# Dynamics near the origin of the long range scattering for the one-
dimensional Schrödinger equation

Rémi Carles  Univ Rennes, CNRS
IRMAR - UMR 6625
F-35000 Rennes, France<EMAIL_ADDRESS>
###### Abstract.

We consider the cubic Schrödinger equation on the line, for which the
scattering theory requires modifications due to long range effects. We revisit
the construction of the modified wave operator, and recall the construction of
its inverse, in order to describe the asymptotic behavior of these operators
near the origin. At leading order, these operators, whose definition includes
a nonlinear modification in the phase compared to the linear dynamics,
correspond to the identity. We compute explicitly the first corrector in the
asymptotic expansion, and justify this expansion by error estimates.

This work was supported by Centre Henri Lebesgue, program ANR-11-LABX-0020-0.
A CC-BY public copyright license has been applied by the author to the present
document and will be applied to all subsequent versions up to the Author
Accepted Manuscript arising from this submission.

##  1\. Introduction

We consider the cubic Schrödinger equation on the line,

(1.1) $i{\partial}_{t}u+\frac{1}{2}{\partial}_{x}^{2}u=\lambda|u|^{2}u,\quad
x\in{\mathbb{R}},$

with $\lambda\in{\mathbb{R}}$. It is well known that in terms of scattering,
this equation corresponds to a borderline case, where long range effects
appear (see [1]). The goal of this paper is to analyze the modified scattering
map near the origin, as well as the modified wave operator and its inverse. We
first recall a few aspects of scattering theory for the nonlinear Schrödinger
equation in the short range case, then turn to the specificities of this long
range setting.

###  1.1. Short range scattering

Denote by $U(t)$ the Schrödinger group,

$U(t):=e^{i\frac{t}{2}{\partial}_{x}^{2}}.$

In view of the explicit formula for $U(t)$ as a convolution operator, we have

(1.2)
$U(t)f(x)\mathop{\sim}\limits_{t\rightarrow\pm\infty}e^{i\frac{x^{2}}{2t}}\frac{1}{(it)^{1/2}}\widehat{f}\left(\frac{x}{t}\right),$

where we normalize the Fourier transform as follows,

${\mathcal{F}}f(\xi)=\widehat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{{\mathbb{R}}}f(x)e^{-ix\xi}dx.$

See Lemma 2.2 for a more precise statement regarding (1.2). Consider, for the
sake of comparison with (1.1), the case of a quintic, defocusing nonlinearity

(1.3) $i{\partial}_{t}u+\frac{1}{2}{\partial}_{x}^{2}u=|u|^{4}u,\quad
x\in{\mathbb{R}}.$

The discussion for, e.g., the two-dimensional or the three-dimensional cubic
Schrödinger equation would be similar. Given

(1.4) $u_{-}\in\Sigma:=\left\\{f\in H^{1}({\mathbb{R}});\
\|f\|_{\Sigma}:=\|f\|_{L^{2}}+\|{\partial}_{x}f\|_{L^{2}}+\|xf\|_{L^{2}}<\infty\right\\},$

there exists a unique $u_{0}\in\Sigma=H^{1}\cap\mathcal{F}(H^{1})$ such that
the (unique, global) solution $u$ to (1.3) with $u_{\mid t=0}=u_{0}$ satisfies

$\|U(-t)u(t)-u_{-}\|_{\Sigma}\mathop{\longrightarrow}\limits_{t\rightarrow-\infty}0.$

We recall that $U(t)$ is unitary on $H^{1}$, but not on $\mathcal{F}(H^{1})$,
this is why the quantity measured above is not $u(t)-U(t)u_{-}$. The map
$u_{-}\mapsto u_{0}$ is classically referred to as _wave operator_ (see e.g.
[6]).

Conversely, given $u_{0}\in\Sigma$, there exists $u_{+}\in\Sigma$ such that

$\|U(-t)u(t)-u_{+}\|_{\Sigma}\mathop{\longrightarrow}\limits_{t\rightarrow+\infty}0.$

The map $u_{-}\mapsto u_{+}$ is called the _scattering operator_. It can be
defined for other (defocusing) nonlinearities, but strictly supercubic, in
view of [1], where it is proved that, typically for (1.1), it is not possible
to compare the nonlinear dynamics with the linear one for large time (see also
[6]). We will see in the next subsection that in the cubic case (1.1),
nontrivial long range effects must be taken into account, and that these
effects are explicit.

In general, rather little is known regarding properties of the scattering map
$S$. For instance, it is proven in [3] that for smooth, power-like
nonlinearities such that short range scattering is known, the wave and
scattering operators are analytic. The formula for the associated Taylor
series is given, and the formula differs whether the expansion is considered
at the origin or at a nontrivial state. Such asymptotic expansions (not
necessarily in the analytic case) have proven useful in the context of inverse
problems, see e.g. [20, 21] and references therein.

###  1.2. Long range case: modified scattering

The picture is different in the case of (1.1): the nonlinear dynamics cannot
be compared to the linear one, unless one considers the trivial case $u\equiv
0$ ([1]), so a modified scattering theory has been developed in order to
describe the large time behavior of $u$, sharing some similarities with the
linear case, presented in e.g. [5].

Given asymptotic states $u_{-}$ and $u_{+}$, we introduce the long range phase
corrections (different whether $t\to-\infty$ of $t\to+\infty$),

(1.5)
$S_{\pm}(t,x):=\mp\lambda\left|\widehat{u}_{\pm}\left(\frac{x}{t}\right)\right|^{2}\log|t|.$

Loosely speaking, the existence of modified wave operator reads as follows:
given $u_{-}$ (sufficiently small), there exists a solution $u$ to (1.1) such
that

$u(t,x)\mathop{\sim}\limits_{t\rightarrow-\infty}e^{iS_{-}(t,x)}U(t)u_{-}(x)\mathop{\sim}\limits_{t\rightarrow-\infty}e^{iS_{-}(t,x)+i\frac{x^{2}}{2t}}\frac{1}{(it)^{1/2}}\widehat{u}_{-}\left(\frac{x}{t}\right),$

where the second approximation stems from (1.2). Note that the phase
modification $S_{-}$ is by no means negligible as $t\to-\infty$: it accounts
for long range effects, as established initially in [26]. We denote by
$u_{\mid t=0}={W_{-}^{\rm mod}}(u-)$ the modified wave operator.

The modified asymptotic completeness is similar: given $u_{0}$ (sufficiently
small), there exists $u_{+}$ such that the solution $u$ to (1.1) with $u_{\mid
t=0}=u_{0}$ satisfies

$u(t)\mathop{\sim}\limits_{t\rightarrow+\infty}e^{iS_{+}(t)}U(t)u_{+}.$

Such a result was proven initially in [14]. The modified scattering map is
given by ${S^{\rm mod}}(u_{-})=u_{+}$. At this stage, we have not addressed
the function spaces in which the above asymptotics have been proven.

In [26], the existence of modified wave operators was established for
$u_{-}\in\mathcal{F}(H^{2})$, with $\|\widehat{u}_{-}\|_{L^{\infty}}$
sufficiently small, and the solution $u$ to (1.1) has an $L^{2}$ regularity,
$u_{0}\in L^{2}({\mathbb{R}})$. We emphasize, as the notation will be used
many times, that the space $\mathcal{F}(H^{s})$ is characterized by

$\mathcal{F}(H^{s})=\\{f\in{\mathcal{S}}^{\prime}(R),\
\|f\|_{\mathcal{F}(H^{s})}^{2}:=\int_{{\mathbb{R}}}\left\langle
x\right\rangle^{2s}|f(x)|^{2}dx<\infty\\},\quad\left\langle
x\right\rangle=\sqrt{1+x^{2}}.$

The result of [26] also addresses the case where $u_{-}\in{\mathcal{H}}$
(defined below, see (1.6)) where, provided again that
$\|\widehat{u}_{-}\|_{L^{\infty}}$ sufficiently small, the solution $u$ to
(1.1) has an $H^{1}$ regularity, and convergence holds in this space.

In [14], the asymptotic completeness was proven for $u_{0}\in
H^{\gamma}\cap\mathcal{F}(H^{\gamma})$ with $\gamma>1/2$ and
$\|u_{0}\|_{H^{\gamma}\cap\mathcal{F}(H^{\gamma})}$ sufficiently small. The
obtained asymptotic state $u_{+}$ is such that $\widehat{u}_{+}\in L^{2}\cap
L^{\infty}$.

Denote

(1.6) $\displaystyle{{\mathcal{H}}}$ $\displaystyle:=\\{f\in
H^{1}({\mathbb{R}});\ \left\langle x\right\rangle{\partial}_{x}f,\left\langle
x\right\rangle^{3}f\in L^{2}({\mathbb{R}})\\}$
$\displaystyle=\\{f\in{\mathcal{S}}^{\prime}({\mathbb{R}});\|f\|_{{\mathcal{H}}}:=\|\left\langle
x\right\rangle{\partial}_{x}f\|_{L^{2}}+\|\left\langle
x\right\rangle^{3}f\|_{L^{2}}<\infty\\}.$

In [2], the main result of [26] was adapted for $u_{-}\in{\mathcal{H}}$ with
$\|\widehat{u}_{-}\|_{L^{\infty}}$ sufficiently small, and the regularity of
the solution $u$ to (1.1) was proven to be at least $\Sigma$, hence
$u_{0}\in\Sigma$, making it possible to connect this result with the
asymptotic completeness from [14], thus defining a map $u_{-}\mapsto u_{+}$
from (a subset of) ${\mathcal{H}}$ to $L^{2}\cap L^{\infty}$. When invoking
the modified scattering operator, we refer to this notion.

This gap in regularity between $u_{-}$ and $u_{+}$ was considerably diminished
in [15], where, with the same notations as above, the authors consider the
setting (along with smallness conditions)

$u_{-}\in\mathcal{F}(H^{\alpha}),\quad u_{0}\in\mathcal{F}(H^{\beta}),\quad
u_{+}\in\mathcal{F}(H^{\delta}),$

with the constraints $1/2<\delta<\beta<\alpha<1$, allowing these three indices
to be arbitrarily close one from another. This is achieved by adapting the
notion of (modified) asymptotic completeness in order to avoid loss of
differentiability issues, in space dimensions two and three, where the
borderline nonlinearity in terms of scattering is $|u|^{2/d}u$.

We also emphasize that there are many references addressing the theory of long
range scattering for nonlinear Schrödinger equations (e.g. [4, 17, 18, 23, 24,
25]), as well as for Schrödinger-like equations, like Hartree equation (e.g.
[7, 11, 12, 27]), derivative nonlinear Schrödinger equation (e.g. [16, 13]),
Maxwell-Schrödinger system (e.g. [9, 8]), wave-Schrödinger system (e.g. [10]),
not to mention other dispersive equations.

###  1.3. Main result

In the same spirit as what has been achieved for the wave and scattering
operators in the short range case, we consider the asymptotic behavior of the
modified wave and scattering operators, with two restrictions compared to [3].
First, we shall confine ourselves to the asymptotic expansion near the origin.
Second, we compute only the first two terms of this asymptotic expansion. We
will see that this already requires some amount of work, but that the same
method should provide some, if not all, higher order terms in the expansion at
the origin. On the other hand, the description of these operators near a
nontrivial state certainly requires a different approach.

We emphasize that the question addressed here is different from the (higher
order) asymptotic expansion of the large time behavior of $u(t)$, as studied
in [22]. Our main results are gathered in the following statement:

######  Theorem 1.1.

Let $v_{-}\in{\mathcal{H}}$ and $v_{0}\in\Sigma$.

  * •

For any $0<\eta<2$, we have, in $\Sigma$ and as $\varepsilon\to 0$,

${W_{-}^{\rm mod}}(\varepsilon v_{-})=\varepsilon
v_{-}+\varepsilon^{3}w_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right),$

where $w_{2}\in\Sigma$ is defined by

$\displaystyle w_{2}$
$\displaystyle=-i\lambda\int_{-\infty}^{-1}\left(U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)+\frac{1}{|\tau|}\mathcal{F}^{-1}\left(|\widehat{v}_{-}|^{2}\widehat{v}_{-}\right)\right)d\tau$
$\displaystyle\quad-i\lambda\int_{-1}^{0}U(t-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau.$

  * •

For any $0<\eta<2$, we have, in $L^{2}$ and as $\varepsilon\to 0$,

$\left(W_{+}^{\rm mod}\right)^{-1}(\varepsilon v_{0})=\varepsilon
v_{0}+\varepsilon^{3}\mu_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right),$

where $\mu_{2}=\mu_{2}(v_{0})\in L^{2}$ is defined by

$\displaystyle\mu_{2}$
$\displaystyle=-i\lambda\int_{0}^{1}\left(U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)\right)ds$
$\displaystyle\quad+\lambda\int_{1}^{\infty}\left(M(-\tau)\mathcal{F}^{-1}\left(\left|\widehat{M(\tau)v_{0}}\right|^{2}\widehat{M(\tau)v_{0}}\right)-\mathcal{F}^{-1}\left(|\widehat{v}_{0}|^{2}\widehat{v}_{0}\right)\right)\frac{d\tau}{\tau},$

and $M(t)$ stands for the multiplication by $e^{i\frac{x^{2}}{2t}}$.

  * •

For any $0<\eta<2$, we have, in $L^{2}$ and as $\varepsilon\to 0$,

${S^{\rm mod}}(\varepsilon v_{-})=\varepsilon
v_{-}+\varepsilon^{3}\nu_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right),$

where $\nu_{2}\in L^{2}$ is defined by $\nu_{2}=w_{2}+\mu_{2}(v_{-}),$ where
$w_{2}$ is given by the first point, and $\mu_{2}$ by the second.

######  Remark 1.2.

The functions $\mu_{2}$ and $\nu_{2}$ are more regular than merely $L^{2}$, as
we will see in Section 6 that they belong to
$H^{\gamma}\cap\mathcal{F}(H^{\gamma})$ for any $0<\gamma<1$.

It may be surprising that the leading order behavior of the modified wave and
scattering operators at the origin is the identity: we recall that from their
definition, these operators are already nontrivial, and account for long range
effects. We also emphasize that the first corrector has a more involved
expression than in the short range case, a case where we would simply have
(typically for the two-dimensional and three-dimensional cubic Schrödinger
equations, see e.g. [3])

$\displaystyle w_{2}$
$\displaystyle=-i\lambda\int_{-\infty}^{0}\left(U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)\right)ds,$
$\displaystyle\mu_{2}$
$\displaystyle=-i\lambda\int_{0}^{\infty}\left(U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)\right)ds.$

Note that in the (one-dimensional) long range case, these integrals diverge.

As evoked above, in principle, the method of proof that we present allows to
compute (some) higher order terms in the asymptotic expansions near the
origin.

###  1.4. Outline

In Section 2, we recall several properties which are classical in the context
of scattering theory for nonlinear Schrödinger equations, and which are of
constant use in this paper. In Section 3, we revisit the construction of the
modified wave operator ${W_{-}^{\rm mod}}$, in such a way that the leading
order behavior of this operator at the origin, presented in Section 4, is
rather straightforward. In Section 4, we also consider the first corrector in
the asymptotic expansion of ${W_{-}^{\rm mod}}$ at the origin, an aspect which
requires some extra work. In Section 5, we recall the modified asymptotic
completeness result established in [14], and infer the leading of behavior of
${S^{\rm mod}}$ at the origin. The first corrector is derived in Section 6,
where the last two error estimates announced in Theorem 1.1 are proved.

###  1.5. Notations

We recall the classical factorization of the Schrödinger group,

$U(t)=e^{i\frac{t}{2}{\partial}_{x}^{2}}=M(t)D(t)\mathcal{F}M(t),$

where the multiplication $M(t)$, the dilation $D(t)$ and the Fourier transform
$\mathcal{F}$ are defined by

(1.7) $\displaystyle M(t)=e^{i\frac{x^{2}}{2t}},\quad
D(t)f(x)=\frac{1}{(it)^{1/2}}f\left(\frac{x}{t}\right),$
$\displaystyle{\mathcal{F}}f(\xi)=\widehat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int_{{\mathbb{R}}}f(x)e^{-ix\xi}dx.$

Note that each of these three operators is unitary on $L^{2}({\mathbb{R}})$,
and that (1.2) reads $U(t)\approx M(t)D(t)\mathcal{F}$, see also Lemma 2.2. We
recall that the space $\Sigma$ is defined by

$\Sigma=H^{1}\cap\mathcal{F}(H^{1})=\left\\{f\in H^{1}({\mathbb{R}});\
\|f\|_{\Sigma}:=\|f\|_{L^{2}}+\|{\partial}_{x}f\|_{L^{2}}+\|xf\|_{L^{2}}<\infty\right\\}.$

We note that $\Sigma$ is a Banach algebra, invariant under the Fourier
transform, and such that $\Sigma\hookrightarrow L^{1}\cap L^{\infty}$. In
addition, $\Sigma$ is invariant under the action of $U(t)$, for any
$t\in{\mathbb{R}}$.

For functions $f^{\varepsilon},g^{\varepsilon}\geqslant 0$ depending on time
$t$ and $\varepsilon$, the notation

$f^{\varepsilon}\lesssim g^{\varepsilon}$

means that there exists $C$ independent of $t$ and $\varepsilon$ such that

$f^{\varepsilon}\leqslant Cg^{\varepsilon}.$

##  2\. Technical preliminaries

In this section, we gather classical estimates which can be found in several
references cited in the introduction.

######  Lemma 2.1.

The operator

$J(t)=x+it{\partial}_{x}$

satisfies the following properties:

  * •

$J(t)=U(t)xU(-t)$, and therefore $J$ commutes with the linear part of (1.1),

(2.1) $\left[J(t),i{\partial}_{t}+\frac{1}{2}{\partial}_{x}^{2}\right]=0\,.$

  * •

It can be factorized as

$J(t)=it\,e^{i\frac{x^{2}}{2t}}{\partial}_{x}\Big{(}e^{-i\frac{x^{2}}{2t}}\,\cdot\Big{)}\,.$

As a consequence, $J$ yields weighted Gagliardo-Nirenberg inequalities. For
$2\leqslant r\leqslant\infty$, there exists $C(r)$ depending only on $r$ such
that

(2.2)
$\left\|f\right\|_{L^{r}}\leqslant\frac{C(r)}{|t|^{\delta(r)}}\left\|f\right\|_{L^{2}}^{1-\delta(r)}\left\|J(t)f\right\|_{L^{2}}^{\delta(r)},\quad\delta(r):=\frac{1}{2}-\frac{1}{r}\,.$

Also, if $F(z)=G(|z|^{2})z$ is $C^{1}$, then $J(t)$ acts like a derivative on
$F(w)$:

(2.3)
$J(t)\left(F(w)\right)={\partial}_{z}F(w)J(t)w-{\partial}_{\overline{z}}F(w)\overline{J(t)w}\,.$

######  Lemma 2.2.

Denote by

${\mathcal{R}}(t)=M(t)D(t)\mathcal{F}\left(M(t)-1\right)\mathcal{F}^{-1},$

where the above terms are defined in (1.7). The following estimates hold for
$|t|\geqslant 1$:

  1. (1)

For all $s>1/2$ and $0\leqslant\theta\leqslant 1$,

$\|{\mathcal{R}}(t)f\|_{L^{\infty}_{x}}\lesssim\frac{1}{|t|^{1/2+\theta}}\|f\|_{H^{s+2\theta}},\quad\forall
f\in{\mathcal{S}}({\mathbb{R}}).$

  2. (2)

For all $0\leqslant\theta\leqslant 1$,

$\|{\mathcal{R}}(t)f\|_{L^{2}_{x}}\lesssim\frac{1}{|t|^{\theta}}\|f\|_{H^{2\theta}},\quad\forall
f\in{\mathcal{S}}({\mathbb{R}}).$

  3. (3)

For all $0\leqslant\theta\leqslant 1$,

$\|J(t){\mathcal{R}}(t)f\|_{L^{2}_{x}}\lesssim\frac{1}{|t|^{\theta}}\|f\|_{H^{1+2\theta}},\quad\forall
f\in{\mathcal{S}}({\mathbb{R}}).$

  4. (4)

We have

$\|{\partial}_{x}{\mathcal{R}}(t)f\|_{L^{2}_{x}}\lesssim\frac{1}{|t|}\|f\|_{H^{1}}+\frac{1}{|t|}\|xf\|_{H^{2}},\quad\forall
f\in{\mathcal{S}}({\mathbb{R}}).$

######  Remark 2.3.

We emphasize that ${\mathcal{R}}(t)$ does not map $L^{\infty}({\mathbb{R}})$
into itself, since
$\mathcal{F}\left(M(t)-1\right)\mathcal{F}^{-1}=U(\frac{1}{t})-1$ and
$U(\frac{1}{t})(e^{-it\frac{x^{2}}{2}})=\delta_{0}$.

Even though such estimates can be found in the existing literature, we give a
proof, as the ideas will be resumed when $M(t)-1$ is present.

###### Proof.

In view of the definition of ${\mathcal{R}}$, we readily have

$\|{\mathcal{R}}(t)f\|_{L^{\infty}_{x}}=\frac{1}{|t|^{1/2}}\|\mathcal{F}(M(t)-1)\mathcal{F}^{-1}f\|_{L^{\infty}}$

Using Hausdorff-Young inequality and the general estimate

(2.4)
$|M(t)-1|=2\left|\sin\left(\frac{x^{2}}{4t}\right)\right|\lesssim\left|\frac{x^{2}}{t}\right|^{\theta},\quad\forall\,0\leqslant\theta\leqslant
1,$

we find

$\|{\mathcal{R}}(t)f\|_{L^{\infty}_{x}}\lesssim\frac{1}{|t|^{1/2}}\left\|\left|\frac{x^{2}}{t}\right|^{\theta}\mathcal{F}^{-1}f\right\|_{L^{1}}.$

Using the easy property $\mathcal{F}(H^{s})\hookrightarrow L^{1}$ for $s>1/2$
(the same is true on ${\mathbb{R}}^{d}$ provided that $s>d/2$, from Cauchy-
Schwarz inequality),

$\|{\mathcal{R}}(t)f\|_{L^{\infty}_{x}}\lesssim\frac{1}{|t|^{1/2+\theta}}\left\|\left\langle
x\right\rangle^{s}|x|^{2\theta}\mathcal{F}^{-1}f\right\|_{L^{2}}\lesssim\frac{1}{|t|^{1/2+\theta}}\|f\|_{H^{s+2\theta}},$

hence the first inequality. For the second one, since $D(t)$ is unitary on
$L^{2}$, Plancherel formula and (2.4) yield

$\|{\mathcal{R}}(t)f\|_{L^{2}_{x}}=\|(M(t)-1)\mathcal{F}^{-1}f\|_{L^{2}}\lesssim\frac{1}{|t|^{\theta}}\left\||x|^{2\theta}\mathcal{F}^{-1}f\right\|_{L^{2}}\lesssim\frac{1}{|t|^{\theta}}\|f\|_{H^{2\theta}}.$

For the third inequality, we use the formula $J(t)=U(t)xU(-t)$, along with the
factorization

(2.5) $U(-t)=iM(-t)\mathcal{F}^{-1}D\left(\frac{1}{t}\right)M(-t),$

to obtain

$\|J(t){\mathcal{R}}(t)f\|_{L^{2}_{x}}=\|x(M(t)-1)\mathcal{F}^{-1}f\|_{L^{2}}\lesssim\frac{1}{|t|^{\theta}}\left\||x|^{1+2\theta}\mathcal{F}^{-1}f\right\|_{L^{2}}\lesssim\frac{1}{|t|^{\theta}}\|f\|_{H^{1+2\theta}}.$

To estimate ${\partial}_{x}{\mathcal{R}}(t)f$, two terms appear, whether the
derivative hits the first factor $M(t)$ or not, and we have

$\|{\partial}_{x}{\mathcal{R}}(t)f\|_{L^{2}_{x}}\leqslant\left\|\frac{x}{t}D(t)\mathcal{F}(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}+\frac{1}{|t|}\left\|{\partial}_{x}\mathcal{F}(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}.$

The first term on the right hand side is equal to

$\displaystyle\left\|x\mathcal{F}(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}$
$\displaystyle=\left\|{\partial}_{x}(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}$
$\displaystyle\leqslant\left\|\frac{x}{t}\mathcal{F}^{-1}f\right\|_{L^{2}}+\left\|(M(t)-1){\partial}_{x}\mathcal{F}^{-1}f\right\|_{L^{2}}$
$\displaystyle\lesssim\frac{1}{|t|}\|f\|_{H^{1}}+\frac{1}{|t|^{\theta}}\|xf\|_{H^{2\theta}},$

for any $\theta\in[0,1]$. Note that

$\frac{1}{|t|}\left\|{\partial}_{x}\mathcal{F}(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}=\frac{1}{|t|}\left\|x(M(t)-1)\mathcal{F}^{-1}f\right\|_{L^{2}}\lesssim\frac{1}{|t|}\left\|x\mathcal{F}^{-1}f\right\|_{L^{2}},$

hence the result, by choosing $\theta=1$. ∎

##  3\. Modified wave operator

A key aspect in the proofs of the existence of modified wave operators for
(1.1), in particular to get minimal regularity assumptions on the asymptotic
state $u_{-}$, is to consider a suitable approximate solution near
$t=-\infty$. Introduce two such approximate solutions, $u_{1}$ and $u_{2}$,
defined for $t\leqslant-10$, given by

$u_{1}(t,x):=\frac{1}{(it)^{1/2}}e^{i\frac{x^{2}}{2t}}{\widehat{u}_{-}}\left(\frac{x}{t}\right)e^{iS_{-}(t,x)}=M(t)D(t)\widehat{w}(t,x),$

where

$\widehat{w}(t):={\widehat{u}_{-}}e^{i\lambda|{\widehat{u}_{-}}|^{2}\log|t|},$

and

$u_{2}(t):=U(t)\mathcal{F}^{-1}\widehat{w}.$

As $t\to-\infty$, $u_{1}$ and $u_{2}$ are close, in view of Lemma 2.2, since
$u_{2}(t)-u_{1}(t)={\mathcal{R}}(t)\widehat{w}$. Our goal is to keep track of
the smallness of the asymptotic state as precisely as possible, in the sense
that we consider $u_{-}^{\varepsilon}=\varepsilon v_{-}$ for some fixed
$v_{-}$, assume that $\varepsilon>0$ is small, and we want to get larger
powers of $\varepsilon$ in the error terms whenever we can. It turns out that
apparently, $u_{2}$ is a better candidate than $u_{1}$ in this direction. More
explicitly, the Duhamel formula for $u-u_{1}$ (see [15]) contains a linear
term ($R\widehat{w}$), which we want to remove. We first gather some estimates
on $u_{2}$.

######  Lemma 3.1.

Let ${u_{-}}\in{\mathcal{H}}$: $\|u_{2}(t)\|_{L^{2}}=\|{u_{-}}\|_{L^{2}}$ for
all $t\leqslant-10$. There exists $C>0$ independent of ${u_{-}}$ such that for
all $t\leqslant-10$,

$\displaystyle\|u_{2}(t)\|_{L^{\infty}}\leqslant
C\frac{\|{\widehat{u}_{-}}\|_{H^{1}}}{\sqrt{|t|}}\left(1+\|{\widehat{u}_{-}}\|_{H^{1}}^{2}\right),$
$\displaystyle\|{\partial}_{x}u_{2}(t)\|_{L^{2}}\leqslant
C\frac{\|{u_{-}}\|_{{\mathcal{H}}}}{|t|}\left(1+(\log|t|)^{3}\|{u_{-}}\|_{{\mathcal{H}}}^{6}\right),$
$\displaystyle\|J(t)u_{2}(t)\|_{L^{2}}\leqslant
C\|{u_{-}}\|_{{\mathcal{H}}}\left(1+\|{u_{-}}\|_{{\mathcal{H}}}^{6}\right).$

###### Proof.

The conservation of the $L^{2}$-norm is obvious. For the other norms, we write
$u_{2}=u_{1}+R\widehat{w}$, and estimate the last two terms independently.

The estimate

$\|u_{1}(t)\|_{L^{\infty}}\leqslant\frac{\|{\widehat{u}_{-}}\|_{L^{\infty}}}{\sqrt{|t|}}$

is straightforward, so we focus on the control of
$\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}$. If we use the first point
of Lemma 2.2 with $\theta=0$ and $s>1/2$, the phase factor causes the
appearance of a $\log|t|$ term, since the $H^{1}$-norm of $\widehat{w}$ is
involved. So we rather take $\theta>0$ and $s>1/2$ such that $s+2\theta=1$,
and so, for $t\leqslant-10$,

$\displaystyle\|{\mathcal{R}}(t)\widehat{w}\|_{L^{\infty}}\lesssim\frac{\|\widehat{w}\|_{H^{1}}}{|t|^{1/2+\theta}}$
$\displaystyle\lesssim\frac{1}{|t|^{1/2+\theta}}\left(\|{\widehat{u}_{-}}\|_{H^{1}}+(\log|t|)\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\widehat{u}_{-}}\|_{H^{1}}\right)$
$\displaystyle\lesssim\frac{1}{\sqrt{|t|}}\left(\|{\widehat{u}_{-}}\|_{H^{1}}+\|{\widehat{u}_{-}}\|_{H^{1}}^{3}\right),$

where the logarithmic factor was left out since $\theta>0$. It is actually at
the level of this estimate that the $H^{1}$-norm of ${\widehat{u}_{-}}$
appears, instead of merely its $L^{\infty}$-norm.

In view of Lemma 2.1,

$\|U(-t){\mathcal{R}}\widehat{w}\|_{\Sigma}=\|{\mathcal{R}}\widehat{w}\|_{L^{2}}+\|{\partial}_{x}\c{R}\widehat{w}\|_{L^{2}}+\|J(t){\mathcal{R}}\widehat{w}\|_{L^{2}},$

and Lemma 2.2 yields

$\|U(-t){\mathcal{R}}(t)\widehat{w}\|_{\Sigma}\lesssim\frac{1}{|t|}\|\widehat{w}\|_{H^{3}}+\frac{1}{|t|}\|x\widehat{w}\|_{H^{2}}.$

Using the fact that $H^{s}({\mathbb{R}})$ is an algebra for $s>1/2$, we infer

$\displaystyle\|U(-t){\mathcal{R}}(t)\widehat{w}\|_{\Sigma}$
$\displaystyle\lesssim\frac{1}{|t|}\left(\|{\widehat{u}_{-}}\|_{H^{3}}+(\log|t|)^{3}\|{\widehat{u}_{-}}\|_{H^{3}}^{7}\right)$
$\displaystyle\quad+\frac{1}{|t|}\left(\|x{\widehat{u}_{-}}\|_{H^{2}}+(\log|t|)^{2}\|x{\widehat{u}_{-}}\|_{H^{2}}^{5}\right)$
$\displaystyle\lesssim\frac{1}{|t|}\left(\|{u_{-}}\|_{{\mathcal{H}}}+\sum_{j=0,1}(\log|t|)^{2+j}\|{u_{-}}\|_{{\mathcal{H}}}^{5+2j}\right).$

The expression of $u_{1}$ yields directly

$\|{\partial}_{x}u_{1}(t)\|_{L^{2}}\leqslant\frac{1}{|t|}\|{\partial}_{x}{\widehat{u}_{-}}\|_{L^{2}}+|\lambda|\frac{\log|t|}{|t|}\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\partial}_{x}{\widehat{u}_{-}}\|_{L^{2}}.$

In view of the factorization for $J$ stated in Lemma 2.1,

$\displaystyle J(t)u_{1}(t)$
$\displaystyle=itM(t){\partial}_{x}\left(D(t){\widehat{u}_{-}}e^{iS_{-}(t)}\right)$
$\displaystyle=e^{i\frac{x^{2}}{2t}+iS_{-}(t)}\left(iD(t)({\partial}_{x}{\widehat{u}_{-}})-\lambda
D(t){\widehat{u}_{-}}\times\frac{\log|t|}{t}\times{\partial}_{x}\left|{\widehat{u}_{-}}\right|^{2}\left(\frac{x}{t}\right)\right),$

thus

$\|J(t)u_{1}(t)\|_{L^{2}}\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}+\frac{\log|t|}{|t|}\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\widehat{u}_{-}}\|_{H^{1}}.$

The lemma follows, by noticing that $\frac{\log|t|}{|t|}$ is uniformly bounded
for $t\leqslant-10$, and that $J(t){\mathcal{R}}(t)\widehat{w}$ is bounded in
$L^{2}$. ∎

######  Proposition 3.2.

There exist $\delta_{0}>0$, $C$, and a polynomial $P$, such that the following
holds. For any $u_{-}\in L^{2}$ with ${u_{-}}\in{\mathcal{H}}$, where
${\mathcal{H}}$ is defined in (1.6), and such that
$\|{\widehat{u}_{-}}\|_{H^{1}}\leqslant\delta_{0}$, there exists a unique
$u_{0}\in\Sigma$ such that the solution $u\in C({\mathbb{R}},\Sigma)$ to (1.1)
with $u_{\mid t=0}=u_{0}$ satisfies, for $t\leqslant-10$,

$\left\|U(-t)\left(u(t)-u_{2}(t)\right)\right\|_{\Sigma}\leqslant
C\|{u_{-}}\|_{{\mathcal{H}}}^{3}P\left(\|{u_{-}}\|_{{\mathcal{H}}}\right)\frac{(\log|t|)^{3}}{|t|}.$

######  Remark 3.3.

The choice of the approximate solution $u_{2}$ is tailored to ensure that,
with the proof presented below, the error term is superlinear in some norm of
$u_{-}$, uniformly in the limit $t\to-\infty$. On the other hand, the
smallness assumption, which is a consequence of the first estimate in Lemma
3.1, is stronger than in the cited references, since Sobolev embedding yields
$H^{1}({\mathbb{R}})\subset L^{\infty}({\mathbb{R}})$.

###### Proof.

Like in [26] or [15], the proof relies on a fixed point argument near
$t=-\infty$. Resuming the computations from [15] (and keeping the same
notations),

$i{\partial}_{t}\widehat{w}=\frac{\lambda}{t}|\widehat{w}|^{2}\widehat{w},$

and so we check that $u_{2}$ solves the equation

$i{\partial}_{t}\mathcal{F}U(-t)u_{2}=\lambda\mathcal{F}U(-t)\left(\frac{1}{t}U(t)\mathcal{F}^{-1}\left(|\widehat{w}|^{2}\widehat{w}\right)\right).$

Therefore, we want to solve

$i{\partial}_{t}\mathcal{F}U(-t)\left(u-u_{2}\right)=\lambda\mathcal{F}U(-t)\left(|u|^{2}u-\frac{1}{t}U(t)\mathcal{F}^{-1}\left(|\widehat{w}|^{2}\widehat{w}\right)\right).$

We insert the term $|u_{2}|^{2}u_{2}$ into the above equation, and use the
identity

$\displaystyle\frac{1}{t}U(t)\mathcal{F}^{-1}\left(|\widehat{w}|^{2}\widehat{w}\right)$
$\displaystyle=\frac{1}{t}MD\mathcal{F}M\mathcal{F}^{-1}\left(|\widehat{w}|^{2}\widehat{w}\right)$
$\displaystyle=\frac{1}{t}{\mathcal{R}}(t)\left(|\widehat{w}|^{2}\widehat{w}\right)+\frac{1}{t}M(t)D(t)\left(|\widehat{w}|^{2}\widehat{w}\right),$

with

$\displaystyle\frac{1}{t}M(t)D(t)\left(|\widehat{w}|^{2}\widehat{w}\right)=M(t)|M(t)D(t)\widehat{w}|^{2}D(t)\widehat{w}=|u_{1}|^{2}u_{1},$

so Duhamel’s formula reads, along with the requirement
$U(-t)\left(u(t)-u_{2}(t)\right)\to 0$ as $t\to-\infty$,

(3.1) $\displaystyle u(t)=u_{2}(t)$
$\displaystyle-i\lambda\int_{-\infty}^{t}U(t-\tau)\left(|u|^{2}u-|u_{2}|^{2}u_{2}\right)(\tau)d\tau$
$\displaystyle\quad+i\lambda\int_{-\infty}^{t}U(t-\tau)\left(|u_{2}|^{2}u_{2}-|u_{1}|^{2}u_{1}\right)(\tau)d\tau$
$\displaystyle\quad-i\lambda\int_{-\infty}^{t}U(t-\tau)\left(\frac{1}{\tau}{\mathcal{R}}(\tau)\left(|\widehat{w}|^{2}\widehat{w}\right)\right)(\tau)d\tau.$

The last two lines correspond to source terms, involving only the various
approximate solutions, and can be estimated thanks to Lemma 2.2, as
$u_{2}=u_{1}+{\mathcal{R}}\widehat{w}$.

Denote by $\Phi(u)=u_{2}+\Phi_{1}(u)+\Phi_{2}+\Phi_{3}$ the right hand side of
(3.1), where $\Phi_{j}$ corresponds to the $j$-th line (note that $\Phi_{2}$
and $\Phi_{3}$ do not depend on $u$). We have

$\displaystyle\left|{\partial}_{x}\left(|u_{2}|^{2}u_{2}-|u_{1}|^{2}u_{1}\right)\right|$
$\displaystyle=\left|{\partial}_{x}\left(|{\mathcal{R}}\widehat{w}+u_{1}|^{2}({\mathcal{R}}\widehat{w}+u_{1})-|u_{1}|^{2}u_{1}\right)\right|$
$\displaystyle\leqslant\left(|{\mathcal{R}}\widehat{w}|^{2}+|u_{1}|^{2}\right)|{\partial}_{x}{\mathcal{R}}\widehat{w}|+\left(|{\mathcal{R}}\widehat{w}|+|u_{1}|\right)|{\mathcal{R}}\widehat{w}||{\partial}_{x}u_{1}|.$

Using the formula $J(t)=U(t)xU(-t)$ from Lemma 2.1, as well as (2.3) that
state that $J(t)$ can be thought of as a derivative here, we find

$\displaystyle\left\|U(-t)\Phi_{2}(t)\right\|_{\Sigma}$
$\displaystyle\leqslant\int_{-\infty}^{t}\left(\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}^{2}+\|u_{1}(\tau)\|_{L^{\infty}}^{2}\right)\|U(-\tau){\mathcal{R}}\widehat{w}\|_{\Sigma}d\tau$
$\displaystyle+\int_{-\infty}^{t}\left(\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}+\|u_{1}(\tau)\|_{L^{\infty}}\right)\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}\|U(-\tau)u_{1}(\tau)\|_{\Sigma}d\tau.$

Most of the terms are estimated like in the proof of Lemma 3.1. To estimate
the single factor $\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}$ in the
second line, we choose $s>1/2$ and $\theta>1/2$ so that $s+2\theta=3$, and
Lemma 2.2 yields, since $H^{3}({\mathbb{R}})$ is an algebra,

$\displaystyle\|{\mathcal{R}}(\tau)\widehat{w}\|_{L^{\infty}}\lesssim\frac{\|\widehat{w}\|_{H^{3}}}{|\tau|^{1/2+\theta}}$
$\displaystyle\lesssim\frac{1}{|\tau|^{1/2+\theta}}\left(\|{\widehat{u}_{-}}\|_{H^{3}}+(\log|\tau|)^{3}\|{\widehat{u}_{-}}\|_{H^{3}}^{7}\right)$
$\displaystyle\lesssim\frac{1}{|\tau|}\left(\|{u_{-}}\|_{{\mathcal{H}}}+\|{u_{-}}\|_{{\mathcal{H}}}^{7}\right).$

For $\|{\widehat{u}_{-}}\|_{H^{1}}\leqslant 1$, invoking Lemma 3.1, we
therefore obtain, for $t\leqslant-10$,

$\|U(-t)\Phi_{2}(t)\|_{\Sigma}\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}^{2}\|{u_{-}}\|_{{\mathcal{H}}}P_{2}\left(\|{u_{-}}\|_{{\mathcal{H}}}\right)\int_{-\infty}^{t}\frac{(\log|\tau|)^{3}}{\tau^{2}}d\tau,$

for some polynomial $P_{2}$.

The term $\Phi_{3}$ is estimated thanks to Lemma 2.2, and the choice of
parameter is motivated by the previous estimate, keeping in mind that a factor
$\frac{1}{\tau}$ is already present in the definition of $\Phi_{3}$. For the
$L^{2}$-estimate, we thus choose $\theta=1$ in Lemma 2.2, so that

$\displaystyle\|\Phi_{3}(t)\|_{L^{2}}$
$\displaystyle\lesssim\int_{-\infty}^{t}\left\||\widehat{w}|^{2}\widehat{w}\right\|_{H^{2}}\frac{d\tau}{\tau^{2}}\lesssim\int_{-\infty}^{t}\|\widehat{w}\|_{L\infty}^{2}\|\widehat{w}\|_{H^{2}}\frac{d\tau}{\tau^{2}}$
$\displaystyle\lesssim\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\widehat{u}_{-}}\|_{H^{2}}\int_{-\infty}^{t}\left(1+(\log|\tau|)^{2}\|{\widehat{u}_{-}}\|_{H^{2}}^{4}\right)\frac{d\tau}{\tau^{2}}.$

The norm $\|xU(-t)\Phi_{3}(t)\|_{L^{2}}=\|J(t)\Phi_{3}(t)\|_{L^{2}}$ is
estimate similarly, by setting again $\theta=1$ (in the third point of Lemma
2.2, and we get, since the $H^{2}$-norm is replaced by an $H^{3}$-norm in the
above computation,

$\|J(t)\Phi_{3}(t)\|_{L^{2}}\lesssim\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\widehat{u}_{-}}\|_{H^{3}}\int_{-\infty}^{t}\left(1+(\log|\tau|)^{3}\|{\widehat{u}_{-}}\|_{H^{3}}^{6}\right)\frac{d\tau}{\tau^{2}}.$

The choice of suitable parameters for the $\dot{H}^{1}$-norm was already made
in the statement of Lemma 2.2, and we have

$\|{\partial}_{x}\Phi_{3}(t)\|_{L^{2}}\lesssim\int_{-\infty}^{t}\left(\||\widehat{w}|^{2}\widehat{w}\|_{H^{1}}+\|x|\widehat{w}|^{2}\widehat{w}\|_{H^{2}}\right)\frac{d\tau}{\tau^{2}}.$

The term involving the $H^{1}$-norm has been estimated above, so we focus on
the other norm. Direct computations yield

$\left|{\partial}_{x}^{2}\left(x|\widehat{w}|^{2}\widehat{w}\right)\right|\lesssim|x||\widehat{w}|^{2}|{\partial}_{x}^{2}\widehat{w}|+|\widehat{w}|^{2}|{\partial}_{x}\widehat{w}|+|x||\widehat{w}||{\partial}_{x}\widehat{w}|^{2},$

and we get

$\displaystyle\|{\partial}_{x}\Phi_{3}(t)\|_{L^{2}}$
$\displaystyle\lesssim\|{\widehat{u}_{-}}\|_{L^{\infty}}^{2}\|{\widehat{u}_{-}}\|_{H^{1}}\int_{-\infty}^{t}\left(1+(\log|\tau|)\|{\widehat{u}_{-}}\|_{H^{1}}^{2}\right)\frac{d\tau}{\tau^{2}}$
$\displaystyle\quad+\|{\widehat{u}_{-}}\|_{L^{\infty}}\|{u_{-}}\|_{{\mathcal{H}}}^{2}\int_{-\infty}^{t}\left(1+(\log|\tau|)^{2}\|{u_{-}}\|_{{\mathcal{H}}}^{4}\right)\frac{d\tau}{\tau^{2}}.$

The source term is therefore controlled, for $t\leqslant-10$, by:

(3.2)
$\|U(-t)(\Phi_{2}(t)+\Phi_{3}(t))\|_{\Sigma}\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}\|{u_{-}}\|_{{\mathcal{H}}}^{2}P\left(\|{u_{-}}\|_{{\mathcal{H}}}\right)\int_{-\infty}^{t}\frac{\left(\log|\tau|\right)^{3}}{\tau^{2}}d\tau,$

for some polynomial $P$ whose precise expression is irrelevant.

The end of the proof consists of a fixed point argument. Mimicking [26], for
$\alpha\in]1/2,1[$ arbitrary (but fixed), and $T\gg 1$, introduce the space

$\displaystyle X^{\alpha}(T)$ $\displaystyle=\Bigl{\\{}u\in
L^{\infty}(]-\infty,-T],L^{2})\text{ such that }t\mapsto U(-t)u(t)\in
L^{\infty}(]-\infty,-T],\Sigma),$
$\displaystyle\qquad\sup_{t\leqslant-T}|t|^{\alpha}\|U(-t)(u(t)-u_{2}(t))\|_{\Sigma}\leqslant\|{\widehat{u}_{-}}\|_{H^{1}}\Bigr{\\}}.$

and define on $X^{\alpha}(T)$ the metric

$d(u,v)=\sup_{t\leqslant-T}|t|^{\alpha}\|u(t)-v(t)\|_{L^{2}}.$

Note that here, we choose to measure distance by considering the $L^{2}$ norm
only: $X^{\alpha}(T)$, equipped with this distance, is a complete metric
space.

For $u\in X^{\alpha}(T)$, $\Phi_{1}(u)$ is estimated by

$\displaystyle\|U(-t)\Phi_{1}(u(t))\|_{L^{2}}$
$\displaystyle\lesssim\int_{-\infty}^{t}\left(\|u(\tau)\|_{L^{\infty}}^{2}+\|u_{2}(\tau)\|_{L^{\infty}}^{2}\right)\|u(\tau)-u_{2}(\tau)\|_{L^{2}}d\tau$
$\displaystyle\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}^{2}\int_{-\infty}^{t}\frac{d\tau}{\tau^{1+\alpha}}\lesssim\frac{\|{\widehat{u}_{-}}\|_{H^{1}}^{2}}{|t|^{\alpha}}.$

For $A\in\\{{\partial}_{x},J(t)\\}$, we have

(3.3)
$\left|A\left(|u|^{2}u-|u_{2}|^{2}u_{2}\right)\right|\lesssim|u|^{2}|A(u-u_{2})|+\left(|u|+|u_{2}|\right)|Au_{2}||u-u_{2}|.$

Therefore,

$\displaystyle\|U(-t)\Phi_{1}(u(t))\|_{\Sigma}$
$\displaystyle\lesssim\int_{-\infty}^{t}\|u(\tau)\|_{L^{\infty}}^{2}\|U(-\tau)(u(\tau)-u_{2}(\tau))\|_{\Sigma}d\tau$
$\displaystyle+\int_{-\infty}^{t}(\|u(\tau)\|_{L^{\infty}}$
$\displaystyle+\|u_{2}(\tau)\|_{L^{\infty}})\|U(-\tau)u_{2}(\tau)\|_{\Sigma}\|U(-\tau)(u(\tau)-u_{2}(\tau))\|_{L^{2}}d\tau$
$\displaystyle\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}^{2}P_{3}\left(\|{u_{-}}\|_{{\mathcal{H}}}\right)\int_{-\infty}^{t}\frac{d\tau}{\tau^{1+\alpha}},$

for some polynomial $P_{3}$ whose precise expression is irrelevant, where we
have used Lemma 3.1 to estimate $\|U(-\tau)u_{2}(\tau)\|_{\Sigma}$. Therefore,
by choosing $T$ sufficiently large, we check that $\Phi$ maps $X^{\alpha}(T)$
to itself.

To conclude by a fixed point argument, we show that $\Phi$ is a contraction
provided that $\|{\widehat{u}_{-}}\|_{H^{1}}$ is sufficiently small. Indeed,
for $u,v\in X^{\alpha}(T)$,

$\Phi(u)(t)-\Phi(v)(t)=-i\lambda\int_{-\infty}^{t}U(t-\tau)\left(|u|^{2}u-|v|^{2}v\right)(\tau)d\tau,$

and so, for $t\leqslant-10$,

$\displaystyle\left\|\Phi(u)(t)-\Phi(v)(t)\right\|_{L^{2}}$
$\displaystyle\lesssim\int_{-\infty}^{t}\left(\|u(\tau)\|_{L^{\infty}}^{2}+\|v(\tau)\|_{L^{\infty}}^{2}\right)\|u(\tau)-v(\tau)\|_{L^{2}}d\tau$
$\displaystyle\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}^{2}d(u,v)\int_{-\infty}^{t}\frac{d\tau}{\tau^{1+\alpha}},$

where we have used

$\displaystyle\|u(\tau)\|_{L^{\infty}}+\|v(\tau)\|_{L^{\infty}}$
$\displaystyle\leqslant
2\|u_{2}(\tau)\|_{L^{\infty}}+C\|u(\tau)-u_{2}(\tau)\|_{H^{1}}+C\|v(\tau)-u_{2}(\tau)\|_{H^{1}}$
$\displaystyle\lesssim\frac{\|{\widehat{u}_{-}}\|_{H^{1}}}{\sqrt{|\tau|}},$

since $\alpha>1/2$. Therefore,

$d\left(\Phi(u),\Phi(v)\right)\lesssim\|{\widehat{u}_{-}}\|_{H^{1}}^{2}d(u,v),$

hence the result provided that $\|{\widehat{u}_{-}}\|_{H^{1}}$ is sufficiently
small. ∎

##  4\. Behavior of the modified wave operator near the origin

###  4.1. Leading order asymptotic behavior

Let $u_{-}^{\varepsilon}=\varepsilon v_{-}$ with $v_{-}\in{\mathcal{H}}$
independent of $\varepsilon$. Denote by $u^{\varepsilon}$ the solution
provided by Proposition 3.2. Then like $u_{-}^{\varepsilon}$,
$u^{\varepsilon}$ is of order $\varepsilon$, and the remainder in Proposition
3.2 is $\mathcal{O}(\varepsilon^{3})$. The long range phase correction
$S_{-}^{\varepsilon}$ is $\mathcal{O}(\varepsilon^{2}\log|t|)$, for
$t\leqslant-10$: its contribution is negligible for times $t$ such that
$\varepsilon^{2}|\log|t||\ll 1$, so in particular we can match with a linear
solution at $t^{\varepsilon}_{\gamma}=-1/\varepsilon^{\gamma}$,

$i{\partial}_{t}{\mathbf{u}}^{\varepsilon}_{1}+\frac{1}{2}{\partial}_{x}^{2}{\mathbf{u}}^{\varepsilon}_{1}=0,$

with

${\mathbf{u}}^{\varepsilon}_{1\mid
t=-1/\varepsilon^{\gamma}}=\frac{1}{(it)^{1/2}}e^{i\frac{x^{2}}{2t}}\mathcal{F}(u_{-}^{\varepsilon})\left(\frac{x}{t}\right)\big{|}_{t=-1/\varepsilon^{\gamma}}.$

The right hand side is $M(t)D(t)\mathcal{F}u_{-}^{\varepsilon}$ evaluated at
$t=-1/\varepsilon^{\gamma}$ In view of Lemma 2.2, up to a small error term,

${\mathbf{u}}_{1}^{\varepsilon}(t)=U(t)u_{-}^{\varepsilon},\text{ or,
equivalently, }{\mathbf{u}}_{1}^{\varepsilon}=\varepsilon v_{1},\text{ with
}v_{1}(t)=U(t)v_{-}.$

We choose this definition for the first order approximation
${\mathbf{u}}_{1}^{\varepsilon}$. Let $t\geqslant-1/\varepsilon^{\gamma}$.
Duhamel’s formula implies

$\displaystyle U(-t)\left(u^{\varepsilon}(t)-v^{\varepsilon}_{1}(t)\right)$
$\displaystyle=U\left(-s\right)\left(u^{\varepsilon}(s)-u^{\varepsilon}_{2}(s)+u_{2}^{\varepsilon}(s)-{\mathbf{u}}^{\varepsilon}_{1}(s)\right)\Big{|}_{s=-1/\varepsilon^{\gamma}}$
$\displaystyle\quad-i\lambda\int_{-1/\varepsilon^{\gamma}}^{t}U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}\right)(\tau)d\tau,$

that is

$\displaystyle U(-t)u^{\varepsilon}(t)-u_{-}^{\varepsilon}$
$\displaystyle=U\left(-s\right)\left(u^{\varepsilon}(s)-u^{\varepsilon}_{2}(s)\right)\Big{|}_{s=-1/\varepsilon^{\gamma}}+U(-s)u_{2}^{\varepsilon}(s)\Big{|}_{s=-1/\varepsilon^{\gamma}}-u_{-}^{\varepsilon}$
$\displaystyle\quad-i\lambda\int_{-1/\varepsilon^{\gamma}}^{t}U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}\right)(\tau)d\tau,$

The first term on the right hand side is estimated thanks to Proposition 3.2:

$\left\|U\left(-s\right)\left(u^{\varepsilon}(s)-u^{\varepsilon}_{2}(s)\right)\Big{|}_{s=-1/\varepsilon^{\gamma}}\right\|_{\Sigma}\lesssim\varepsilon^{3+\gamma}|\log\varepsilon|.$

On the other hand,

$\left\|U(-s)u_{2}^{\varepsilon}(s)\Big{|}_{s=-1/\varepsilon^{\gamma}}-u_{-}^{\varepsilon}\right\|_{\Sigma}=\|\mathcal{F}^{-1}\widehat{w}_{\mid
s=-1/\varepsilon^{\gamma}}-u_{-}^{\varepsilon}\|_{\Sigma}=\|\widehat{w}_{\mid
s=-1/\varepsilon^{\gamma}}-\mathcal{F}(u_{-}^{\varepsilon})\|_{\Sigma}.$

Recalling that

$\widehat{w}=\mathcal{F}(u_{-}^{\varepsilon})\exp\left(i|\mathcal{F}(u_{-}^{\varepsilon})|^{2}\log|t|\right),$

we have directly

$\displaystyle\|\widehat{w}-\mathcal{F}(u_{-}^{\varepsilon})\|_{\Sigma}$
$\displaystyle\lesssim\|\left\langle
x\right\rangle\left(\widehat{w}-\mathcal{F}(u_{-}^{\varepsilon})\right)\|_{L^{2}}+\left\|{\partial}_{x}\widehat{u_{-}^{\varepsilon}}\left(\exp\left(i|\mathcal{F}(u_{-}^{\varepsilon})|^{2}\log|t|\right)-1\right)\right\|_{L^{2}}$
$\displaystyle\quad+|\log\varepsilon|\left\|\mathcal{F}(u_{-}^{\varepsilon}){\partial}_{x}|\mathcal{F}(u_{-}^{\varepsilon})|^{2}\right\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon^{3}|\log\varepsilon|.$

The integral term can be estimated in $\Sigma$ by

$\int_{-1/\varepsilon^{\gamma}}^{t}\|u^{\varepsilon}(\tau)\|_{L^{\infty}}^{2}\|U(-\tau)u^{\varepsilon}(\tau)\|_{\Sigma}d\tau.$

Since $\|U((T)u^{\varepsilon}(-T)\|_{\Sigma}\lesssim\varepsilon$ from Lemma
3.1 and Proposition 3.2 (where $T$ is independent of $\varepsilon$), we have
the following uniform estimate from [14, Theorem 1.1] (see also [19]),
provided that $\varepsilon>0$ is sufficiently small,

(4.1)
$\|u^{\varepsilon}(\tau)\|_{L^{\infty}}\lesssim\frac{\varepsilon}{\left\langle\tau\right\rangle^{1/2}},\quad\forall\tau\in{\mathbb{R}}.$

Writing

$\|U(-\tau)u^{\varepsilon}(\tau)\|_{\Sigma}\leqslant\|U(-\tau)\left(u^{\varepsilon}(\tau)-{\mathbf{u}}_{1}^{\varepsilon}(\tau)\right)\|_{\Sigma}+\|U(-\tau){\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{\Sigma},$

and using the obvious fact that
$\|U(-\tau){\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{\Sigma}=\varepsilon\|v_{-}\|_{\Sigma}$,
we have, for $t\geqslant-1/\varepsilon^{\gamma}$,

$\left\|U(-t)u^{\varepsilon}(t)-u_{-}^{\varepsilon}\right\|_{\Sigma}\lesssim\varepsilon^{3}|\log\varepsilon|+\varepsilon^{2}\int_{-1/\varepsilon^{\gamma}}^{t}\left\|U(-\tau)u^{\varepsilon}(\tau)-u_{-}^{\varepsilon}\right\|_{\Sigma}\frac{d\tau}{\left\langle\tau\right\rangle}+\varepsilon^{3}\int_{-1/\varepsilon^{\gamma}}^{t}\frac{d\tau}{\left\langle\tau\right\rangle}.$

Therefore, Gronwall lemma yields

$\sup_{-1/\varepsilon^{\gamma}\leqslant t\leqslant
T}\left\|U(-t)u^{\varepsilon}(t)-u_{-}^{\varepsilon}\right\|_{\Sigma}\lesssim\varepsilon^{3}|\log\varepsilon|e^{C\varepsilon^{2}|\log\varepsilon|+C\varepsilon^{2}\log\left\langle
T\right\rangle},$

for some constant $C$ independent of $\varepsilon$ and $T$. In particular, for
any $\delta\in]0,1]$, the right hand side is
$\mathcal{O}(\varepsilon^{3-\delta})$ on
$[-1/\varepsilon^{\gamma},1/\varepsilon^{\beta}]$ for any $\beta>0$. This
estimate implies the property

${W_{-}^{\rm mod}}(\varepsilon v_{-})=\varepsilon
v_{-}+\mathcal{O}(\varepsilon^{3-\eta}),$

for any $\eta>0$. In the next subsection, we improve this estimate by
describing the first corrector term.

###  4.2. Next term in the asymptotic expansion

Following e.g. [3], a refined asymptotic expansion for $u^{\varepsilon}$, at
least on bounded time intervals, is given by the solution
${\mathbf{u}}_{2}^{\varepsilon}$ to

$i{\partial}_{t}{\mathbf{u}}_{2}^{\varepsilon}+\frac{1}{2}{\partial}_{x}^{2}{\mathbf{u}}_{2}^{\varepsilon}=\lambda|{\mathbf{u}}_{1}^{\varepsilon}|^{2}{\mathbf{u}}_{1}^{\varepsilon},$

where the Cauchy data must be chosen carefully. Set

${\mathbf{u}}_{\rm
app}^{\varepsilon}={\mathbf{u}}_{1}^{\varepsilon}+{\mathbf{u}}^{\varepsilon}_{2}.$

We now have

$\displaystyle U(-t)\left(u^{\varepsilon}(t)-{\mathbf{u}}^{\varepsilon}_{\rm
app}(t)\right)$
$\displaystyle=U\left(-s\right)\left(u^{\varepsilon}(s)-u^{\varepsilon}_{2}(s)+u_{2}^{\varepsilon}(s)-{\mathbf{u}}_{\rm
app}^{\varepsilon}(s)\right)\Big{|}_{s=-1/\varepsilon^{\gamma}}$
$\displaystyle\quad-i\lambda\int_{-1/\varepsilon^{\gamma}}^{t}U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}-|{\mathbf{u}}_{1}^{\varepsilon}|^{2}{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)d\tau.$

We have seen before that

$\left\|U\left(-s\right)\left(u^{\varepsilon}(s)-u^{\varepsilon}_{2}(s)\right)\Big{|}_{s=-1/\varepsilon^{\gamma}}\right\|_{\Sigma}\lesssim\varepsilon^{3+\gamma}|\log\varepsilon|=o(\varepsilon^{3}).$

So if we want to catch the $\varepsilon^{3}$ term, this is good. Making
$U(-s)(u_{2}^{\varepsilon}(s)-{\mathbf{u}}_{\rm app}^{\varepsilon}(s))$ small
($o(\varepsilon^{3})$ in $\Sigma$) at $s=-1/\varepsilon^{\gamma}$ is what
should tell us how to choose the data for ${\mathbf{u}}_{2}^{\varepsilon}$.
Indeed, the integral term is estimated in $L^{2}$ ($\Sigma$ will require more
care) by

$\int_{-1/\varepsilon^{\gamma}}^{t}\left(\|u^{\varepsilon}(\tau)\|_{L^{\infty}}^{2}+\|{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{\infty}}^{2}\right)\|u^{\varepsilon}(\tau)-v_{1}^{\varepsilon}(\tau)\|_{L^{2}}d\tau\lesssim\varepsilon^{2}\int_{-1/\varepsilon^{\gamma}}^{t}\|u^{\varepsilon}(\tau)-{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{2}}\frac{d\tau}{\left\langle\tau\right\rangle},$

and we now write

$\|u^{\varepsilon}(\tau)-{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{2}}\leqslant\|u^{\varepsilon}(\tau)-{\mathbf{u}}_{\rm
app}^{\varepsilon}(\tau)\|_{L^{2}}+\|{\mathbf{u}}_{2}^{\varepsilon}(\tau)\|_{L^{2}},$

so the last term yield a smaller contribution (in terms of $\varepsilon$ at
least) than in the previous subsection.

Thus, everything seems to boil down to choosing
${\mathbf{u}}_{2}^{\varepsilon}$ at time $-1/\varepsilon^{\gamma}$ in an
efficient way, so that

$\left\|U(-s)(u_{2}^{\varepsilon}(s)-{\mathbf{u}}_{\rm
app}^{\varepsilon}(s))\Big{|}_{s=-1/\varepsilon^{\gamma}}\right\|_{\Sigma}=o(\varepsilon^{3}).$

By construction,

$U(-s)(u_{2}^{\varepsilon}(s))\Big{|}_{s=-1/\varepsilon^{\gamma}}=\mathcal{F}^{-1}\left(\mathcal{F}(u_{-}^{\varepsilon})\exp\left(-i\lambda\gamma|\mathcal{F}(u_{-}^{\varepsilon})|^{2}\log\varepsilon\right)\right).$

Taking into account the second term in the asymptotic expansion of the
exponential in $\widehat{w}$, we define ${\mathbf{u}}_{2}^{\varepsilon}$ by

(4.2) $\displaystyle U(-t){\mathbf{u}}_{2}^{\varepsilon}(t)$
$\displaystyle=-i\lambda\gamma\varepsilon^{3}(\log\varepsilon)\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-}|^{2}\mathcal{F}(v_{-})\right)$
$\displaystyle\quad-i\lambda\varepsilon^{3}\int_{-1/\varepsilon^{\gamma}}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau.$

######  Lemma 4.1.

Let $v_{-}\in{\mathcal{H}}$. For every $t\in{\mathbb{R}}$,
$U(-t){\mathbf{u}}^{\varepsilon}_{2}(t)/\varepsilon^{3}$ converges in
$L^{2}\cap L^{\infty}({\mathbb{R}})$, and the limit is independent of
$\gamma$. This limit is given by

$\displaystyle U(-t)v_{2}(t)$
$\displaystyle=-i\lambda\int_{-\infty}^{-1}\left(U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)+\frac{1}{|\tau|}\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)\right)d\tau$
$\displaystyle\quad-i\lambda\int_{-1}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau.$

The limit also holds in $\Sigma$, and for all $t\in{\mathbb{R}}$,

$\left\|U(-t)\left(\frac{{\mathbf{u}}^{\varepsilon}_{2}(t)}{\varepsilon^{3}}-v_{2}(t)\right)\right\|_{\Sigma}\lesssim\varepsilon^{\gamma}.$

######  Remark 4.2.

We emphasize that even though the limit $v_{2}$ is independent of $\gamma$,
the error estimate improves for large values of $\gamma$. In view of the
definition of ${\mathbf{u}}_{2}^{\varepsilon}$, the matching condition at
$t=-1/\varepsilon^{\gamma}$ between $u_{2}^{\varepsilon}$ and
${\mathbf{u}}_{\rm app}^{\varepsilon}$ is not better than
$\mathcal{O}\left(\varepsilon^{5}(\log\varepsilon)^{2}\right)$, so it is no
use (in this paper) to consider $\gamma>2$.

###### Proof.

The proof consists in writing a precise asymptotic expansion of the argument
of the integral involved in (4.2). Writing, for any $\theta\in]0,1]$,

$M(t)=e^{i\frac{x^{2}}{2t}}=1+\mathcal{O}\left(\left|\frac{x^{2}}{t}\right|^{\theta}\right),$

we have, in the same vein as in Lemma 2.2,

$U(\tau)v_{-}(x)=M(\tau)D(\tau)\mathcal{F}v_{-}+R_{1}(\tau,x),$

where, as $\tau\to-\infty$

$\|R_{1}(\tau)\|_{L^{2}}=\mathcal{O}\left(\frac{1}{|\tau|^{\theta}}\right),\quad\|R_{1}(\tau)\|_{L^{\infty}}=\mathcal{O}\left(\frac{1}{|\tau|^{\theta+1/2}}\right).$

We infer, as $\tau\to-\infty$,

$|U(\tau)v_{-}|^{2}U(\tau)v_{-}=e^{i\frac{\pi}{4}}\frac{e^{i\frac{x^{2}}{2\tau}}}{|\tau|^{3/2}}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)\left(\frac{x}{\tau}\right)+R_{2}(\tau,x),$

with

$\|R_{2}(\tau)\|_{L^{2}}=\mathcal{O}\left(\frac{1}{|\tau|^{1+\theta}}\right),\quad\|R_{2}(\tau)\|_{L^{\infty}}=\mathcal{O}\left(\frac{1}{|\tau|^{3/2+\theta}}\right).$

In view of (2.5), we next compute

$\displaystyle U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)$
$\displaystyle=ie^{-i\frac{x^{2}}{2\tau}}\mathcal{F}^{-1}D\left(\frac{1}{\tau}\right)\left(\frac{e^{i\frac{\pi}{4}}}{|\tau|^{3/2}}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)\left(\frac{x}{\tau}\right)\right)$
$\displaystyle\quad+ie^{-i\frac{x^{2}}{2\tau}}\mathcal{F}^{-1}D\left(\frac{1}{\tau}\right)\left(e^{-i\frac{x^{2}}{2\tau}}R_{2}(\tau,x)\right)$
$\displaystyle=-\frac{e^{-i\frac{x^{2}}{2\tau}}}{|\tau}\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)(x)+R_{3}(\tau,x)$
$\displaystyle=-\frac{1}{|\tau|}\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)(x)+R_{4}(\tau,x),$

where

$\|R_{4}(\tau)\|_{L^{2}\cap
L^{\infty}}=\mathcal{O}\left(\frac{1}{|\tau|^{1+\theta}}\right).$

Set $\theta=1$, and write

$\displaystyle\frac{1}{\lambda\varepsilon^{3}}U(-t){\mathbf{u}}_{2}^{\varepsilon}(t)$
$\displaystyle=-i\gamma(\log\varepsilon)\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-}|^{2}\mathcal{F}(v_{-})\right)-i\int_{-1/\varepsilon^{\gamma}}^{-1}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau$
$\displaystyle\quad-i\int_{-1}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau$
$\displaystyle=-i\gamma(\log\varepsilon)\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-}|^{2}\mathcal{F}(v_{-})\right)$
$\displaystyle\quad-i\int_{-1/\varepsilon^{\gamma}}^{-1}\left(-\frac{1}{|\tau|}\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right)+R_{4}(\tau)\right)d\tau$
$\displaystyle\quad-i\int_{-1}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau$
$\displaystyle=-i\int_{-\infty}^{-1}R_{4}(\tau)d\tau+\underbrace{\mathcal{O}\left(\int_{-\infty}^{-1/\varepsilon^{\gamma}}\frac{d\tau}{\tau^{2}}\right)}_{=\mathcal{O}(\varepsilon^{\gamma})}-i\int_{-1}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau.$

Therefore,

$U(-t)v_{2}(t)=-i\lambda\int_{-\infty}^{-1}R_{4}(\tau)d\tau-i\lambda\int_{-1}^{t}U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)d\tau,$

and, by construction,

$R_{4}(\tau)=U(-\tau)\left(|U(\tau)v_{-}|^{2}U(\tau)v_{-}\right)+\frac{1}{|\tau|}\mathcal{F}^{-1}\left(|\mathcal{F}(v_{-})|^{2}\mathcal{F}(v_{-})\right),$

hence the formula of the lemma. In particular, by the same arguments as above,
the second integral diverges logarithmically as $t\to+\infty$. We leave out
the convergence in $\Sigma$ (since $v_{-}\in{\mathcal{H}}$, so we can pay some
momentum estimate when controlling $M-1$), which relies on similar ideas, like
the proof of Lemma 2.2. Note that the present argument ∎

We can now improve the error estimate proven in the previous subsection with
only the leading order approximation ${\mathbf{u}}_{1}^{\varepsilon}$. By
construction,

$\left\|U(-s)(u_{2}^{\varepsilon}(s)-{\mathbf{u}}_{\rm
app}^{\varepsilon}(s))\Big{|}_{s=-1/\varepsilon^{\gamma}}\right\|_{\Sigma}=\mathcal{O}\left(\varepsilon^{5}(\log\varepsilon)^{2}\right),$

and we have, if $0<\gamma<2$,

$\displaystyle\|U(-t)\left(u^{\varepsilon}(t)-{\mathbf{u}}^{\varepsilon}_{\rm
app}(t)\right)\|_{\Sigma}$
$\displaystyle\lesssim\varepsilon^{3+\gamma}|\log\varepsilon|$
$\displaystyle\quad+\int_{-1/\varepsilon^{\gamma}}^{t}\|U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}-|{\mathbf{u}}_{1}^{\varepsilon}|^{2}{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}d\tau.$

In view of (3.3),

$\displaystyle\|U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}-|{\mathbf{u}}_{1}^{\varepsilon}|^{2}{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}\lesssim\|$
$\displaystyle
u^{\varepsilon}(\tau)\|_{L^{\infty}}^{2}\|U(-\tau)\left(u^{\varepsilon}-{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}$
$\displaystyle+\left(\|u^{\varepsilon}(\tau)\|_{L^{\infty}}+\|{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{\infty}}\right)$
$\displaystyle\|U(-\tau){\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{\Sigma}\|u^{\varepsilon}(\tau)-{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{\infty}}.$

Recalling (4.1), we readily have

$\|u^{\varepsilon}(\tau)\|_{L^{\infty}}\lesssim\frac{\varepsilon}{\left\langle\tau\right\rangle^{1/2}}.$

On the other hand,

$\|U(-\tau){\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{\Sigma}=\varepsilon\|v_{-}\|_{\Sigma},$

and Sobolev embedding yields, along with (2.2),

$\|{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{\infty}}\lesssim\frac{\varepsilon}{\left\langle\tau\right\rangle^{1/2}},$

as well as

$\|u^{\varepsilon}(\tau)-{\mathbf{u}}_{1}^{\varepsilon}(\tau)\|_{L^{\infty}}\lesssim\frac{1}{\left\langle\tau\right\rangle^{1/2}}\|U(-\tau)\left(u^{\varepsilon}-v_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}.$

Therefore,

$\displaystyle\|U(-\tau)\left(|u^{\varepsilon}|^{2}u^{\varepsilon}-|{\mathbf{u}}_{1}^{\varepsilon}|^{2}{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}$
$\displaystyle\lesssim\frac{\varepsilon^{2}}{\left\langle\tau\right\rangle}\|U(-\tau)\left(u^{\varepsilon}-{\mathbf{u}}_{1}^{\varepsilon}\right)(\tau)\|_{\Sigma}$
$\displaystyle\lesssim\frac{\varepsilon^{2}}{\left\langle\tau\right\rangle}$
$\displaystyle\left(\|U(-\tau)\left(u^{\varepsilon}-{\mathbf{u}}_{\rm
app}^{\varepsilon}\right)(\tau)\|_{\Sigma}+\|U(-\tau){\mathbf{u}}_{2}^{\varepsilon}(\tau)\|_{\Sigma}\right).$

We see that there exists $C$ such that for all $T\geqslant 1$,

$\|U(-\tau){\mathbf{u}}_{2}^{\varepsilon}(\tau)\|_{\Sigma}\lesssim\varepsilon^{3}\log\left\langle
T\right\rangle,\quad\forall-\frac{1}{\varepsilon^{\gamma}}\leqslant\tau\leqslant
T,$

where the logarithmic correction is necessarily present for large positive
$\tau$, as pointed out above. Gronwall lemma now yields, for all $0<\gamma<2$,

$\sup_{-1/\varepsilon^{\gamma}\leqslant t\leqslant
T}\left\|U(-t)\left(u^{\varepsilon}(t)-{\mathbf{u}}_{\rm
app}^{\varepsilon}(t)\right)\right\|_{\Sigma}\lesssim\varepsilon^{3+\gamma}|\log\varepsilon|e^{C\varepsilon^{2}|\log\varepsilon|+C\varepsilon^{2}\log\left\langle
T\right\rangle},$

for some constant $C$ independent of $\varepsilon$ and $T$. In particular, the
right hand side is $o(\varepsilon^{3})$ on
$[-1/\varepsilon^{\gamma},1/\varepsilon^{\beta}]$ for any $\beta>0$.

The first point of Theorem 1.1 follows, by considering the above error
estimate at time $t=0$ and setting $w_{2}=v_{2\mid t=0}$.

##  5\. Asymptotic completeness

###  5.1. Main steps of the construction of $u_{+}$

We recall the main steps from the proof of [14, Theorem 1.2], a result which
we state in the particular case that we shall consider (the initial data may
have a different regularity there, and an extra short range nonlinearity can
be incorporated). In passing, we keep track of the size of the remainders more
precisely.

######  Theorem 5.1 (From [14]).

Let $u_{0}\in\Sigma$, with
$\|u_{0}\|_{\Sigma}=\varepsilon^{\prime}\leqslant\delta$, where $\delta>0$ is
sufficiently small. There exist unique functions $W\in L^{\infty}\cap L^{2}$
and $\Phi\in L^{\infty}$ such that, for $t\geqslant 1$ and
$C\delta<\alpha<1/4$,

(5.1)
$\left\|\mathcal{F}\left(U(-t)u(t)\right)\exp\left(i\lambda\int_{1}^{t}|\widehat{u}(\tau)|^{2}\frac{d\tau}{\tau}\right)-W\right\|_{L^{2}\cap
L^{\infty}}\leqslant
C(\varepsilon^{\prime})^{3}t^{-\alpha+C(\varepsilon^{\prime})^{2}},$

and

(5.2)
$\left\|\lambda\int_{1}^{t}|\widehat{u}(\tau)|^{2}\frac{d\tau}{\tau}-\lambda|W|^{2}\log
t-\Phi\right\|_{L^{\infty}}\leqslant
C(\varepsilon^{\prime})^{3}t^{-\alpha+C(\varepsilon^{\prime})^{2}}.$

In particular,

$u(t,x)=\frac{1}{(it)^{1/2}}W\left(\frac{x}{t}\right)\exp\left(i\frac{x^{2}}{2t}-i\lambda\left|W\left(\frac{x}{t}\right)\right|^{2}\log
t-i\Phi\left(\frac{x}{t}\right)\right)+\rho(t,x),$

with

$\|\rho(t)\|_{L^{2}}\leqslant
C\varepsilon^{\prime}t^{-\alpha+C(\varepsilon^{\prime})^{2}},\quad\|\rho(t)\|_{L^{\infty}}\leqslant
C\varepsilon^{\prime}t^{-1/2-\alpha+C(\varepsilon^{\prime})^{2}}.$

In [14], the time decay in (5.2) is raised to the power $2/3$, because the
setting is more general and includes the case of dimension three. We explain
below why this power can be discarded in the above statement. The asymptotic
state $u_{+}$ is naturally given by

$u_{+}=\mathcal{F}^{-1}\left(We^{-i\Phi}\right).$

###### Sketch of the proof.

As announced above, we recall the main steps from [14], and slightly improve
some estimates in terms of the powers of $\varepsilon^{\prime}$.

As evoked before in the case of (4.1), (the proof of) [14, Theorem 1.1]
provides global estimates for $u$ (proven in [14, Lemma 3.2 and 3.3], see also
[19]):

(5.3) $\|u(t)\|_{L^{\infty}}\lesssim\frac{\varepsilon^{\prime}}{\left\langle
t\right\rangle^{1/2}},\quad\|U(-t)u(t)\|_{\Sigma}\lesssim\varepsilon^{\prime}\left\langle
t\right\rangle^{C(\varepsilon^{\prime})^{2}}.$

Denote

$v(t)=U(-t)u(t),\quad\widehat{w}=\widehat{v}\underbrace{\exp\left(i\lambda\int_{1}^{t}|\widehat{v}(\tau)|^{2}\frac{d\tau}{\tau}\right)}_{=:B(t)}.$

Then (1.1) is recasted as

(5.4)
$i{\partial}_{t}\widehat{w}=\frac{\lambda}{t}B(t)\left(I_{1}+I_{2}\right),$

where

$I_{1}(t)=\mathcal{F}\left(M(-t)-1\right)\mathcal{F}^{-1}\left(|\widehat{M(t)v}|^{2}\widehat{M(t)v}\right),\quad
I_{2}(t)=\left|\widehat{M(t)v}\right|^{2}\widehat{M(t)v}-|\widehat{v}|^{2}\widehat{v}.$

These source terms are controlled as follows, for $t\geqslant 1$:

$\|I_{1}(t)\|_{L^{2}\cap L^{\infty}}+\|I_{2}(t)\|_{L^{2}\cap
L^{\infty}}\lesssim\frac{\|v\|_{\Sigma}^{3}}{t^{\alpha}},$

provided that $\alpha<1/2$. In view of (5.3) and the definition of $v$, this
entails

$\|I_{1}(t)\|_{L^{2}\cap L^{\infty}}+\|I_{2}(t)\|_{L^{2}\cap
L^{\infty}}\lesssim(\varepsilon^{\prime})^{3}t^{-\alpha+3C(\varepsilon^{\prime})^{2}}.$

Renaming $3C$ to $C$, and integrating (5.4), we readily find that there exists
$W\in L^{2}\cap L^{\infty}$ such that

(5.5) $\|\widehat{w}(t)-W\|_{L^{2}\cap
L^{\infty}}\lesssim(\varepsilon^{\prime})^{3}t^{-\alpha+C(\varepsilon^{\prime})^{2}},$

thus proving (5.1), since

$|\widehat{v}(t,\xi)|=\left|\mathcal{F}\left(U(-t)u(t)\right)(\xi)\right|=\left|e^{-it\frac{\xi^{2}}{2}}\widehat{u}(t,\xi)\right|=|\widehat{u}(t,\xi)|.$

The phase corrector $\Phi$ is obtained by introducing the function

(5.6)
$\Psi(t)=\lambda\int_{1}^{t}\left(|\widehat{w}(\tau)|^{2}-|\widehat{w}(t)|^{2}\right)\frac{d\tau}{\tau}.$

We have, for $t>s>2$,

$\Psi(t)-\Psi(s)=\lambda\int_{s}^{t}\left(|\widehat{w}(\tau)|^{2}-|\widehat{w}(t)|^{2}\right)\frac{d\tau}{\tau}+\lambda\left(|\widehat{w}(t)|^{2}-|\widehat{w}(s)|^{2}\right)\log
s.$

In view of the above estimates, for $t_{2}>t_{1}>2$,

$\displaystyle\left||\widehat{w}(t_{2})|^{2}-|\widehat{w}(t_{1})|^{2}\right|$
$\displaystyle\lesssim\left(|\widehat{w}(t_{2})|+|\widehat{w}(t_{1})|\right)|\widehat{w}(t_{2})-\widehat{w}(t_{1})|$
$\displaystyle\lesssim\left(|\widehat{w}(t_{2})|+|\widehat{w}(t_{1})|\right)(\varepsilon^{\prime})^{3}t_{1}^{-\alpha+C(\varepsilon^{\prime})^{2}}.$

On the other hand, we have

$\displaystyle|\widehat{w}(t)|\lesssim\varepsilon^{\prime},$

from (5.5). Therefore,

(5.7)
$\left||\widehat{w}(t_{2})|^{2}-|\widehat{w}(t_{1})|^{2}\right|\lesssim(\varepsilon^{\prime})^{4}t_{1}^{-\alpha+C(\varepsilon^{\prime})^{2}},\quad
t_{2}>t_{1}>2,$

and so

$|\Psi(t)-\Psi(s)|\lesssim(\varepsilon^{\prime})^{4}s^{-\alpha+C(\varepsilon^{\prime})^{2}}\log
s.$

In particular, there exists $\Phi\in L^{\infty}$ such that, for $t>2$,

(5.8)
$|\Phi-\Psi(t)|\lesssim(\varepsilon^{\prime})^{4}t^{-\alpha+C(\varepsilon^{\prime})^{2}}\log
t.$

The estimate (5.2) then follows from the identity

$\lambda\int_{1}^{t}|\widehat{w}(\tau)|^{2}\frac{d\tau}{\tau}=\lambda|W|^{2}\log
t+\Phi+\Psi(t)-\Phi+\lambda\left(|\widehat{w}(t)|^{2}-|W|^{2}\right)\log t,$

using (5.5) and (5.8), possibly modifying the constant $C$, and using the fact
that the condition on $\alpha$ in Theorem 5.1 is open.

In view of (5.1) and (5.2), we find

$\mathcal{F}\left(U(-t)u(t)\right)=We^{-i\lambda|W|^{2}\log t-i\Phi}+r(t),$

where

$\|r(t)\|_{L^{2}\cap
L^{\infty}}\lesssim(\varepsilon^{\prime})^{3}t^{-\alpha+C(\varepsilon^{\prime})^{2}}.$

Writing

$1=U(t)U(-t)=M(t)D(t)\mathcal{F}M(t)U(-t)=M(t)D(t)\mathcal{F}U(-t)+{\mathcal{R}}(t)\mathcal{F}U(-t),$

we infer

$u(t,x)=M(t)D(t)\mathcal{F}U(-t)u(t)+\rho_{1}(t,x),$

with, in view of Lemma 2.2,

$\|\rho_{1}(t)\|_{L^{2}}\lesssim\frac{\varepsilon^{\prime}}{\sqrt{t}}\|\mathcal{F}U(-t)u(t)\|_{L^{2}}\lesssim\frac{\varepsilon^{\prime}}{\sqrt{t}},$

and, since $\alpha<1/4$, using the first point of Lemma 2.2 with
$\theta=\alpha$ and $s=1-2\alpha$,

$\|\rho_{1}(t)\|_{L^{\infty}}\lesssim\frac{\varepsilon^{\prime}}{t^{1/2+\alpha}}\|\mathcal{F}U(-t)u(t)\|_{H^{1}}\lesssim\varepsilon^{\prime}t^{-1/2-\alpha+C(\varepsilon^{\prime})^{2}}.$

We conclude thanks to the estimates

$\displaystyle\|M(t)D(t)r(t)\|_{L^{2}}=\|r(t)\|_{L^{2}}\lesssim(\varepsilon^{\prime})^{3}t^{-\alpha+C(\varepsilon^{\prime})^{2}},$
$\displaystyle\|M(t)D(t)r(t)\|_{L^{\infty}}=\frac{1}{\sqrt{t}}\|r(t)\|_{L^{\infty}}\lesssim(\varepsilon^{\prime})^{3}t^{-1/2-\alpha+C(\varepsilon^{\prime})^{2}},$

since $u(t)=M(t)D(t)\left(We^{-i\lambda|W|^{2}\log
t-i\Phi}\right)+M(t)D(t)r(t)+\rho_{1}(t)$. ∎

###  5.2. Modified scattering operator: leading order behavior near the origin

In view of Theorem 5.1, the asymptotic state that we consider is
$u_{+}^{\varepsilon}=\mathcal{F}^{-1}\left(W^{\varepsilon}e^{-i\Phi^{\varepsilon}}\right)$,
with $u_{0}^{\varepsilon}=u^{\varepsilon}_{\mid t=0}$. The main remark at this
stage is that the reduction presented in the proof of Theorem 5.1 boils down
the analysis of the asymptotic behavior of $u_{+}^{\varepsilon}$ as
$\varepsilon\to 0$ to a regular asymptotic expansion. In order to treat the
last two cases of Theorem 1.1, we suppose that, for $0<\eta<2$,

$u_{0}^{\varepsilon}=\varepsilon
v_{0}+\varepsilon^{3}w_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right),$

with $v_{0},w_{2}\in\Sigma$ and the remainder term is (measured) in $\Sigma$.
For the last point of Theorem 1.1, we assume $w_{2}=\rho^{\varepsilon}=0$,
while for the second point, $v_{0}=v_{-}$ and $w_{2}=v_{2\mid t=0}$ as in the
first point.

We keep the same notations as the proof of Theorem 5.1, with now $\varepsilon$
instead of $\varepsilon^{\prime}$. The definition of $\widehat{w}$, in
particular the term $B$, requires as a first step the asymptotic description
of $u^{\varepsilon}$ at time $t=1$ instead of only $t=0$.

In view of [14, Theorem 1.1], as in (5.3), we have

$\|u^{\varepsilon}(t)\|_{L^{\infty}}\lesssim\frac{\varepsilon}{\left\langle
t\right\rangle^{1/2}},\quad\|U(-t)u^{\varepsilon}(t)\|_{\Sigma}\lesssim\varepsilon\left\langle
t\right\rangle^{C\varepsilon^{2}}.$

In view of the proof of (5.1) and (5.2), we directly know that

$\|{\partial}_{t}\widehat{w}\|_{L^{1}([1,\infty[,L^{2}\cap
L^{\infty})}\lesssim\varepsilon^{3},$

and so

$\widehat{w}(t)=\widehat{w}(1)+\mathcal{O}(\varepsilon^{3})\text{ in
}L^{\infty}([1,\infty[,L^{2}\cap L^{\infty}).$

By definition,
$\widehat{w}(1)=\widehat{v}(1)=\mathcal{F}\left(U(-1)u^{\varepsilon}(1)\right)=\varepsilon\widehat{v}_{0}+\mathcal{O}(\varepsilon^{3})$
in $\Sigma$.

Regarding $\Phi$, the definition(5.6), and (5.7) yield

$\|\Phi\|_{L^{\infty}}\leqslant\|\Phi-\Psi(t)\|_{L^{\infty}}+\|\Psi(t)\|_{L^{\infty}}\lesssim\varepsilon^{4},$

where the right hand side is estimated uniformly in $t\geqslant 1$. Therefore,

$u_{+}^{\varepsilon}=\mathcal{F}^{-1}\left(We^{-i\Phi}\right)=\varepsilon
v_{0}+\mathcal{O}(\varepsilon^{3})\text{ in }L^{2}.$

Therefore, at leading order, we have

${S^{\rm mod}}(\varepsilon v_{-})=\varepsilon
v_{-}+\mathcal{O}(\varepsilon^{3}),\quad\left(W_{+}^{\rm
mod}\right)^{-1}(\varepsilon v_{0})=\varepsilon
v_{0}+\mathcal{O}(\varepsilon^{3})\quad\text{in }L^{2}.$

##  6\. Higher order asymptotic expansion of the final state

The higher order asymptotic expansion for $u_{+}^{\varepsilon}$, involving an
$\varepsilon^{3}$ term, requires more work, even at the formal level. We first
show how to derive this term, then, in a final subsection, prove the error
estimates announced in Theorem 1.1.

###  6.1. Derivation of the first corrector

We first examine the value of $u^{\varepsilon}$ at time $t=1$, in view of
future connexions with Theorem 5.1. The idea is the same as in Section 4.2:
for finite time, since we consider small data, the asymptotic expansion is
given by Picard’s scheme, and we have directly from Section 4,

$u^{\varepsilon}_{\mid t=1}=\varepsilon
U(1)v_{0}+\varepsilon^{3}\widetilde{w}_{2}+\varepsilon^{5-\eta}\widehat{\rho}^{\varepsilon},$

with

$\widetilde{w}_{2}=U(1)w_{2}-i\lambda\int_{0}^{1}U(1-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)ds,$

and $\|\rho^{\varepsilon}\|_{\Sigma}\lesssim 1$ as $\varepsilon\to 0$.

To derive the first corrector, we plug the asymptotic expansion

$\widehat{w}(t)=\varepsilon\widehat{v}_{0}+\varepsilon^{3}\widehat{\nu}_{2}(t)+\varepsilon^{5-\eta}\widehat{r^{\varepsilon}}(t)$

into (5.4), with $r^{\varepsilon}=\mathcal{O}(1)$ in some topology we shall
precise. We first proceed formally, and then check that the above ansatz
indeed provides a corrector of $u^{\varepsilon}_{+}$ in $L^{2}$. We compute
successively

$|\widehat{v}|^{2}=|\widehat{w}|^{2}=\varepsilon^{2}|\widehat{v}_{0}|^{2}+\mathcal{O}(\varepsilon^{4}),$

hence

$B(t)=e^{i\lambda\int_{1}^{t}|\widehat{v}(\tau)|^{2}\frac{d\tau}{\tau}}=1+i\lambda\varepsilon^{2}|\widehat{v}_{0}|^{2}\log
t+\mathcal{O}(\varepsilon^{4}),$

and

$\widehat{v}=\widehat{w}\overline{B}=\varepsilon\widehat{v}_{0}+\varepsilon^{3}\left(\widehat{\nu}_{2}-i\lambda|\widehat{v}_{0}|^{2}\widehat{v}_{0}\log
t\right)+\mathcal{O}(\varepsilon^{5-\eta}).$

We then expand the factor $I_{1}^{\varepsilon}$ and $I_{2}^{\varepsilon}$ from
(5.4):

$\displaystyle I_{1}^{\varepsilon}(t)$
$\displaystyle=\varepsilon^{3}\mathcal{F}\left(M(-t)-1\right)\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right)+\mathcal{O}(\varepsilon^{5}),$
$\displaystyle I_{2}^{\varepsilon}(t)$
$\displaystyle=\varepsilon^{3}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}-|\widehat{v}_{0}|^{2}\widehat{v}_{0}\right)+\mathcal{O}(\varepsilon^{5}).$

Therefore, the natural candidate for $\nu_{2}$ satisfies:

${\partial}_{t}\widehat{\nu}_{2}=\frac{\lambda}{t}(J_{1}+J_{2}),$

where

$\displaystyle J_{1}$
$\displaystyle=\mathcal{F}\left(M(-t)-1\right)\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right),$
$\displaystyle J_{2}$
$\displaystyle=\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}-|\widehat{v}_{0}|^{2}\widehat{v}_{0}.$

The Cauchy datum for $\nu_{2}$ is given at $t=1$: on the one hand, the above
asymptotic expansion implies

$\widehat{v}_{\mid
t=1}=\varepsilon\widehat{v}_{0}+\varepsilon^{3}\widehat{\nu}_{2\mid
t=1}+\mathcal{O}(\varepsilon^{5-\eta}),$

while on the other hand, the analysis of Section 4 yields

$U(-t)u^{\varepsilon}(t)_{\mid t=1}=\varepsilon
v_{0}+\varepsilon^{3}U(-1)\widetilde{w}_{2}+\mathcal{O}(\varepsilon^{5-\eta}),$

so we infer

$\nu_{2\mid
t=1}=U(-1)\widetilde{w}_{2}=w_{2}-i\lambda\int_{0}^{1}U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)ds.$

As $v_{0}\in\Sigma$, the proof of Theorem 5.1 readily yields
$\frac{1}{t}(J_{1}+J_{2})\in L^{1}([1,\infty[;L^{2}\cap L^{\infty})$. Since
$\Sigma$ is a Banach algebra, invariant under the Fourier transform, and such
that $\Sigma\hookrightarrow L^{1}\cap L^{\infty}$, we have $\nu_{2\mid
t=1}\in\Sigma$, and so $\nu_{2}\in L^{\infty}([1,\infty[;L^{2}\cap
L^{\infty})$ and

$\widehat{\nu}_{2}(t)\mathop{\longrightarrow}\limits_{t\rightarrow\infty}\widehat{\nu}_{2\mid
t=1}+\lambda\int_{1}^{\infty}(J_{1}(\tau)+J_{2}(\tau))\frac{d\tau}{\tau}\quad\text{in
}L^{2}\cap L^{\infty}.$

However, in the asymptotic expansion of $B$, $I_{1}$ and $I_{2}$, the more
precise information that we need is that the remainder term is
$\mathcal{O}(\varepsilon^{5})$ together with some algebraic decay in time, so
we obtain an asymptotic expansion for $\widehat{w}$ in $L^{2}$. This requires
more effort, and the proof is presented in the next subsection.

###  6.2. Justification of the second order asymptotic expansion

As a first step, we analyze the regularity of the corrector $\nu_{2}$. To do
so, we introduce an extra function space: for $0<\gamma<1$, set

$\Sigma^{\gamma}=H^{\gamma}\cap\mathcal{F}(H^{\gamma})=\\{f\in
H^{\gamma}({\mathbb{R}}),\
\|f\|_{\Sigma^{\gamma}}:=\|f\|_{H^{\gamma}}+\|\left\langle
x\right\rangle^{\gamma}f\|_{L^{2}}<\infty\\}.$

We note that like $\Sigma$, $\Sigma^{\gamma}$ is invariant under the Fourier
transform, as well as the action of $U(t)$, and, when $\gamma>1/2$, it is an
algebra embedded into $L^{1}\cap L^{\infty}$.

######  Lemma 6.1.

Let $v_{0},w_{2}\in\Sigma$. Recall that the first corrector is defined by

$\widehat{\nu}_{2}(t)=\widehat{w}_{2}-i\lambda\int_{0}^{1}\mathcal{F}U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)ds+\lambda\int_{1}^{t}(J_{1}(\tau)+J_{2}(\tau))\frac{d\tau}{\tau},$

where $J_{1}$ and $J_{2}$ are defined by

$\displaystyle J_{1}(t)$
$\displaystyle=\mathcal{F}\left(M(-t)-1\right)\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right),$
$\displaystyle J_{2}(t)$
$\displaystyle=\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}-|\widehat{v}_{0}|^{2}\widehat{v}_{0}.$

Then $J_{1},J_{2}\in L^{\infty}([1,\infty[,\Sigma)$, with

$\sup_{t\geqslant 1}\|J_{1}(t)\|_{\Sigma}+\sup_{t\geqslant
1}\|J_{2}(t)\|_{\Sigma}\lesssim\|v_{0}\|_{\Sigma}^{3},$

and, for any $1/2<\gamma<1$,

$\|J_{1}(t)\|_{\Sigma^{\gamma}}+\|J_{2}(t)\|_{\Sigma^{\gamma}}\lesssim\frac{\|v_{0}\|_{\Sigma}^{3}}{t^{(1-\gamma)/2}},$

and therefore $\nu_{2}\in L^{\infty}_{\rm loc}([1,\infty[,\Sigma)\cap
L^{\infty}([1,\infty[,\Sigma^{\gamma})$. Finally,
$\widehat{\nu}_{2}(t)\to\widehat{\nu}_{2}^{\infty}$ in $\Sigma^{\gamma}$ as
$t\to\infty$, where

$\widehat{\nu}_{2}^{\infty}=\widehat{w}_{2}-i\lambda\int_{0}^{1}\mathcal{F}U(-s)\left(|U(s)v_{0}|^{2}U(s)v_{0}\right)ds+\lambda\int_{1}^{\infty}(J_{1}(\tau)+J_{2}(\tau))\frac{d\tau}{\tau}.$

###### Proof of Lemma 6.1.

Since $w_{2},v_{0}\in\Sigma$, the first two terms defining $\widehat{\nu}_{2}$
belong to $\Sigma$ (as it is an algebra). For any (fixed) $t$,
$J_{1},J_{2}\in\Sigma$, and $\nu_{2}(t)\in\Sigma$: the uniform bound in time
is straightforward, but to get some time decay, we pay a little regularity.
Let $0<\gamma<1$: like in the proof of Lemma 2.2, write, for $t\geqslant 1$,

$\displaystyle\|J_{1}(t)\|_{\Sigma^{\gamma}}$
$\displaystyle=\left\|\left(M(-t)-1\right)\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right)\right\|_{\Sigma^{\gamma}}$
$\displaystyle\lesssim\frac{1}{t^{(1-\gamma)/2}}\left\||x|^{1-\gamma}\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right)\right\|_{\Sigma^{\gamma}}$
$\displaystyle\lesssim\frac{1}{t^{(1-\gamma)/2}}\left\|\mathcal{F}^{-1}\left(\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right)\right\|_{\Sigma}\lesssim\frac{\|v_{0}\|_{\Sigma}^{3}}{t^{(1-\gamma)/2}},$

since $\Sigma$ is an algebra. The assumption $\gamma>1/2$ simplifies the
computations in the case of $J_{2}$, as $\Sigma^{\gamma}$ is an algebra, and
the first inequality below is straightforward:

$\displaystyle\|J_{2}(t)\|_{\Sigma^{\gamma}}$
$\displaystyle\lesssim\left(\|M(t)v_{0}\|_{\Sigma^{\gamma}}^{2}+\|v_{0}\|_{\Sigma^{\gamma}}^{2}\right)\|(M(t)-1)v_{0}\|_{\Sigma^{\gamma}}$
$\displaystyle\lesssim\|v_{0}\|_{\Sigma}^{2}\frac{1}{t^{(1-\gamma)/2}}\||x|^{1-\gamma}v_{0}\|_{\Sigma^{\gamma}}\lesssim\frac{\|v_{0}\|_{\Sigma}^{3}}{t^{(1-\gamma)/2}}.$

The lemma follows, since the extra decay in time, $t^{(\gamma-1)/2}$, ensures
the convergence of the integral in the last term defining $\widehat{\nu}_{2}$.
∎

We recall two uniform estimates which will be of constant use in the course of
the proof:

(6.1)
$\|\widehat{v}(t)\|_{L^{\infty}}+\left\|\widehat{M(t)v}\right\|_{L^{\infty}}\lesssim\varepsilon.$

The first quantity is estimated in the proof of Theorem 5.1, thanks to (5.5).
The second one is controlled thanks to (5.3), since, using (2.5),

$\left\|\widehat{M(t)v}\right\|_{L^{\infty}}=\left\|\mathcal{F}M(t)U(-t)u(t)\right\|_{L^{\infty}}=\left\|D\left(\frac{1}{t}\right)M(-t)u(t)\right\|_{L^{\infty}}=\sqrt{t}\left\|u(t)\right\|_{L^{\infty}}.$

The justification of the formal asymptotic expansion derived in the previous
subsection relies on a Gronwall type argument. The error that we want to
control is
$\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}$,
which solves, by construction (keeping in mind that $v_{0}$ does not depend on
time),

${\partial}_{t}\left(\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\right)=\frac{\lambda}{t}B(I_{1}^{\varepsilon}+I_{2}^{\varepsilon})-\frac{\lambda}{t}\varepsilon^{3}(J_{1}+J_{2}).$

We rewrite the right hand side as

$\frac{\lambda}{t}B\left(I_{1}^{\varepsilon}+I_{2}^{\varepsilon}-\varepsilon^{3}J_{1}-\varepsilon^{3}J_{2}\right)+\frac{\lambda}{t}(B-1)\varepsilon^{3}(J_{1}+J_{2}),$

and the last term will be considered as a source term. Indeed, from the
definition

$B(t)=e^{i\lambda\int_{1}^{t}|\widehat{v}(\tau)|^{2}\frac{d\tau}{\tau}}=e^{i\lambda\int_{1}^{t}|\widehat{w}(\tau)|^{2}\frac{d\tau}{\tau}},$

so from (6.1),

(6.2) $\|B(t)-1\|_{L^{\infty}}\lesssim\varepsilon^{2}\log t,$

and Lemma 6.1 yields

(6.3)
$\left\|\varepsilon^{3}(B-1)(J_{1}+J_{2})\right\|_{L^{2}}\lesssim\varepsilon^{5}\frac{\log
t}{t^{(1-\gamma)/2}},$

for any $1/2<\gamma<1$.

We now focus on the term
$B(I_{1}^{\varepsilon}+I_{2}^{\varepsilon}-\varepsilon^{3}J_{1}-\varepsilon^{3}J_{2})$,
to estimate it in $L^{2}$, and examine successively
$B(I_{1}^{\varepsilon}-\varepsilon^{3}J_{1})$ and
$B(I_{2}^{\varepsilon}-\varepsilon^{3}J_{2})$. First, by definition,

$I_{1}^{\varepsilon}-\varepsilon^{3}J_{1}=\mathcal{F}\left(M(-t)-1\right)\mathcal{F}^{-1}\left(\left|\widehat{M(t)v}\right|^{2}\widehat{M(t)v}-\varepsilon^{3}\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\right),$

so the term $v-\varepsilon v_{0}$ is naturally factored out, and we must
relate it to the left hand side, involving $w-\varepsilon
v_{0}-\varepsilon^{3}\nu_{2}$. Since $B\overline{B}=1$, we can write

$\widehat{\rho}:=\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}=B\widehat{v}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}=B\left(\widehat{v}-\varepsilon\overline{B}\widehat{v}_{0}-\varepsilon^{3}\overline{B}\widehat{\nu}_{2}\right),$

so we have

(6.4)
$\widehat{v}-\varepsilon\widehat{v}_{0}=\overline{B}\left(\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\right)+\varepsilon(\overline{B}-1)\widehat{v}_{0}+\varepsilon^{3}\overline{B}\widehat{\nu}_{2}.$

The first term on the right hand side will be treated differently from the
last two terms, which will be considered as source terms. Using the formula

$|z_{2}|^{2}z_{2}-|z_{1}|^{2}z_{1}=|z_{2}|^{2}(z_{2}-z_{1})+z_{1}\operatorname{Re}(z_{2}-z_{1})(\overline{z}_{2}+\overline{z}_{1}),$

with $z_{2}=\mathcal{F}(Mv)$ and $z_{1}=\varepsilon\mathcal{F}(Mv_{0})$, as
well as (6.4), yielding

(6.5) $\mathcal{F}\left(M(v-\varepsilon
v_{0})\right)=\mathcal{F}M\mathcal{F}^{-1}\left(\overline{B}\left(\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\right)+\varepsilon(\overline{B}-1)\widehat{v}_{0}+\varepsilon^{3}\overline{B}\widehat{\nu}_{2}\right),$

we write

$\|I_{1}^{\varepsilon}-\varepsilon^{3}J_{1}\|_{L^{2}}\leqslant\|G_{1}\|_{L^{2}}+\|S_{1}\|_{L^{2}}+\|S_{2}\|_{L^{2}},$

where, simplifying the algebraic structure for the sake of presentation,

$\displaystyle G_{1}$
$\displaystyle=\mathcal{F}\left(M^{-1}-1\right)\mathcal{F}^{-1}\left(\left(|\mathcal{F}(Mv)|^{2}+\varepsilon^{2}|\mathcal{F}(Mv_{0})|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\overline{B}\widehat{\rho}\right)\right),$
$\displaystyle S_{1}$
$\displaystyle=\varepsilon\mathcal{F}\left(M^{-1}-1\right)\mathcal{F}^{-1}\left(\left(|\mathcal{F}(Mv)|^{2}+\varepsilon^{2}|\mathcal{F}(Mv_{0})|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left((\overline{B}-1)\widehat{v}_{0}\right)\right),$
$\displaystyle S_{2}$
$\displaystyle=\varepsilon^{3}\mathcal{F}\left(M^{-1}-1\right)\mathcal{F}^{-1}\left(\left(|\mathcal{F}(Mv)|^{2}+\varepsilon^{2}|\mathcal{F}(Mv_{0})|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\overline{B}\widehat{\nu}_{2}\right)\right).$

The Gronwall term $G_{1}$ is estimated by merely using the boundedness on
$L^{2}$ of $\mathcal{F}\left(M^{-1}-1\right)\mathcal{F}^{-1}$, and the
unitarity of $\mathcal{F}M\mathcal{F}^{-1}$, and we write

$\|G_{1}\|_{L^{2}}\lesssim\left(\|\mathcal{F}(Mv)\|_{L^{\infty}}^{2}+\varepsilon^{2}\|\mathcal{F}(Mv_{0})\|_{L^{\infty}}^{2}\right)\|\widehat{\rho}\|_{L^{2}}\lesssim\varepsilon^{2}\|\widehat{\rho}\|_{L^{2}},$

by using (6.1). For the source term $S_{1}$ and $S_{2}$, we invoke the same
arguments as in the proof of Lemma 2.2, and the fact that
$\mathcal{F}M\mathcal{F}^{-1}$ is unitary on $H^{s}$ for any $s\geqslant 0$:

$\displaystyle\|S_{1}\|_{L^{2}}$
$\displaystyle\lesssim\frac{\varepsilon}{\sqrt{t}}\left\|\left(|\mathcal{F}(Mv)|^{2}+\varepsilon^{2}|\mathcal{F}(Mv_{0})|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left((\overline{B}-1)\widehat{v}_{0}\right)\right\|_{H^{1}}$
$\displaystyle\lesssim\frac{\varepsilon}{\sqrt{t}}\left(\|v\|_{\mathcal{F}(H^{1})}^{2}+\varepsilon^{2}\|v_{0}\|_{\mathcal{F}(H^{1})}^{2}\right)\|(\overline{B}-1)\widehat{v}_{0}\|_{H^{1}}$
$\displaystyle\lesssim\frac{\varepsilon^{3}}{\sqrt{t}}t^{2C\varepsilon^{2}}\|(\overline{B}-1)\widehat{v}_{0}\|_{H^{1}},$

where we have used (5.3) for the last inequality. The last term is controlled
by

$\|(\overline{B}-1)\widehat{v}_{0}\|_{H^{1}}\leqslant\|\overline{B}-1\|_{L^{\infty}}\|v_{0}\|_{\Sigma}+\|\widehat{v}_{0}{\partial}_{x}\overline{B}\|_{L^{2}}\lesssim\varepsilon^{2}\log
t+\|\widehat{v}_{0}{\partial}_{x}\overline{B}\|_{L^{2}},$

in view of (6.2). Writing

$\widehat{v}_{0}{\partial}_{x}\overline{B}=-2i\lambda\widehat{v}_{0}\int_{1}^{t}\operatorname{Re}\left(\overline{\widehat{v}}{\partial}_{x}\widehat{v}\right)\frac{d\tau}{\tau},$

(5.3) now yields

$\|\widehat{v}_{0}{\partial}_{x}\overline{B}\|_{L^{2}}\lesssim\|v_{0}\|_{L^{2}}\int_{1}^{t}\|\widehat{v}(\tau)\|_{L^{\infty}}\|{\partial}_{x}\widehat{v}(\tau)\|_{L^{2}}\frac{d\tau}{\tau}\lesssim\varepsilon^{2}\int_{1}^{t}\frac{d\tau}{\tau^{1-C\varepsilon^{2}}}\lesssim\varepsilon^{2}t^{C\varepsilon^{2}}.$

The estimate for $S_{2}$ is rather similar:

$\displaystyle\|S_{2}\|_{L^{2}}$
$\displaystyle\lesssim\frac{\varepsilon^{3}}{t^{1/2}}\left\|\left(|\mathcal{F}(Mv)|^{2}+\varepsilon^{2}|\mathcal{F}(Mv_{0})|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\overline{B}\widehat{\nu}_{2}\right)\right\|_{H^{1}}$
$\displaystyle\lesssim\frac{\varepsilon^{3}}{t^{1/2}}\left(\|\mathcal{F}(Mv)\|_{H^{1}}^{2}+\varepsilon^{2}\|\mathcal{F}(Mv_{0})\|_{H^{1}}^{2}\right)\|\overline{B}\widehat{\nu}_{2}\|_{H^{1}}\lesssim\frac{\varepsilon^{5}}{t^{1/2-2C\varepsilon^{2}}}\|\overline{B}\widehat{\nu}_{2}\|_{H^{1}}.$

The last term is controlled via Leibniz formula, invoking Lemma 6.1,

$\|\overline{B}\widehat{\nu}_{2}\|_{H^{1}}\leqslant\|\widehat{\nu}_{2}\|_{H^{1}}+\|\widehat{\nu}_{2}{\partial}_{x}\overline{B}\|_{L^{2}}\lesssim\log
t+\varepsilon^{2}t^{C\varepsilon^{2}},$

and the source terms are estimated by

$\|S_{1}\|_{L^{2}}+\|S_{2}\|_{L^{2}}\lesssim\varepsilon^{5}\frac{\log
t}{t^{1/2-3C\varepsilon^{2}}}.$

We now turn to the $L^{2}$ estimate of
$I_{2}^{\varepsilon}-\varepsilon^{3}J_{2}$, and proceed along the same spirit.

$I_{2}^{\varepsilon}-\varepsilon^{3}J_{2}=\left|\widehat{M(t)v}\right|^{2}\widehat{M(t)v}-|\widehat{v}|^{2}\widehat{v}-\varepsilon^{3}\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}+\varepsilon^{3}|\widehat{v}_{0}|^{2}\widehat{v}_{0}.$

We distinguish
$\left|\widehat{M(t)v}\right|^{2}\widehat{M(t)v}-\varepsilon^{3}\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}$
and
$|\widehat{v}|^{2}\widehat{v}-\varepsilon^{3}|\widehat{v}_{0}|^{2}\widehat{v}_{0}$.
Discarding the precise algebraic structure like before,

$\left|\widehat{M(t)v}\right|^{2}\widehat{M(t)v}-\varepsilon^{3}\left|\widehat{M(t)v_{0}}\right|^{2}\widehat{M(t)v_{0}}\approx\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\mathcal{F}M(v-\varepsilon
v_{0}).$

The last factor is again rewritten thanks to (6.5), and we estimate
$I_{2}^{\varepsilon}-\varepsilon^{3}J_{2}$ as

$\|I_{2}^{\varepsilon}-\varepsilon^{3}J_{2}\|_{L^{2}}\leqslant\|G_{2}\|_{L^{2}}+\|S_{3}\|_{L^{2}}+\|S_{4}\|_{L^{2}},$

where

$\displaystyle G_{2}$
$\displaystyle=\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\overline{B}\left(\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\right)\right)$
$\displaystyle\quad-\left(\left|\widehat{v}\right|^{2}+\varepsilon^{2}\left|\widehat{v_{0}}\right|^{2}\right)\overline{B}\left(\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\right),$
$\displaystyle S_{3}$
$\displaystyle=\varepsilon\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\left(\overline{B}-1\right)\widehat{v}_{0}\right)$
$\displaystyle\quad-\varepsilon\left(\left|\widehat{v}\right|^{2}+\varepsilon^{2}\left|\widehat{v_{0}}\right|^{2}\right)\left(\overline{B}-1\right)\widehat{v}_{0},$
$\displaystyle S_{4}$
$\displaystyle=\varepsilon^{3}\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\widehat{\nu}_{2}-\varepsilon^{3}\left(\left|\widehat{v}\right|^{2}+\varepsilon^{2}\left|\widehat{v_{0}}\right|^{2}\right)\widehat{\nu}_{2}.$

For the Gronwall term $G_{2}$, we proceed like for $G_{1}$, and write

$\|G_{2}\|_{L^{2}}\lesssim\varepsilon^{2}\|\widehat{w}-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}\|_{L^{2}}=\varepsilon^{2}\|\widehat{\rho}\|_{L^{2}}.$

The source terms $S_{3}$ and $S_{4}$ are readily of size
$\mathcal{O}(\varepsilon^{5})$; we recover some decay in time by making the
quantity $M-1$ appear systematically. In the case of $S_{3}$, we write

$\displaystyle S_{3}$
$\displaystyle=\varepsilon\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\mathcal{F}M\mathcal{F}^{-1}\left(\left(\overline{B}-1\right)\widehat{v}_{0}\right)$
$\displaystyle\quad-\varepsilon\left(\left|\widehat{v}\right|^{2}+\varepsilon^{2}\left|\widehat{v_{0}}\right|^{2}\right)\left(\overline{B}-1\right)\widehat{v}_{0}\pm\varepsilon\left(\left|\widehat{M(t)v}\right|^{2}+\varepsilon^{2}\left|\widehat{M(t)v_{0}}\right|^{2}\right)\left(\overline{B}-1\right)\widehat{v}_{0},$

and estimate as follows:

$\displaystyle\|S_{3}\|_{L^{2}}$
$\displaystyle\lesssim\varepsilon^{3}\|(M-1)\mathcal{F}^{-1}\left(\left(\overline{B}-1\right)\widehat{v}_{0}\right)\|_{L^{2}}$
$\displaystyle\quad+\|\left(\overline{B}-1\right)\widehat{v}_{0}\|_{L^{\infty}}\|(M-1)\widehat{v}\|_{L^{2}}\left(\|\widehat{Mv}\|_{L^{\infty}}+\|\widehat{v}\|_{L^{\infty}}\right)$
$\displaystyle\lesssim\frac{\varepsilon^{3}}{\sqrt{t}}\|x\mathcal{F}^{-1}\left(\left(\overline{B}-1\right)\widehat{v}_{0}\right)\|_{L^{2}}+\varepsilon^{5}\frac{\log
t}{\sqrt{t}}\times\|x\widehat{v}\|_{L^{2}},$

where we have used (6.1) and (6.2). We have already estimated the $H^{1}$-norm
of $\left(\overline{B}-1\right)\widehat{v}_{0}$,

$\|\left(\overline{B}-1\right)\widehat{v}_{0}\|_{H^{1}}\lesssim\varepsilon^{2}\left(\log
t+t^{C\varepsilon^{2}}\right),$

and therefore

$\|S_{3}\|_{L^{2}}\lesssim\varepsilon^{5}\frac{\log
t}{t^{1/2-C\varepsilon^{2}}}.$

The term $S_{4}$ is controlled similarly, by using the same ideas as above,
and we come up with:

$\frac{d}{dt}\|\widehat{\rho}\|_{L^{2}}\lesssim\frac{\varepsilon^{2}}{t}\|\widehat{\rho}\|_{L^{2}}+\varepsilon^{5}\frac{\log
t}{t^{3/2-3C\varepsilon^{2}}}.$

Gronwall lemma then implies, provided that $\varepsilon>0$ is sufficiently
small:

######  Proposition 6.2.

Suppose that $u^{\varepsilon}$ solves (1.1), with $u^{\varepsilon}_{\mid
t=0}=u_{0}^{\varepsilon}\in\Sigma$ such that

$u_{0}^{\varepsilon}=\varepsilon
v_{0}+\varepsilon^{3}w_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right),$

for $v_{0},w_{2}\in\Sigma$, and some $0<\eta<2$. Then we have

$\sup_{t\geqslant
1}\|\widehat{w}(t)-\varepsilon\widehat{v}_{0}-\varepsilon^{3}\widehat{\nu}_{2}(t)\|_{L^{2}}\lesssim\varepsilon^{5-\eta},$

where $\nu_{2}$ is defined in Lemma 6.1.

In particular, letting $t$ go to infinity, we infer

$W=\varepsilon\widehat{v}_{0}+\varepsilon^{3}\widehat{\nu}_{2}^{\infty}+\mathcal{O}\left(\varepsilon^{5-\eta}\right)\quad\text{in
}L^{2}.$

As $u_{+}^{\varepsilon}=\mathcal{F}^{-1}\left(We^{-i\Phi}\right)$, and we have
seen at the end of Section 5 that
$\|\Phi\|_{L^{\infty}}=\mathcal{O}(\varepsilon^{4})$, we infer

$u_{+}^{\varepsilon}=\varepsilon
v_{0}+\varepsilon^{3}\nu_{2}+\mathcal{O}\left(\varepsilon^{5-\eta}\right)\quad\text{in
}L^{2},$

thus completing the proof of Theorem 1.1.

### Acknowledgments.

The author wishes to thank Léo Bigorgne for discussions on this topic, as well
as Satoshi Masaki and Pavel Naumkin for useful explanations on Theorem 5.1.

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