N_{\rm max}}{N}\big)^{1-} \|u_1\|_{L^4}
\|v_2\|_{L^4}\|u_3\|_{L^4}\|v_4\|_{L^4}\\
&\lesssim  \big(\frac{N_{\rm max}}{N}\big)^{1-}
\frac{\delta^{\frac{1}{4}+ \frac{1}{3}-}}{N_1 N_2 N_3 N_4}
\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}
\|u_3\|_{X^{1,1/2}}\|v_4\|_{Y^{1,1/2}}\\
&\lesssim  N^{-2+}\delta^{\frac{7}{12}-}\|u_1\|_{X^{1,1/2}}\|v_2\|_{Y^{1,1/2}}
\|u_3\|_{X^{1,1/2}}\|v_4\|_{Y^{1,1/2}}.
\end{align*}
Now, we treat the term $\int_0^{\delta}E_{10}$. It is sufficient to prove
\begin{equation}\label{e.aE10}
\begin{aligned}
&\int_0^{\delta}\sum
\Big|\frac{m(n_4+n_5+n_6)-m(n_4)m(n_5)m(n_6)}{m(n_4)m(n_5)m(n_6)}\Big|
\prod_{j=1}^6\widehat{u_j}(n_j,t)\\
&\lesssim  N^{-2+}\delta^{\frac{1}{2}-}\prod_{j=1}^6\|u_j\|_{X^1}.
\end{aligned}
\end{equation}
This follows easily from the facts that the multiplier is bounded by
$(N_{\rm max}/N)^{3/2}$, at least two frequencies are bigger than
$N/3$, say $|n_{i_1}|\geq |n_{i_2}|\gtrsim N$, the Strichartz bound
$X^{0,3/8}\subset L^4$ and the inclusion\footnote{This inclusion is
an easy consequence of Sobolev embedding.} $X^{\frac{1}{2}+}\subset
L_{xt}^{\infty}$. Indeed, if we combine these informations, it is
not hard to get
\begin{align*}
\int_0^{\delta}E_{10}&\lesssim \big(\frac{N_{\rm max}}{N}\big)^{\frac{3}{2}}
\frac{1}{N_{i_1} N_{i_2} N_{i_3}
N_{i_4}}\delta^{\frac{1}{2}-}\frac{1}{(N_{i_5}N_{i_6})^{1/2-}}
\prod_{j=1}^6\|u_j\|_{X^1}\\
&\lesssim N^{-2+}\delta^{\frac{1}{2}-}N_{\rm max}^{0-}
\prod_{j=1}^6\|u_j\|_{X^1}
\end{align*}

For the expression $\int_0^{\delta}E_{11}$, we use again that the
multiplier is bounded by $(N_{\rm max}/N)^{3/2}$, at least two
frequencies are bigger than $N/3$ (say $|n_{i_1}|\geq
|n_{i_2}|\gtrsim N$), the Strichartz bounds in lemma
\ref{l.Strichartz} and the inclusions
$X^{\frac{1}{2}+},Y^{\frac{1}{2}+}\subset L^{\infty}_{xt}$ to obtain
\begin{equation}
\begin{aligned}
&\int_0^{\delta}\sum
\Big|\frac{m(n_1+n_2+n_3)-m(n_1)m(n_2)m(n_3)}{m(n_1)m(n_2)m(n_3)}\Big|
\prod_{j=1}^4\widehat{u_j}(n_j,t)\widehat{v_5}(n_5,t)\\
&\lesssim \big(\frac{N_{\rm max}}{N}\big)^{\frac{3}{2}}
\frac{1}{N_{i_1} N_{i_2} N_{i_3} N_{i_4}}
\frac{\delta^{\frac{1}{2}-}}{N_{i_5}^{1/2-}}
\prod_{j=1}^4\|u_j\|_{X^1} \|v_5\|_{Y^1}\\
&\lesssim  N^{-2+}\delta^{\frac{1}{2}-}\prod_{j=1}^4\|u_j\|_{X^1} \|v_5\|_{Y^1}.
\end{aligned}
\end{equation}
The analysis of $\int_0^{\delta}E_{12}$ is similar to the
$\int_0^{\delta}E_{11}$. This completes the proof.
\end{proof}

\section{Global well-posedness below the energy space}\label{s.global}

In this section we combine the variant local well-posedness result in
proposition \ref{p.local} with the two almost conservation results in the
propositions \ref{p.al} and \ref{p.ae} to prove the theorem \ref{t.A}.

\begin{remark} \label{rmk5.1}\rm
Note that the spatial mean $\int_{\mathbb{T}} v(t,x) dx$ is
preserved during the evolution \eqref{e.nls-kdv}.
Thus, we can assume that the initial data $v_0$ has zero-mean, since otherwise
we make the change $w= v-\int_{\mathbb{T}}v_0 dx$ at the expense of two harmless linear
terms (namely, $u\int_{\mathbb{T}}v_0 dx$ and $\partial_x v \int_{\mathbb{T}}v_0$).
\end{remark}

The definition of the I-operator implies that the initial data satisfies
$\|Iu_0\|_{H^1}^2 +\|Iv_0\|_{H^1}^2\lesssim N^{2(1-s)}$ and
$\|Iu_0\|_{L^2}^2 +\|Iv_0\|_{L^2}^2\lesssim 1$. By the estimates (\ref{e.L1})
and (\ref{e.E3}), we get that $|L(Iu_0, Iv_0)|\lesssim N^{1-s}$ and $|E(Iu_0,
Iv_0)|\lesssim N^{2(1-s)}$.

Also, any bound for $L(Iu,Iv)$ and $E(Iu,Iv)$ of the form
$|L(Iu,Iv)|\lesssim N^{1-s}$ and $|E(Iu,Iv)|\lesssim N^{2(1-s)}$ implies that
$\|Iu\|_{L^2}^2\lesssim M$, $\|Iv\|_{L^2}^2\lesssim N^{1-s}$ and
$\|Iu\|_{H^1}^2+\|Iv\|_{H^1}^2\lesssim N^{2(1-s)}$.

Given a time $T$, if we can uniformly bound the $H^1$-norms of the solution at
times $t=\delta$, $t=2\delta$, etc., the local existence result in
proposition \ref{p.local} says that the solution can be extended up to any time
interval where such a uniform bound holds. On the other hand, given a time $T$,
if we can interact $T\delta^{-1}$ times the local existence result, the
solution exists in the time interval $[0,T]$. So, in view of the
propositions \ref{p.al} and \ref{p.ae}, it suffices to show
\begin{equation}\label{e.L}
(N^{-1+}\delta^{\frac{19}{24}-}N^{3(1-s)} +
N^{-2+}\delta^{\frac{1}{2}-}N^{4(1-s)})T\delta^{-1}\lesssim
N^{1-s}
\end{equation}
and
\begin{equation}\label{e.E}
\begin{aligned}
&\big\{(N^{-1+}\delta^{\frac{1}{6}-}
+ N^{-\frac{2}{3}+}\delta^{\frac{3}{8}-}
+N^{-\frac{3}{2}+}\delta^{\frac{1}{8}-})N^{3(1-s)}\\
&+N^{-1+}\delta^{\frac{1}{2}-}N^{4(1-s)}
+N^{-2+}\delta^{\frac{1}{2}-}N^{6(1-s)}\big\}\frac{T}{\delta}
\lesssim  N^{2(1-s)}
\end{aligned}
\end{equation}
At this point, we recall that the proposition \ref{p.local} says that
$\delta\sim N^{-\frac{16}{3}(1-s)-}$ if $\beta\neq 0$ and $\delta\sim
N^{-8(1-s)-}$ if $\beta=0$. Hence,

\noindent $\bullet$ $\beta\neq 0$. The condition (\ref{e.L}) holds for
\begin{equation*}
-1+\frac{5}{24}\frac{16}{3}(1-s)+3(1-s)< (1-s), \quad \text{i.e. },
s>19/28
\end{equation*}
and
\begin{equation*}
-2+\frac{1}{2}\frac{16}{3}(1-s)+4(1-s)< (1-s), \quad\text{i.e. },
s>11/17;
\end{equation*}
Similarly,  condition (\ref{e.E}) is satisfied if
\begin{gather*}
-1+\frac{5}{6}\frac{16}{3}(1-s) + 3(1-s)< 2(1-s), \quad\text{i.e. }, s>40/49;
\\
-\frac{2}{3}+\frac{5}{6}\frac{16}{3}(1-s)+3(1-s)< 2(1-s), \quad\text{i.e. },
s>11/13;
\\
-\frac{3}{2}+\frac{7}{8}\frac{16}{3}(1-s)+3(1-s)< 2(1-s), \quad\text{i.e. },
s>25/34;
\\
-1+\frac{1}{2}\frac{16}{3}(1-s)+4(1-s)< 2(1-s), \quad\text{i.e. },
s>11/14;
\\
-2+\frac{1}{2}\frac{16}{3}(1-s)+6(1-s)< 2(1-s), \quad\text{i.e. },
s>7/10.
\end{gather*}
Thus, we conclude that the non-resonant NLS-KdV system is globally well-posed
for any $s>11/13$.

\noindent $\bullet$ $\beta=0$. Condition (\ref{e.L}) is fulfilled when
\begin{equation*}
-1+\frac{5}{24}8(1-s)+3(1-s)< (1-s), \quad\text{i.e. },
s>8/11
\end{equation*}
and
\begin{equation*}
-2+\frac{1}{2}8(1-s)+4(1-s)< (1-s), \quad\text{i.e. },
s>5/7;
\end{equation*}
Similarly, the condition (\ref{e.E}) is verified for
\begin{gather*}
-1+\frac{5}{6}8(1-s) + 3(1-s)< 2(1-s), \quad\text{i.e. }, s>20/23;
\\
-\frac{2}{3}+\frac{5}{6}8(1-s)+3(1-s)< 2(1-s), \quad\text{i.e. },
s>8/9;
\\
-\frac{3}{2}+\frac{7}{8}8(1-s)+3(1-s)< 2(1-s), \quad\text{i.e. },
s>13/16;
\\
-1+\frac{1}{2}8(1-s)+4(1-s)< 2(1-s), \quad\text{i.e. },
s>5/6;
\\
-2+\frac{1}{2}8(1-s)+6(1-s)< 2(1-s), \quad\text{i.e. },
s>3/4.
\end{gather*}
Hence, we obtain that the resonant NLS-KdV system is globally well-posed
for any $s>8/9$.

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\end{document}