# Blow-up for the 1D cubic NLS

Valeria Banica  Sorbonne Université, CNRS, Université de Paris, Laboratoire
Jacques-Louis Lions (LJLL), F-75005 Paris, France, and Institut Universitaire
de France (IUF<EMAIL_ADDRESS>,  Renato Lucà  BCAM -
Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E48009 Bilbao,
Basque Country - Spain and Ikerbasque, Basque Foundation for Science, 48011
Bilbao, Basque Country - Spain<EMAIL_ADDRESS>,  Nikolay Tzvetkov  Ecole
Normale Supérieure de Lyon, UMPA, UMR CNRS-ENSL 5669, 46, allée d’Italie,
69364-Lyon Cedex 07, France<EMAIL_ADDRESS>and  Luis Vega  BCAM
- Basque Center for Applied Mathematics, Alameda de Mazarredo 14, E48009
Bilbao, and Dpto. Matemáticas UPV/EHU Apto. 644 48080 Bilbao, Basque Country -
Spain<EMAIL_ADDRESS>
###### Abstract.

We consider the 1D cubic NLS on $\mathbb{R}$ and prove a blow-up result for
functions that are of borderline regularity, i.e. $H^{s}$ for any
$s<-\frac{1}{2}$ for the Sobolev scale and $\mathcal{F}L^{\infty}$ for the
Fourier-Lebesgue scale. This is done by identifying at this regularity a
certain functional framework from which solutions exit in finite time. This
functional framework allows, after using a pseudo-conformal transformation, to
reduce the problem to a large-time study of a periodic Schrödinger equation
with non-autonomous cubic nonlinearity. The blow-up result corresponds to an
asymptotic completeness result for the new equation. We prove it using
Bourgain’s method and exploiting the oscillatory nature of the coefficients
involved in the time-evolution of the Fourier modes. Finally, as an
application we exhibit singular solutions of the binormal flow. More
precisely, we give conditions on the curvature and the torsion of an initial
smooth curve such that the constructed solutions generate several
singularities in finite time.

##  1\. Introduction

We consider the Cauchy problem for the focusing 1D cubic NLS on $\mathbb{R}$:

(1) $iu_{t}+u_{xx}+|u|^{2}u=0,\quad u|_{t=0}=u_{0}.$

The results in this paper hold also in the defocusing case but for simplicity
we will work only with the focusing case.

We recall that with respect to the scaling $\lambda u(\lambda^{2}t,\lambda x)$
that leaves invariant the equation, the invariant space on the Sobolev scale
is $\dot{H}^{-\frac{1}{2}}$ and the one on the Fourier-Lebesgue scale is
$\mathcal{F}L^{\infty}$, i.e. Fourier transform in $L^{\infty}$.

We shall briefly review the initial value problem results for (1) on Sobolev
and Fourier-Lebesgue spaces. The equation is well-posed in $H^{s}$ for $s\geq
0$ (see [34, 15, 10]) and the flow map is uniformly continuous on bounded sets
of $H^{s}$. It turned out that the last property cannot hold in $H^{s}$ for
$s<0$ (see [25, 11]). In [28, 26], it was shown that the Sobolev norms of
Schwartz solutions have controlled growth for $-\frac{1}{2}<s<0$. In [20], the
authors prove the global well-posedness of (1) in $H^{s}$ for $s>-1/2$ in the
sense that the solution map on the Schwartz class admits a unique continuous
extension to $H^{s}$, $s>-1/2$. Therefore in [20] the assumption on the
initial data is

(2) $\langle\xi\rangle^{-\frac{1}{2}^{+}}\widehat{u_{0}}(\xi)\in L^{2}.$

The result of [20] is sharp in the sense that for $s<-1/2$ a norm inflation
with loss of regularity appear [9, 27, 31], in particular (1) is ill-posed in
the Hadamard sense in $H^{s}$, $s<-1/2$. In what concerns the Fourier-Lebesgue
spaces, the problem is locally well-posed on $\mathcal{F}L^{p}$ for
$p<+\infty$ (see [16, 35]). Therefore one can prove local well-posedness of
(1) for data satisfying

(3) $\widehat{u_{0}}(\xi)\in L^{p},\quad\forall\,p<\infty.$

Recall that for $c_{0}\in\mathbb{R}$, the Galilean transformation
$e^{-ic_{0}^{2}t+ic_{0}x}u(t,x-2c_{0}t)$ leaves invariant the set of solutions
of (1), as well as the space $\mathcal{F}L^{p}$ for all $p\leq\infty$.

In summary, we presently have a well-posedness theory that misses the critical
spaces with respect to scaling. Let us notice that the Dirac mass and its
linear Schrödinger evolution
$e^{it\Delta}\delta_{0}=\frac{e^{i\frac{x^{2}}{4t}}}{\sqrt{t}}$ are of
borderline regularity with respect to the scaling and the Galilean symmetries
and therefore do not fit in the scope of applicability of the previously
described results. Actually there are simple explicit solutions of (1) for
$t>0$, having the same kind of borderline regularity, given by

$u_{\alpha}(t,x)=\alpha e^{i|\alpha|^{2}\log
t}\,\,\frac{e^{i\frac{x^{2}}{4t}}}{\sqrt{t}},\quad\alpha\in\mathbb{C}.$

In this spirit recently a new insight has been given to solutions of (1) of
critical regularity. Solutions which are smooth perturbations of $u_{\alpha}$
were constructed in [1, 2, 3, 4, 7], solutions of type superpositions of
$u_{\alpha}$ were constructed in [5, 7] and smooth perturbations of the latter
are given in [18].

Our goal in this work is to prove a suitable form of local well-posedness of
(1) for initial data satisfying

(4) $e^{i\xi^{2}}\widehat{u_{0}}(\xi)\mbox{ is a }2\pi\mbox{-periodic function
in }L^{2}(0,2\pi).$

More importantly, the solutions blow-up in finite time due to a loss of
information of some phases. Clearly if $u_{0}\neq 0$ satisfies (4) then it
does not satisfy neither (2) nor (3).

Let us next give a motivation behind the assumption (4). It would be more
convenient to assume that the initial data is prescribed at time $t=1$ (this
can always be achieved thanks to the invariance under translations in time).
Let us notice that $u_{\alpha}(1,x)$ satisfies (4) and Theorem 1.1 will
include $u_{\alpha}(t,x)$ in the framework of a suitable local well-posedness
and blow-up results. More precisely, here is a soft statement of our first
main result.

######  Theorem 1.1.

Let $s\in(0,1/2)$ and let $u_{0}$ satisfying (4) be such that

$e^{i\xi^{2}}\widehat{u_{0}}(\xi)\in H^{s}(\mathbb{T})\,,$

where $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. Then there exists a unique, in a
suitable sense, solution of

$iu_{t}+u_{xx}+|u|^{2}u=0$

with datum $u_{0}$ at time $t=1$ in the interval $(0,1]$. The solution $u(t)$
belongs for $t\in(0,1]$ to the functional framework

$e^{it\xi^{2}}\hat{u}(t,\xi)\mbox{ is }2\pi\mbox{-periodic},$

and falls out of this framework at $t=0$. The dependence with respect to the
initial data is continuous in the following sense. Let $(u_{0,n})$ be a
sequence of initial data satisfying (4) such that
$(e^{i\xi^{2}}\widehat{u_{0,n}}(\xi))$ converges to
$e^{i\xi^{2}}\widehat{u_{0}}(\xi)$ in $H^{s}(\mathbb{T})$. Then for every
$t\in(0,1]$ the sequence $(e^{it\xi^{2}}\widehat{u_{n}(t)}(\xi))$ converges to
$e^{it\xi^{2}}\widehat{u(t)}(\xi)$ in $H^{s}(\mathbb{T})$, where $u_{n}$ is
the solution of (1) with data $u_{0,n}$. We have that $u\in
C\big{(}(0,1];{\mathcal{S}}^{\prime}(\mathbb{R})\big{)}$ but it blows-up at
zero in the sense that $\lim_{t\rightarrow 0^{+}}u(t,\cdot)$ does not exist in
${\mathcal{S}}^{\prime}(\mathbb{R})$.

######  Remark 1.2.

The Theorem 1.1 extends to $s\geq\frac{1}{2}$ by invoking Lemma 3.2.
Uniqueness will be specified in Theorem 1.4.

######  Remark 1.3.

A similar statement holds for data satisfying

$\exists\,\theta\in\mathbb{R}\,\,\colon\,\,e^{i\theta\xi^{2}}\widehat{u_{0}}(\xi)\mbox{
is }2\pi\mbox{-periodic}$

with a different blow-up time (depending on $\theta$).

Below, we will give a much more precise description of the lack of convergence
of $u(t,\cdot)$ as $t\rightarrow 0^{+}$.

We next continue the motivation of the assumption (4). Observe that for
solutions of the free Schrödinger equation

$iu_{t}+u_{xx}=0,$

it is immediate to see that if $e^{i\xi^{2}}\widehat{u}(1,\xi)$ is
$2\pi-$periodic then $e^{it\xi^{2}}\widehat{u}(t,\xi)$ is also
$2\pi-$periodic, for every $t\in\mathbb{R}$. In fact

$e^{it\xi^{2}}\widehat{u}(t,\xi)=\widehat{u}_{0}(\xi)=\sum\limits_{k\in\mathbb{Z}}a_{k}e^{ik\xi}$

for all $\xi$ and for some Fourier coefficients $a_{k}$. Moreover, any way to
measure the size of $\widehat{u}(t,\xi)$ described in terms of the size of
$|a_{k}|$ will be preserved for all time. It is not so obvious that the
periodicity of

$w(t,\xi):=e^{it\xi^{2}}\widehat{u}(t,\xi),$

holds, at least formally, for all $t$ for solutions of the nonlinear problem
(1). However let us notice that the equation for $w$ is:

$w_{t}(\eta,t)=\frac{i}{8\pi^{3}}e^{-it\eta^{2}}\int
e^{it(\xi_{1}^{2}-\xi_{2}^{2}+(\eta-\xi_{1}+\xi_{2})^{2})}w(\xi_{1},t)\bar{w}(\xi_{2},t)w(\eta-\xi_{1}+\xi_{2},t)\,d\xi_{1}d\xi_{2}.$

The phase can be factorized as $(\xi_{1}-\xi_{2})(\xi_{1}-\eta)$, so it is
invariant under translations. Thus the equation of $w$ is compatible with
periodicity; the solutions we shall construct are in this framework. Now we
shall give a version of Theorem 1.1 that takes in account the physical
variable point of view.

######  Theorem 1.4.

For initial data

$u_{0}(x)=e^{i\frac{x^{2}}{4}}f(x),$

at time $t=1$, with $f$ a periodic function in $H^{s}(0,4\pi)$ and
$s\in(0,\frac{1}{2})$ there exists a unique solution of the 1D cubic NLS
equation (1) that blows up in finite time in the following sense: the solution
belongs for $t\in(0,1]$ to the functional framework

(5) $w(t,\xi):=e^{it\xi^{2}}\hat{u}(t,\xi)\mbox{ is }2\pi\mbox{-periodic},$

$w\in C\big{(}(0,1];H^{s}(\mathbb{T})\big{)},$ and falls out of this framework
at $t=0$. More precisely, by denoting $A_{j}(t)$ the Fourier coefficients of
$w(t)$, then

$\sup_{t\in(0,1]}\|\\{A_{j}(t)\\}_{j\in\mathbb{Z}}\|_{l^{2.s}}\leq
C(\|f\|_{H^{s}(0,4\pi)}),$

and there exists a sequence $\\{\alpha_{j}\\}_{j\in\mathbb{Z}}\in l^{2,s}$
such that

(6) $|A_{j}(t)-e^{i(|\alpha_{j}|^{2}-2\|f\|_{L^{2}(0,4\pi)}^{2})\log
t}\alpha_{j}|\leq C(\|f\|_{H^{s}(0,4\pi)})\,t,\quad\forall
j\in\mathbb{Z},t\in(0,1).$

Finally the solution can be written as

$u(t,x)=\sum_{j\in\mathbb{Z}}A_{j}(t)\frac{e^{i\frac{(x-j)^{2}}{4t}}}{\sqrt{t}}=\sum_{j\in\mathbb{Z}}A_{j}(t)e^{it\Delta}\delta_{j},$

where $\delta_{j}$ is the Dirac mass at $j$, so in particular $u$ focalises at
$t=0$.

######  Remark 1.5.

As a consequence of the above result even though $|A_{j}(t)|$ have a limit for
all $j$ when $t$ goes to zero, the Fourier coefficient themselves don’t, due
to the the phase loss $(|\alpha_{j}|^{2}-2\|f\|_{L^{2}(0,4\pi)}^{2})\log t$.
This is what we refer as a phase blow up. Loss of phase phenomena for
nonlinear Schrödinger equation have been noticed in [29, 30].

The proof of Theorem 1.4 goes as follows. Using the pseudo-conformal
transformation

(7)
$v(\tau,y):=\frac{e^{i\frac{y^{2}}{4\tau}}}{\sqrt{\tau}}\,\overline{u}(\frac{1}{\tau},\frac{y}{\tau}),$

and imposing the ansatz (5) reduces the problem to constructing solutions
$v(\tau)$ for $\tau\geq 1$ for a $2\pi$-periodic Schrödinger equation with a
non-autonomous cubic nonlinearity:

(8) $iv_{\tau}+v_{yy}+\frac{1}{\tau}|v|^{2}v=0.$

These solutions are obtained by gluing solutions on time-intervals of size one
constructed in turn by using Bourgain’s arguments [6]. In a next step we start
the study of their asymptotic behaviour as $\tau$ goes to infinity. To do so
we first derive pointwise estimates for the Fourier in time transform of the
derivative in time of the Fourier modes of $v(\tau)$. Then, we show that the
$H^{s}$-norms of $v(\tau)$ are bounded in time by using also strongly the
oscillatory nature of the coefficients involved in the time-evolution of the
Fourier modes of $v(\tau)$. Finally the limit of the modulations of the
Fourier modes of $v(\tau)$, that yields the blow-up phenomena of Theorem 1.4,
is obtained by gathering all this previous information and using again the
oscillatory nature of the time-evolutions of the Fourier modes of $v(\tau)$.

Existence of solutions $u$ as in Theorem 1.4 was proved in [5] for more
regular data, and correspond to existence of wave operators for the equation
on $v$. More precisely given $\\{\alpha_{k}\\}_{k\in\mathbb{Z}}\in l^{2,s}$
with $s>\frac{1}{2}$ solutions for (1) of type (5) and satisfying (6) were
obtained, but without any information on the set of functions $f$ reached at
$t=1$. Regular perturbations of such solutions were constructed in [18]. Also,
in [7] the case when $\\{\alpha_{k}\\}_{k\in\mathbb{Z}}\in l^{p}$ was treated,
and local in time small solutions were obtained for $1<p<\infty$, with an
improvement of the time of existence under smallness in $l^{\infty}$, and
global small solutions were obtained for $p=2$. In turn, Theorem 1.4 is an
asymptotic completeness result for the equation on $v$, at regularity
$l^{2,s}$ with $0<s<\frac{1}{2}$, i.e. given any $f\in H^{s}$ we construct a
solution of (1) of type (5) and satisfying (6).

Finally, based on Theorem 1.4 we prove a result for the binormal flow:

$\chi_{t}(t,x)=\chi_{x}\times\chi_{xx}(t,x),$

which is used as a model for the dynamics in time in a 3D fluid of superfluid
of a filament of vorticity concentrated on a 3D-curve $\chi(t)$ parametrized
by arclenght parameter $x$ ([12], [23] and the references therein). The
binormal flow is rigorously connected to the focusing equation (1) due to
Hasimoto’s transform [21]. The self-similar solutions of the binormal flow,
that are smooth curves generating a corner in finite time, correspond to
phenomena that appear in fluids and superfluids (Figures 1 and 2), and were
studied rigorously in [19]. We recall that in [5] it is constructed the
evolution in time, according to the binormal flow, of curves that are
initially polygonal lines and that become instantaneously smooth. As a
consequence, since the binormal flow is time reversible, smooth curves can
generate several singularities in the shape of corners in finite time. This
result is based on the existence of wave operators for the Schrödinger
equation (8). Here, we shall use the asymptotic completeness result for (8),
stemming from Theorem 1.4, to obtain a general criterium for generating
singularities in finite time through the binormal flow.

######  Theorem 1.6.

Let $\chi_{1}$ be a curve with $4\pi-$periodic curvature and torsion equal to
$\frac{x}{2}$ modulo a $H^{\frac{3}{2}^{+}}4\pi$-periodic function. Then there
exists $\chi(t)$, with $\chi(1)=\chi_{1}$, a strong solution of the binormal
flow on $\mathbb{R}^{*}$ and weak solution on $\mathbb{R}$ which generates
several corner-singularities at time $t=0$.

We first note that the hypothesis on the data $\chi_{1}$ is satisfied by the
family of self-similar solutions of the binormal flow, whose profile curvature
is constant and whose torsion equal to $\frac{x}{2}$. In fluids vortex
filaments dynamics having the behavior of the self-similar solutions is
observed for delta-wing planes, see Figure 1. In superfluids this family of
solutions were used by physicists studying the vortex lines reconnections and
the Kelvin waves that emerge from, see Figure 2. We note also that if we want
a blow-up after time $t_{0}$ then we should start with a curve with $4\pi
t_{0}$-periodic curvature and torsion equal to $\frac{x}{2t_{0}}$ modulo a
$H^{\frac{3}{2}^{+}}4\pi t_{0}$-periodic function.

Figure 1. Vortex filaments in a fluid encountering a triangular obstacle of
the delta wing type, H. Werlé, ONERA 63.

Figure 2. Reconnection scaling in quantum fluids, E. Fonda, K. Sreenivasan and
D. Lathrop, PNAS 19

The proof of Theorem 1.6 relies first on a control in time of the
$l^{2,\frac{3}{2}^{+}}$-norms of the sequences
$\\{A_{j}(t)\\}_{j\in\mathbb{Z}}$ in Theorem 1.4 provided that $f\in
H^{\frac{3}{2}^{+}}$, and then on the proof of the persistence of this
regularity in the limit sequence $\\{\alpha_{j}\\}_{j\in\mathbb{Z}}$. Finally,
the construction in [5] for $\\{\alpha_{j}\\}_{j\in\mathbb{Z}}\in
l^{2,\frac{3}{2}^{+}}$ allows to construct the solutions in Theorem 1.6.

The paper is organized as follows: in the next section we give the proof of
Theorem 1.4 while the last section contains the proof of Theorem 1.6. The
proof of Theorem 1.1 is contained in the one of Theorem 1.4.

Acknowledgements: This research is partially supported as follows. VB is
partially supported by the Institut Universitaire de France, by the French ANR
project SingFlows. RL is supported by BERC program 2022-2025 and by MICINN
(Spain) projects Severo Ochoa CEX2021-001142, PID2021-123034NB-I00 and by the
Ramon y Cajal fellowship RYC2021-031981-I. NT is partially supported by the
ANR project Smooth ANR-22-CE40-0017. LV is funded by MICINN (Spain) projects
Severo Ochoa CEX2021-001142, and PID2021-126813NB-I00 (ERDF A way of making
Europe), and by Eusko Jaurlaritza project IT1615-22 and BERC program. The
authors would like to thank the referees for their careful reading and
suggestions.

##  2\. Proof of Theorem 1.4

###  2.1. The equations

To prove Theorem 1.4 we look thus for solutions of (1) on $t>0$ of the type

(9) $u(t,x)=\sum_{j}A_{j}(t)\frac{e^{i\frac{(x-j)^{2}}{4t}}}{\sqrt{t}},$

with initial data

(10)
$u(1,x)=e^{i\frac{x^{2}}{4}}f(x)=\sum_{j}A_{j}(1)e^{i\frac{(x-j)^{2}}{4}}\iff
A_{j}(1)=\hat{f}(j)e^{-i\frac{j^{2}}{4}}.$

After pseudo-conformal transformation (7) such a solution $u(t)$ on $0<t\leq
1$ turns into a solution $v(t)$ of the NLS on $1\leq t<\infty$:

(11) $iv_{t}+v_{xx}+\frac{1}{t}|v|^{2}v=0,$

that is a $2\pi-$periodic function (modulo constants, that we avoid for the
sake of the clearness of the presentation, and that do not affect the
arguments and results) of the type:

(12) $v(t,x)=\sum_{j}B_{j}(t)e^{itj^{2}}e^{ixj},$

where

(13) $B_{j}(t)=\overline{A_{j}}(\frac{1}{t}).$

In particular the Fourier coefficients of $v(t)$ satisfy

$|\hat{v}(t,j)|=|A_{j}(1/t)|,$

thus estimating Sobolev norms of $v(t)$ is equivalent to weighted estimates on
the sequence $A_{j}(1/t)$.

######  Remark 2.1.

In particular from Theorem 1.4 we obtain periodic NLS solutions $v$ with
Fourier modes converging in modulus:

$||\hat{v}(t,j)|^{2}-|\alpha_{j}|^{2}|\leq\frac{C}{t},$

but with fluctuations of the modes.

Also, we can note that as usually for questions concerning the long-time
behavior of cubic NLS, we have the relation:

$B_{j}(t)=\mathcal{F}(e^{-it\Delta}v(t))(j).$

We denote then

$B(t,x):=\sum_{j}B_{j}(t)e^{ijx}=e^{-it\Delta}v(t)(x),$

and noting that we have the following expression for the Fourier transform in
both time and space variable:

$v(t,x)=\sum_{j}(e^{itj^{2}}B_{j}(t))e^{ijx}\Longrightarrow\hat{v}(\lambda,j)=\hat{B}_{j}(\lambda+j^{2}),$

we introduce Bourgain’s norms:

(14) $\|v\|_{X^{s,b}}:=\Big{(}\int\sum_{j}\langle
j\rangle^{2s}\langle\lambda+j^{2}\rangle^{2b}|\widehat{v}(\lambda,j)|^{2}d\lambda\Big{)}^{\frac{1}{2}}=\Big{(}\int\sum_{j}\langle
j\rangle^{2s}\langle\lambda\rangle^{2b}|\widehat{B}_{j}(\lambda)|^{2}d\lambda\Big{)}^{\frac{1}{2}}=:\|B\|_{H^{s,b}},$

and for localizations in time $f_{\nu}(t)=\eta_{\nu}(t)f(t)$ in
$(\nu-1,\nu+2)$, valued $1$ on $[\nu,\nu+1]$:

(15) $\|v\|_{X_{\nu}^{s,b}}:=\|v_{\nu}\|_{X^{s,b}}=\Big{(}\int\sum_{j}\langle
j\rangle^{2s}\langle\lambda\rangle^{2b}|\widehat{B}_{j,\nu}(\lambda)|^{2}d\lambda\Big{)}^{\frac{1}{2}}=:\|B\|_{H^{s,b}_{\nu}}.$

Now we remark that $u(t)$ of ansatz (9) is a solution of (1) if and only if
the sequence $A(t)=\\{A_{j}(t)\\}_{j\in\mathbb{Z}}$ is solution of the system

$i\partial_{t}A_{k}(t)=\frac{1}{t}\sum_{(j_{1},j_{2},j_{3})\in\Sigma_{k}}e^{-i\frac{k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2}}{4t}}A_{j_{1}}(t)\overline{A_{j_{2}}(t)}A_{j_{3}}(t),$

translating into the sequence $B(t)=\\{B_{j}(t)\\}_{j\in\mathbb{Z}}$ being a
solution of the system (modulo constants)

(16)
$i\partial_{t}B_{k}(t)=\frac{1}{t}\sum_{(j_{1},j_{2},j_{3})\in\Sigma_{k}}e^{-it(k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2})}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t),$

where

$\Sigma_{k}:=\\{(j_{1},j_{2},j_{3}),\,k-j_{1}+j_{2}-j_{3}=0\\}.$

Let us introduce the nonresonant set

$NR_{k}=\\{(j_{1},j_{2},j_{3}),\,k-j_{1}+j_{2}-j_{3}=0,\,k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2}\neq
0\\}.$

First we note that if $k-j_{1}+j_{2}-j_{3}=0$ we have

$\omega_{k,j_{1},j_{2}}:=k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2}=(k-j_{1})(j_{1}-j_{2}).$

For any real function $a$, using a standard symmetrization argument, as
$a(k)-a(j_{1})+a(j_{2})-a(j_{3})$ vanishes on the resonant set, we have:

(17) $\partial_{t}\sum_{k}a(k)|B_{k}(t)|^{2}\\\
=\frac{1}{2ti}\sum_{k-j_{1}+j_{2}-j_{3}=0}(a(k)-a(j_{1})+a(j_{2})-a(j_{3}))e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t)\overline{B_{k}(t)}\\\
=\frac{1}{2ti}\sum_{k;NR_{k}}(a(k)-a(j_{1})+a(j_{2})-a(j_{3}))e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t)\overline{B_{k}(t)}.$

Therefore by taking $a\equiv 1$ we obtain that the system conserves the “mass”
:

(18)
$\sum_{k}|B_{k}(t)|^{2}=\|v(1)\|_{L^{2}(0,2\pi)}^{2}=\|f\|_{L^{2}(0,2\pi)}^{2}=:M.$

In particular (16) can be written as

(19)
$i\partial_{t}B_{k}(t)=\frac{1}{t}\sum_{NR_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t)+\frac{1}{t}\left(2M-|B_{k}(t)|^{2}\right)B_{k}(t).$

At this point we are in a situation to apply a modified scattering technique
for 1D cubic NLS as initiated by Ozawa ([32]), Hayashi and Naumkin ([22]) and
Carles ([8]). More precisely, we will show that the first term in the right-
hand-side of (19) is controllable by oscillatory integral techniques, while
the second term will dictate the long time dynamics of (19). Let us notice
that in the proof of modified scattering for 1D cubic NLS in [32] (see (2.4)
in the revisited proof in [24]) the modulus of the perturbative term is
absolutely integrable in time, while in here the perturbative term can be
shown to be integrable in time only in the sense of oscillatory integrals.

###  2.2. Construction à la Bourgain of the solution $v(t)$ on $[1,\infty)$

In view of (10), (12) and (13), we consider the initial condition

$b_{j}:=\overline{A_{j}}(1)=\overline{\hat{f}(j)e^{-ij^{2}}}.$

Let

$\eta\in\mathcal{C}^{\infty}_{0}(-1,2),\quad\eta(t)=1,\quad\forall t\in[0,1],$

such that we have a partition of unity

$\frac{1}{2}\sum_{\nu\in\mathbb{N}}\eta_{\nu}(t)=1,\quad\forall
t\in[1,\infty),$

with

$\eta_{\nu}(t)=\eta(t-\nu).$

For all $\nu\in\mathbb{N}^{*}$ we construct
$\\{B_{k}^{\nu}\\}_{k\in\mathbb{Z}}$ recursively as follows, by using at each
step Bourgain’s method ([6], see also §3.5.1 in [13]). First we construct
$\\{B_{k}^{1}\\}_{k\in\mathbb{Z}}$ solution of:

$i\partial_{t}B_{k}^{1}=\frac{1}{t}\eta_{1}(t)\sum_{(j_{1},j_{2},j_{3})\in\Sigma_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}^{1}\overline{B_{j_{2}}^{1}}B_{j_{3}}^{1}(t),\quad
B_{k}^{1}(1)=b_{k}.$

We note that $\\{B_{k}^{1}\\}_{k\in\mathbb{Z}}$ solves on $[1,2]$ the equation
we are interested in:

$i\partial_{t}B_{k}^{1}=\frac{1}{t}\sum_{(j_{1},j_{2},j_{3})\in\Sigma_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}^{1}\overline{B_{j_{2}}^{1}}B_{j_{3}}^{1}(t).$

Next we construct $\\{B_{k}^{2}\\}_{k\in\mathbb{Z}}$ solution of

$i\partial_{t}B_{k}^{2}=\frac{1}{t}\eta_{2}(t)\sum_{(j_{1},j_{2},j_{3})\in\Sigma_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}^{2}\overline{B_{j_{2}}^{2}}B_{j_{3}}^{2}(t),\quad
B_{k}^{2}(2)=B_{k}^{1}(2).$

And so on. Then

$B_{k}(t):=\left\\{\begin{array}[]{c}B_{k}^{1}(t),\forall t\in[1,2],\\\
B_{k}^{2}(t),\forall t\in[2,3],\\\ \mbox{etc}.\end{array}\right.$

solves (19) on $(1,\infty)$ with $B_{k}(1)=b_{k}$, therefore $v$ defined as in
(12) solves (11) on $(0,1)$ with $v(1,x)=\sum_{j}b_{j}e^{ij^{2}}e^{ixj}$. In
view of the choice of $b_{j}^{\prime}s$ we recover the initial data
$v(1)=\overline{f}$. Therefore we have obtained solutions $u(t)$ on
$t\in(0,1]$ of the type (9), with initial data (10).

Also, arguing as in §3.5.1 in [13] we have the following informations
involving Bourgain’s norms (14)-(15), for $0\leq
s,\frac{1}{2}<b<\frac{5}{8},\nu\in\mathbb{N},t\in[\nu,\nu+1]$:

(20) $\|\\{B_{j}(t)\\}\|_{l^{2,s}}=\|v(t)\|_{H^{s}}\leq
C\|v\|_{X^{s,b}_{\nu}}=C\|B\|_{H^{s,b}_{\nu}}\leq
C\|v(\nu)\|_{H^{s}}=C\|\\{B_{j}(\nu)\\}\|_{l^{2,s}},$

where the constant $C$ is independent of $t$, $\nu$ and depends only of the
$l^{2}$ norm of $\\{b_{k}\\}$.

###  2.3. Extra-properties on $B_{j}$’s

In order to determinate the large time behaviour of $B_{j}(t)$ we shall first
obtain some large time controls. We start with a decay estimate of the Fourier
transform in time of a localization of $\partial_{t}B_{j}(t)$.

######  Lemma 2.2.

For any $\nu\in\mathbb{N}$, and $\eta_{\nu}$ a smooth cutoff supported in
$[\nu,\nu+1]$, we have for any $\epsilon>0$ and $\epsilon^{\prime}>0$ the
following estimate on the Fourier transform in time:

$|\mathcal{F}(\eta_{\nu}(\cdot)\partial_{t}B_{k}(\cdot))(\lambda)|\leq\frac{C(\epsilon^{\prime},\|\\{B_{j}(1)\\}\|_{l^{2}})\||\xi|^{1^{+}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}}{\nu\,\langle\lambda\rangle^{\min\\{\frac{\epsilon}{2},(b-\frac{1}{2})^{-}\\}}}\|\\{B_{j}(\nu)\\}\|_{l^{2,\epsilon+2\epsilon^{\prime}}},\quad\forall
k\in\mathbb{Z},\lambda\in\mathbb{R},$

###### Proof.

We denote

$\Lambda_{k,m}=\\{(j_{1},j_{2})\in\mathbb{Z}^{2},(k-j_{1})(j_{1}-j_{2})=m\\},$

and we have:

(21) $\mathcal{F}(\eta_{\nu}(\cdot)\partial_{t}B_{k}(\cdot))(\lambda)$ $=\int
e^{it\lambda}\eta_{\nu}(t)\frac{1}{t}\sum_{m\neq
0}\sum_{j_{1},j_{2}\in\Lambda_{k,m}}e^{-itm}(B_{j_{1}}\overline{B}_{j_{2}}B_{k-j_{1}+j_{2}})(t)dt$
$+\int
e^{it\lambda}\eta_{\nu}(t)\frac{1}{t}\left(2M-|B_{k}(t)|^{2}\right)B_{k}(t)dt.$

In the following we shall estimate the first term111This term is controllable
directly for $\\{B_{j}\\}\in l^{2,s}$ with $s>\frac{1}{2}$, but not for lower
regularity. , that we denote by $F(\nu,k,\lambda)$:

$\frac{1}{\nu}\int\int\sum_{m\neq
0}e^{-it(\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m)}\tilde{\eta}_{\nu}(t)dt\sum_{j_{1},j_{2}\in\Lambda_{k,m}}\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})d\lambda_{1}d\lambda_{2}d\lambda_{3}$
$=\frac{1}{\nu}\int\sum_{m\neq
0}\widehat{\tilde{\eta}}_{\nu}(\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m)\sum_{j_{1},j_{2}\in\Lambda_{k,m}}\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})d\lambda_{1}d\lambda_{2}d\lambda_{3},$

where $\tilde{\eta}_{\nu}(t)=\frac{\nu}{t}\eta_{\nu}(t)$ and preserve the same
estimates as $\eta_{\nu}$ at Fourier level (see [7]). We perform Cauchy-
Schwarz in $j_{1},j_{2}$ and use the divisor bounds in $\mathbb{Z}$ (which is
an Euclidean division domain), for any $\epsilon^{\prime}>0$:

$\\#\Lambda_{k,m}\leq
C(\epsilon^{\prime})|m|^{\epsilon^{\prime}}=C(\epsilon^{\prime})|(k-j_{1})(j_{1}-j_{2})|^{\epsilon^{\prime}},\quad\forall
j_{1},j_{2}\in\Lambda_{k,m},$

to get

$|F(\nu,k,\lambda)|\leq\frac{C({\epsilon^{\prime}})}{\nu}\int\sum_{m\neq
0}|\widehat{\tilde{\eta}}_{\nu}(\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m)|$
$\times\Big{(}\sum_{j_{1},j_{2}\in\Lambda_{k,m}}|(k-j_{1})(j_{1}-j_{2})|^{2{\epsilon^{\prime}}}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})|^{2}\Big{)}^{\frac{1}{2}}d\lambda_{1}d\lambda_{2}d\lambda_{3}.$

Now we perform Cauchy-Schwarz in $m$, for $N\in\mathbb{N}$:

$|F(\nu,k,\lambda)|\leq\frac{C(\epsilon^{\prime})}{\nu}\int\Big{(}\sum_{m\neq
0}|\widehat{\tilde{\eta}}_{\nu}(\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m)|^{2}\langle\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m\rangle^{N}\Big{)}^{\frac{1}{2}}$
$\times\Big{(}\sum_{m\neq
0,j_{1},j_{2}\in\Lambda_{k,m}}\frac{|(k-j_{1})(j_{1}-j_{2})|^{2\epsilon^{\prime}}}{\langle\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m\rangle^{N}}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})|^{2}\Big{)}^{\frac{1}{2}}d\lambda_{1}d\lambda_{2}d\lambda_{3}$
$\leq\frac{C(\epsilon^{\prime})\||\xi|^{\frac{(N+1)^{+}}{2}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}}{\nu}$
$\times\int\Big{(}\sum_{m\neq
0,j_{1},j_{2}\in\Lambda_{k,m}}\frac{|(k-j_{1})(j_{1}-j_{2})|^{2\epsilon^{\prime}}}{\langle\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+m\rangle^{N}}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})|^{2}\Big{)}^{\frac{1}{2}}d\lambda_{1}d\lambda_{2}d\lambda_{3}.$

Eventually we perform Cauchy-Schwarz in $\lambda_{1},\lambda_{2},\lambda_{3}$
to get using $b>\frac{1}{2}$:

$|F(\nu,k,\lambda)|\leq\frac{C(\epsilon^{\prime})\||\xi|^{\frac{(N+1)^{+}}{2}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}}{\nu}\Big{(}\int\sum_{m\neq
0,j_{1},j_{2}\in\Lambda_{k,m}}\frac{|(k-j_{1})(j_{1}-j_{2})|^{2\epsilon^{\prime}}}{\langle\lambda-\lambda_{1}+\lambda_{2}-\lambda_{3}+(k-j_{1})(j_{1}-j_{2})\rangle^{N}}\times$
$\times\langle\lambda_{1}\rangle^{1^{+}}|\hat{B}_{j_{1},\nu}(\lambda_{1})|^{2}\langle\lambda_{2}\rangle^{1^{+}}|\hat{B}_{j_{2},\nu}(\lambda_{2})|^{2}\langle\lambda_{3}\rangle^{1^{+}}|\hat{B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})|^{2}d\lambda_{1}d\lambda_{2}d\lambda_{3}\Big{)}^{\frac{1}{2}}.$

Even without taking advantage of the denominator, we get the decay in $\nu$:

$|F(\nu,k,\lambda)|\leq\frac{C(\epsilon^{\prime})\||\xi|^{\frac{1}{2}^{+}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}}{\nu}\|B\|_{H^{0,\frac{1}{2}^{+}}_{\nu}}^{2}\|B\|_{H^{2\epsilon^{\prime},\frac{1}{2}^{+}}_{\nu}}$
$\leq\frac{C(\epsilon^{\prime})\||\xi|^{\frac{1}{2}^{+}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}}{\nu}C(\|\\{B_{j}(1)\\}\|_{l^{2}})\|\\{B_{j}(\nu)\\}\|_{l^{2,2\epsilon^{\prime}}},$

where in the last line we have used estimate (20), the “mass” conservation
(18) and

(22) $|(k-j_{1})(j_{1}-j_{2})|^{2\epsilon^{\prime}}\leq\max\\{\langle
j_{1}\rangle,\langle j_{2}\rangle,\langle
k-j_{1}+j_{2}\rangle\\}^{4\epsilon^{\prime}}.$

To get the decay in $\lambda$ we argue as follows. By taking $N=1$ (so the
upper-bound will involve
$\||\xi|^{1^{+}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}})$ we get directly
$\langle\lambda\rangle^{-\frac{1}{2}}$ except in the case

$|\lambda-(\lambda_{1}-\lambda_{2}+\lambda_{3}-(k-j_{1})(j_{1}-j_{2}))|\leq\frac{|\lambda|}{2}.$

If $|\lambda_{1}-\lambda_{2}+\lambda_{3}|\geq\frac{|\lambda|}{4}$ then
$|\lambda_{j}|\geq\frac{|\lambda|}{12}$ at least for one $j\in\\{1,2,3\\}$,
and in this case we get
$\frac{1}{\langle\lambda\rangle^{(b-\frac{1}{2})^{-}}}$ decay by using the
$H_{\nu}^{0,b}$ and $H_{\nu}^{2\epsilon^{\prime},b}$ norms. Eventually, we are
left with the case

$|\lambda+(k-j_{1})(j_{1}-j_{2})|\leq\frac{3}{4}|\lambda|\Longrightarrow|(k-j_{1})(j_{1}-j_{2})|\geq\frac{1}{4}|\lambda|.$

Then we get $\frac{1}{\langle\lambda\rangle^{\frac{\epsilon}{2}}}$ decay with
a $\|\\{B_{j}(\nu)\\}\|_{l^{2,\epsilon+2\epsilon^{\prime}}}$ bound by using
(22) with $\epsilon$ instead of $2\epsilon^{\prime}$.

We note here that the second term in (21) can be treated by one integration by
parts which yields the factor $\frac{1}{\nu\lambda}$, then in a similarly way
as above for $F(\nu,k,\lambda)$. As $(b-\frac{1}{2})^{-}<1$ the analysis of
the second term in (21) is completed and the Lemma follows.

∎

######  Remark 2.3.

We do not get an upper-bound on
$\mathcal{F}(\eta_{\nu}(t)\partial_{t}B_{k}(t))(\lambda)$ by working with
$B_{j}$’s just in $l^{2}$.

Now we will obtain a control in time the weighted norms of $\\{B_{j}(t)\\}$ as
follows.

######  Lemma 2.4.

For any $t\geq 1,r\in[0,\frac{1}{2})$ we have:

$\|\\{B_{j}(t)\\}\|_{l^{2,r}}\leq C(r,\|\\{B_{j}(1)\\}\|_{l^{2,r}}).$

###### Proof.

The case $r=0$ follows by mass conservation (18). By using (17) and an
integration by parts we have

(23) $\sum_{k}\langle k\rangle^{2r}|B_{k}(t)|^{2}-\sum_{k}\langle
k\rangle^{2r}|B_{k}(1)|^{2}$ $=\int_{1}^{t}\sum_{k;NR_{k}}(\langle
k\rangle^{2r}-\langle j_{1}\rangle^{2r}+\langle j_{2}\rangle^{2r}-\langle
j_{3}\rangle^{2r})e^{-i\tau(k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2})}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{k}}(\tau)\frac{d\tau}{\tau}$
$\leq
C\sum_{k;NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{k}}(t)\frac{1}{t}|+C\sum_{k;NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{k}}(1)|$
$+C\int_{1}^{t}\sum_{k;NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{k}}(\tau)|\frac{d\tau}{\tau^{2}}$
$+C\Big{|}\int_{1}^{t}\sum_{k;NR_{k}}\varphi_{k,j_{1},j_{2},j_{3}}e^{i\tau(j_{1}^{2}-j_{2}^{2}+j_{3}^{2}-k^{2})}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{\partial_{\tau}B_{k}}(\tau)\frac{d\tau}{\tau}\Big{|},$

where

$\varphi_{k,j_{1},j_{2},j_{3}}:=\frac{\langle k\rangle^{2r}-\langle
j_{1}\rangle^{2r}+\langle j_{2}\rangle^{2r}-\langle
j_{3}\rangle^{2r}}{k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2}}.$

We aim to prove that for any $t\geq 1$ and $r\in(0,\frac{1}{2})$ we have:

(24) $\sum_{k}\langle k\rangle^{2r}|B_{k}(t)|^{2}\leq\sum_{k}\langle
k\rangle^{2r}|B_{k}(1)|^{2}+C(r,\|\\{B_{j}(1)\\}\|_{l^{2}})\sup_{\tau\in[1,t]}\|\\{B_{j}(\tau)\\}\|_{l^{2,r}}^{0^{+}},$

so by resorbing the last terms in the left-hand-side we obtain the Lemma for
$r\in(0,\frac{1}{2})$:

$\sum_{k}\langle k\rangle^{2r}|B_{k}(t)|^{2}\leq
C(r,\|\\{B_{j}(1)\\}\|_{l^{2}})\sum_{k}\langle k\rangle^{2r}|B_{k}(1)|^{2}.$

To get (24) we shall use the following technical lemma (estimates in the cases
$r\geq\frac{1}{2}$ were obtained in Lemma 2.5 in [14]).

######  Lemma 2.5.

For $r<\frac{1}{2}$ we have

(25) $|\varphi_{k,j_{1},j_{2},j_{3}}|\leq\frac{C}{\max\\{\langle
k\rangle,\langle j_{1}\rangle,\langle j_{2}\rangle,\langle
j_{3}\rangle\\}},\quad\forall k,(j_{1},j_{2},j_{3})\in NR_{k}.$

As a consequence, for two sequences $\\{N_{j}\\}$ and $\\{P_{j}\\}$ of
positive numbers we have

(26) $\sum_{k;(j_{1},j_{2},j_{3})\in
NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}|N_{j_{1}}N_{j_{2}}N_{j_{3}}P_{k}\leq
C\|N\|_{l^{2}}^{2}\|N\|_{l^{2,0^{+}}}\|P\|_{l^{2}}.$

###### Proof.

By symmetry in the $k,j_{1},j_{2},j_{3}$ variables, to get (25) it is enough
to get

$\varphi_{k,j_{1},j_{2},j_{3}}\leq\frac{C}{\langle k\rangle},\quad\forall
k,(j_{1},j_{2},j_{3})\in NR_{k}.$

To get this estimate we shall decompose the summation in $k,j_{1},j_{2},j_{3}$
in two regions as follows.

a) Case $|k|\lesssim\max\\{|k-j_{1}|,|j_{1}-j_{2}|\\}$. As $r<\frac{1}{2}$ we
have $|(\langle x\rangle^{2r})^{\prime}|=|2rx\langle x\rangle^{2r-1}|\leq
C\langle x\rangle^{2r-1}$ and the mean value theorem (used in two manners: on
$\langle k\rangle^{2r}-\langle j_{1}\rangle^{2r}$ and $\langle
j_{2}\rangle^{2r}-\langle j_{3}\rangle^{2r}$ to get $|j_{1}-j_{2}|$-decay, as
well as on $\langle k\rangle^{2r}-\langle j_{3}\rangle^{2r}$ and $\langle
j_{1}\rangle^{2r}-\langle j_{2}\rangle^{2r}$ to get $|k-j_{1}|-$decay) implies
that

$\varphi_{k,j_{1},j_{2},j_{3}}\leq\frac{C}{\max\\{|k-j_{1}|,|j_{1}-j_{2}|\\}\\}}\leq\frac{C}{\langle
k\rangle}.$

b) Case $|k|\gg\max\\{|k-j_{1}|,|j_{1}-j_{2}|\\}$. In this case
$|k|\lesssim|j_{1}|$, $k$ and $j_{1}$ have same sign, then also
$|k|\lesssim|j_{2}|$, and $j_{1}$ and $j_{2}$ have same sign, and we also
directly get $|k|\lesssim|j_{3}|$, and $k$ and $j_{3}$ have same sign. As
$|(\langle x\rangle^{2r})^{\prime\prime}|\leq C\langle x\rangle^{2r-2}$,
applying twice the mean value theorem, which yields at the numerator the
double integral $\int_{j_{1}}^{k}\int_{y}^{y-j_{1}+j_{2}}(\langle
x\rangle^{2r})^{\prime\prime}dxdy$, whose all boundary values are of same
sign, we get that

$\varphi_{k,j_{1},j_{2},j_{3}}\leq\frac{C}{\min\\{\langle k\rangle,\langle
j_{1}\rangle,\langle j_{2}\rangle,\langle
j_{3}\rangle\\}^{2-2r}}\leq\frac{C}{\langle k\rangle}.$

Estimate (26) follows using (25) and Cauchy-Schwarz:

$\sum_{k;(j_{1},j_{2},j_{3})\in
NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}|N_{j_{1}}N_{j_{2}}N_{j_{3}}P_{k}$
$\leq\sum_{k,j_{1}}\frac{C}{\langle j_{1}\rangle^{\frac{1}{2}^{-}}\langle
k\rangle^{\frac{1}{2}^{+}}}N_{j_{1}}P_{k}\sum_{j_{2}\neq
0}(N_{j_{2}}^{2}+N_{k-j_{1}+j_{2}}^{2})\leq
C\|N\|_{l^{2}}^{2}\|N\|_{l^{2,0^{+}}}\|P\|_{l^{2}}.$

∎

By using Lemma 2.5 we can control as wanted in (24) the first three terms in
(23), while for the last term we use the expression of $\partial_{\tau}B_{j}$
from (16) to get:

(27) $\sum_{k}\langle k\rangle^{2r}|B_{k}(t)|^{2}\leq\sum_{k}\langle
k\rangle^{2r}|B_{k}(1)|^{2}+C(r,\|\\{B_{j}(1)\\}\|_{l^{2}})\sup_{\tau\in[1,t]}\|\\{B_{j}(\tau)\\}\|_{l^{2,r}}^{0^{+}}$
$+C\sum_{0\leq\nu\leq
t}\Big{|}\int_{1}^{t}\sum_{N_{j}}\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}e^{i\tau(j_{1}^{2}-j_{2}^{2}+j_{3}^{2}-j_{4}^{2}+j_{5}^{2}-j_{6}^{2})}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{j_{4}}}B_{j_{5}}\overline{B_{j_{6}}}(\tau)\frac{d\tau}{\tau^{2}}\Big{|},$

where

$N_{j}:=\\{(j_{1},j_{2},j_{3},j_{4},j_{5},j_{6}),\,j_{1}-j_{2}+j_{3}-j_{4}+j_{5}-j_{6}=0,\,j_{1}^{2}-j_{2}^{2}+j_{3}^{2}-(j_{4}-j_{5}+j_{6})^{2}\neq
0\\}.$

We are left with the last term, that by using the notation
$v(\tau,x)=\sum_{j}e^{ijx+i\tau j^{2}}B_{j}(\tau)$ from §2.1, the partition of
unity in time of $[1,\infty)$ from §2.2 with $\eta_{\nu}$ supported in
$(\nu-1,\nu+2)$ and a $\nu_{1}$ supported away from zero, and the notation
$v_{\nu}$ for a smooth in time restriction to $(\nu-1,\nu+2)$, can be
rewritten as

$C\sum_{0\leq\nu\leq
t}\Big{|}\int_{1}^{t}\sum_{N_{j}}\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}\eta_{\nu}(\tau)\widehat{v_{\nu}(\tau)}(j_{1})\overline{\widehat{v_{\nu}(\tau)}(j_{2})}\widehat{v_{\nu}(\tau)}(j_{3})\overline{\widehat{v_{\nu}(\tau)}(j_{4})}\widehat{v_{\nu}(\tau)}(j_{5})\overline{\widehat{v_{\nu}(\tau)}(j_{6})}\frac{d\tau}{\tau^{2}}\Big{|}.$

We use now also Fourier transform in time to write the terms in the summation
in $\nu$ as:

$I_{\nu}:=\Big{|}\frac{1}{\nu^{2}}\int_{N_{\lambda}}\sum_{N_{j}}\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}\widehat{\mu_{\nu}v_{\nu}}(\lambda_{1},j_{1})\overline{\hat{v}_{\nu}(\lambda_{2},j_{2})}\hat{v}_{\nu}(\lambda_{3},j_{3})\overline{\hat{v}_{\nu}(\lambda_{4},j_{4})}\hat{v}_{\nu}(\lambda_{5},j_{5})\overline{\hat{v}_{\nu}(\lambda_{6},j_{6})}d\Lambda\Big{|},$

where
$\mu_{\nu}(\tau):=\frac{\nu^{2}}{\tau^{2}}\eta_{\nu}(\tau)\mathbb{I}_{[1,t]}(\tau)$,
$d\Lambda:=d\lambda_{1}...d\lambda_{6}$ and $N_{\lambda}$ is the restriction
set to
$\lambda_{1}-\lambda_{2}+\lambda_{3}-\lambda_{4}+\lambda_{5}-\lambda_{6}=0$.

We collect now some basic results about the spaces $X^{s,b}$. One can easily
check that

(28)
$\|\frac{\nu^{2}}{\tau^{2}}\eta_{\nu}(\tau)F(\tau,x)\|_{X^{s,b}}\lesssim\|F\|_{X^{s,b}}.$

We have (Lemma 2.1 in [33]), for all $\varepsilon>0$ sufficiently small and
$b<1/2$:

(29)
$\|1_{[\alpha,\beta]}(\tau)F(\tau,x)\|_{X^{s,b}}\lesssim_{\varepsilon}\|F\|_{X^{s,b+\varepsilon}},$

with constant that does not depend on the interval $[\alpha,\beta]$.
Interpolating the embeddings $X^{0,3/8}\hookrightarrow L^{4}_{\tau,x}$ and
$X^{0+,\frac{1}{2}+}\hookrightarrow L^{6}_{\tau,x}$ (see for instance
(27)-(28) in [33]), for $\varepsilon>0$ yields

(30)
$\|F\|_{L^{6-\varepsilon^{\prime}}_{\tau,x}}\lesssim_{\varepsilon^{\prime},\varepsilon^{\prime\prime}}\|F\|_{X^{\varepsilon^{\prime\prime},\frac{1}{2}-2\varepsilon^{\prime\prime}}},$

as long as we take
$0<\varepsilon^{\prime\prime}<\varepsilon_{1}<\varepsilon^{\prime}<\varepsilon_{2}$
with $\varepsilon_{1}$ and $\varepsilon_{2}$ sufficiently small thresholds.

Getting back to estimating $I_{\nu}$ we distinguish two cases. First we assume
that

(31) $\max\\{\langle j_{4}\rangle,\langle j_{5}\rangle,\langle
j_{6}\rangle\\}\gtrsim\max\\{\langle j_{4}-j_{5}+j_{6}\rangle,\langle
j_{1}\rangle,\langle j_{2}\rangle,\langle j_{3}\rangle\\},$

We assume, without loss of generality, that the max is attained at $\langle
j_{4}\rangle$ and also at $\langle j_{5}\rangle$, since the alternated sum of
the $j$’s vanishes. It will be clear by the argument that all the other
combination of indexes are treated in the exact same way. We then use the
estimate

$|\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}|\lesssim 1,$

that follows from Lemma 2.5, to bound

(32) $\displaystyle
I_{\nu}\leq\frac{C}{\nu^{2}}\int_{N_{\lambda}}\sum_{N_{j}}|\widehat{\mu_{\nu}v_{\nu}}(\lambda_{1},j_{1})|\,\Pi_{l=2}^{6}|\hat{v}_{\nu}(\lambda_{l},j_{l})|d\Lambda.$

Defining for any $x\in\mathbb{T}$ and $6\varepsilon\ll r$:

$F_{1}(\tau,x):=\int\sum_{j_{1}}e^{ixj_{1}+i\tau\lambda_{1}}\langle
j_{1}\rangle^{-\varepsilon}|\widehat{\mu_{\nu}v_{\nu}}(j_{1},\lambda_{1})|d\lambda_{1},\,F_{4}(\tau,x):=\int\sum_{j_{4}}e^{ixj_{4}+i\tau\lambda_{4}}\langle
j_{4}\rangle^{5\varepsilon}|\widehat{v_{\nu}}(j_{4},\lambda_{4})|d\lambda_{4},$
(33) $F_{j}(\tau,x):=\int\sum_{j_{l}}e^{ixj_{l}+i\tau\lambda_{j}}\langle
j_{l}\rangle^{-\varepsilon}|\widehat{v_{\nu}}(k_{j},\lambda_{j})|d\lambda_{j},\qquad
l=2,3,5,6,$

from (32) we obtain by using the vanishing of the alternated sums of the $j$’s
and $\lambda$’s that

(34) $\displaystyle I_{\nu}\leq\frac{C}{\nu^{2}}\int\int
F_{1}\overline{F_{2}}F_{3}\overline{F_{4}}F_{5}\overline{F_{6}}(\tau,x)dxd\tau.$

Taking suitable
$\varepsilon^{\prime\prime}\ll\varepsilon^{\prime}\ll\varepsilon$ and using
(30), (29) and (28) we note that

$\|F_{1}\|_{L^{6-\varepsilon^{\prime}}_{\tau,x}}\leq
C\|F_{1}\|_{X^{\varepsilon^{\prime\prime},\frac{1}{2}-2\varepsilon^{\prime\prime}}}\leq
C\|\mu_{\nu}v_{\nu}\|_{X^{\varepsilon^{\prime\prime}-\varepsilon,\frac{1}{2}-2\varepsilon^{\prime\prime}}}\leq
C\|\frac{\nu^{2}}{\tau^{2}}\eta_{\nu}v_{\nu}(\tau,x)\|_{X^{\varepsilon^{\prime\prime}-\varepsilon,\frac{1}{2}-\varepsilon^{\prime\prime}}}$
$\leq
C\|v\|_{X_{\nu}^{\varepsilon^{\prime\prime}-\varepsilon,\frac{1}{2}-\varepsilon^{\prime\prime}}}\leq
C\|v\|_{X^{0,b}_{\nu}},$

as $b\in(\frac{1}{2},\frac{5}{8})$. Thus using the Hölder inequality, the
embedding $X^{0+,\frac{1}{2}+}\hookrightarrow L^{6}_{\tau,x}$ and its
consequence by Sobolev embeddings
$X^{\frac{\delta}{2(6+\delta)}+,\frac{1}{2}+}\hookrightarrow
L^{6+\delta}_{\tau,x}$ (see for instance Corolary 2.1 of [17]) we get:

(35) $\displaystyle I_{\nu}$
$\displaystyle\leq\frac{C}{\nu^{2}}\Big{(}\|F_{1}\|_{L^{6-\varepsilon^{\prime}}_{\tau,x}}\|F_{4}\|_{L^{6+\frac{3\varepsilon^{\prime}}{3-\varepsilon^{\prime}}}_{\tau,x}}\prod_{j=2,3,5,6}\|F_{j}\|_{L^{6}_{\tau,x}}\Big{)}\leq\frac{C}{\nu^{2}}\Big{(}\|v\|_{X^{6\varepsilon,b}_{\nu}}\|v\|^{5}_{X^{0,b}_{\nu}}\Big{)}$
$\displaystyle\leq\frac{C}{\nu^{2}}\|v(\nu)\|_{H^{6\varepsilon}}\|v(\nu)\|^{5}_{L^{2}}\leq\frac{C\|\\{B_{j}(1)\\}\|_{l^{2}}^{5}}{\nu^{2}}\sup_{\tau\in[1,t]}\|\\{B_{j}(\tau)\\}\|_{l^{2,6\varepsilon}}$
$\displaystyle\leq\frac{C(r,\|\\{B_{j}(1)\\}\|_{l^{2}})}{\nu^{2}}\sup_{\tau\in[1,t]}\|\\{B_{j}(\tau)\\}\|_{l^{2,r}}^{0^{+}},$

as $6\varepsilon\ll r$. Therefore we get (24) in the case (31).

In the remaining case when (31) does not hold we have

$\max\\{\langle j_{4}\rangle,\langle j_{5}\rangle,\langle
j_{6}\rangle\\}\lesssim\max\\{\langle j_{4}-j_{5}+j_{6}\rangle,\langle
j_{1}\rangle,\langle j_{2}\rangle,\langle j_{3}\rangle\\}.$

Thus we may assume without loss of generality that the maximum is attained on
$\langle j_{1}\rangle$. We also recall that from Lemma 2.5 we get

(36)
$|\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}|\lesssim\frac{1}{\max\\{\langle
j_{4}-j_{5}+j_{6}\rangle,\langle j_{1}\rangle,\langle j_{2}\rangle,\langle
j_{3}\rangle\\}}.$

We define:

$F_{1}(\tau,x):=\sum_{k_{1}}\int e^{ixj_{1}+i\tau\lambda_{1}}\langle
j_{1}\rangle^{5\varepsilon-1}|\widehat{\mu_{\nu}v}(j_{1},\lambda_{1})|d\lambda_{1},$

and $F_{j}$ as in (33) but for $j=2,3,4,5,6$. Proceeding as above (using the
$L^{6+\varepsilon^{\prime}}$ norm for $F_{2}$, for instance) we arrive to the
estimate

(37) $\displaystyle
I_{\nu}\leq\frac{C}{\nu^{2}}\|v\|_{X^{6\varepsilon-1,b}_{\nu}}\|v\|^{5}_{X^{0,b}_{\nu}}\leq\frac{C}{\nu^{2}}\|v\|^{6}_{X^{0,b}_{\nu}}\leq\frac{C}{\nu^{2}}\|v(\nu)\|^{6}_{L^{2}}\leq
C\|v(1)\|^{5}_{L^{2}}=C(\|\\{B_{j}(1)\\}\|_{l^{2}}),$

where here we choose $\varepsilon<\frac{1}{6}$. This is again sufficient for
getting (24).

∎

###  2.4. The asymptotic behavior of $B_{j}$’s

We shall prove now first the following asymptotic behavior.

######  Lemma 2.6.

For data $\\{B_{k}(1)\\}\in l^{2,s}$ with $0<s<\frac{1}{2}$ and for all
$k\in\mathbb{Z}$ we have:

$\exists\,\beta_{k}:=\lim_{t\rightarrow\infty}\tilde{B}_{k}(t),$

where

$\tilde{B}_{k}(t):=e^{i2M\log
t-i\int_{1}^{t}|B_{k}(\tau)|^{2}\frac{d\tau}{\tau}}B_{k}(t),$

with the following estimate on the rate of convergence:

$\sup_{k\in\mathbb{Z}}|\tilde{B}_{k}(t)-\beta_{k}|\leq\frac{C(\|\\{B_{j}(1)\\}\|_{l^{2,s}})}{t}.$

###### Proof.

We first note that in view of (19) we get

$i\partial_{t}\tilde{B}_{k}(t)=\frac{e^{i2M\log
t-i\int_{1}^{t}|B_{k}(\tau)|^{2}\frac{d\tau}{\tau}}}{t}\sum_{NR_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t),$

so for $t_{2}\geq t_{1}\geq 1$ we have by integration by parts:

$\tilde{B}_{k}(t_{2})-\tilde{B}_{k}(t_{1})=\int_{t_{1}}^{t_{2}}\frac{1}{\tau}\sum_{NR_{k}}e^{-i\tau\omega_{k,j_{1},j_{2}}}e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}(B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}})(\tau)d\tau$
$=\Big{[}\sum_{NR_{k}}\frac{e^{-i\tau\omega_{k,j_{1},j_{2}}}}{\omega_{k,j_{1},j_{2}}}\frac{e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}}{\tau}(B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}})(\tau)\Big{]}_{t_{1}}^{t_{2}}$
$-\int_{t_{1}}^{t_{2}}\sum_{NR_{k}}\frac{e^{-i\tau\omega_{k,j_{1},j_{2}}}}{\omega_{k,j_{1},j_{2}}}\Big{(}\frac{e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}}{\tau}(B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}})(\tau)\Big{)}_{\tau}d\tau.$

The boundary term can be easily upper-bounded by Cauchy-Schwarz in
$j_{1},j_{2}$ by

$\sup_{\tau\in[t_{1},t_{2}]}\frac{1}{\tau}\sum_{NR_{k}}\frac{|B_{j_{1}}B_{j_{2}}B_{j_{3}}|)(\tau)}{|(k-j_{1})(j_{1}-j_{2})|}\leq\frac{C(\|\\{B_{j}(1)\\}\|_{l^{2}})}{t_{1}}.$

So can be the terms in the integral when the derivative in $\tau$ falls on
$\frac{e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}}{\tau}$.
Remains to estimate just one kind of term, when the derivative falls on
$B_{j_{3}}$ for instance, on which we use the partition of unity in time:

$\sum_{\nu}\frac{1}{\nu}\int_{t_{1}}^{t_{2}}\tilde{\eta}_{\nu}(\tau)\sum_{NR_{k}}\frac{e^{-i\tau\omega_{k,j_{1},j_{2}}}}{(k-j_{1})(j_{1}-j_{2})}e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}B_{j_{1}}\overline{B_{j_{2}}}\partial_{\tau}B_{k-j_{1}+j_{2}}(\tau)d\tau.$

We use Fourier transform in time to write this term as:

$J_{k}:=\sum_{t_{1}\leq\nu\leq
t_{2}}\Big{|}\frac{1}{\nu}\int\int\int\int\sum_{m\neq
0;\,j_{1},j_{2}\in\Lambda_{k,m}}\frac{e^{i\tau(m-\lambda_{1}+\lambda_{2}-\lambda_{3})}}{m}e^{i2M\log\tau-i\int_{1}^{\tau}|B_{k}(\tau^{\prime})|^{2}\frac{d\tau^{\prime}}{\tau^{\prime}}}\tilde{\eta}_{\nu}(\tau)$
$\left.\times\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})\widehat{\partial_{\tau}B}_{k-j_{1}+j_{2},\nu}(\lambda_{3})d\lambda_{1}d\lambda_{2}d\lambda_{3}d\tau\right|.$

Now we use Lemma 2.2:

$J_{k}\leq\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C}{\nu^{2}}\int\int\int\sum_{m\neq
0;\,j_{1},j_{2}\in\Lambda_{k,m}}\frac{|\widehat{\tilde{\eta}_{\nu,k}}(m-\lambda_{1}+\lambda_{2}-\lambda_{3})|}{|m|}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})|d\lambda_{1}d\lambda_{2}d\lambda_{3}$
$\leq\sum_{t_{1}\leq\nu\leq t_{2}}\frac{\tilde{C}}{\nu^{2}}\int\int\sum_{m\neq
0;\,j_{1},j_{2}\in\Lambda_{k,m}}\frac{1}{|m|}|B_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})|d\lambda_{1}d\lambda_{2}.$

Here the constant is the one of Lemma 2.2 (note that for the sequence of
functions $\eta_{\nu}$ that we are considering we have
$\||\xi|^{1^{+}}\hat{\eta}_{\nu}(\xi)\|_{L^{\infty}}$ uniformly bounded in
$\nu$):

(38)
$\tilde{C}=C(\epsilon,\|\\{B_{j}(1)\\}\|_{l^{2}})\|\\{B_{j}(\nu)\\}\|_{l^{2,\epsilon}},$

Now we perform successively Cauchy-Schwarz in $j_{1},j_{2}$, for which we use
again, for $m\neq 0$ and $\epsilon>0$:

$\\#\Lambda_{k,m}\leq C(\epsilon)m^{\epsilon},$

then in $m$, and eventually in $\lambda_{1},\lambda_{2}$:

$J_{k}\leq\tilde{C}\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C(\epsilon)}{\nu^{2}}\int\int\sum_{m\neq
0}\frac{1}{|m|^{1-\epsilon}}\Big{(}\sum_{j_{1},j_{2}\in\Lambda_{k,m}}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})|^{2}\Big{)}^{\frac{1}{2}}d\lambda_{1}d\lambda_{2}$
$\leq\tilde{C}\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C(\epsilon)}{\nu^{2}}\int\int\Big{(}\sum_{m\neq
0;\,j_{1},j_{2}\in\Lambda_{k,m}}\frac{1}{|m|^{1-4\epsilon}}|\hat{B}_{j_{1},\nu}(\lambda_{1})\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})|^{2}\Big{)}^{\frac{1}{2}}d\lambda_{1}d\lambda_{2}$
$\leq\tilde{C}\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C(\epsilon)}{\nu^{2}}\Big{(}\int\int\sum_{j_{1},j_{2}}|\langle\lambda_{1}\rangle^{b}\hat{B}_{j_{1},\nu}(\lambda_{1})\langle\lambda_{2}\rangle^{b}\widehat{\overline{B}}_{j_{2},\nu}(\lambda_{2})|^{2}d\lambda_{1}d\lambda_{2}\Big{)}^{\frac{1}{2}}$
$\leq\tilde{C}\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C(\epsilon)}{\nu^{2}}\|B\|_{H^{b,0}_{\nu}}^{2}\leq\tilde{C}\sum_{t_{1}\leq\nu\leq
t_{2}}\frac{C(\epsilon,\|\\{B_{j}(1)\\}\|_{L^{2}})}{\nu^{2}}\leq\frac{C(\epsilon,\|\\{B_{j}(1)\\}\|_{l^{2}})}{t_{1}}\sup_{\tau\in[t_{1},t_{2}]}\|\\{B_{j}(\tau)\\}\|_{l^{2,\epsilon}}$
$\leq\frac{C(\epsilon,\|\\{B_{j}(1)\\}\|_{l^{2}})}{t_{1}}\|\\{B_{j}(1)\\}\|_{l^{2,\epsilon}}\leq\frac{C(\epsilon,\|\\{B_{j}(1)\\}\|_{l^{2}})}{t_{1}}\|\\{B_{j}(1)\\}\|_{l^{2,s}}^{0^{+}},$

where we have used the expression of the constant from (38), Lemma 2.4 and
interpolation.

∎

We can now prove Theorem 1.4. In view of (19) we have

$i\partial_{t}\Big{(}e^{i(2M-|\beta_{k}|^{2})\log
t}B_{k}(t)\Big{)}=\frac{e^{i(2M-|\beta_{k}|^{2})\log
t}}{t}\sum_{NR_{k}}e^{-it\omega_{k,j_{1},j_{2}}}B_{j_{1}}(t)\overline{B_{j_{2}}(t)}B_{j_{3}}(t)$
$-\frac{e^{i(2M-|\beta_{k}|^{2})\log
t}}{t}(|B_{k}(t)|^{2}-|\beta_{k}|^{2})B_{k}(t).$

Then, by integrating in time the first term as for the proof of Lemma 2.6, and
by integrating the second term by using the convergence result of Lemma 2.6 we
obtain

$\exists\,\alpha_{k}:=\lim_{t\rightarrow\infty}e^{i(2M-|\beta_{k}|^{2})\log
t}B_{k}(t),$

with the following estimate on the rate of convergence:

$|B_{k}(t)-\alpha_{k}e^{i(2M-|\alpha_{k}|^{2})\log
t}|\leq\frac{C(\|\\{B_{j}(1)\\}\|_{l^{2}})}{t}.$

Recalling definition (18) of the “mass” we obtain the conclusion (6) of
Theorem 1.4.

######  Remark 2.7.

Theorem 1.4 corresponds to asymptotic completeness for the $B_{j}$’s. For
proving existence of wave operators at the same level of regularity one may
proceed as follows. Let $\\{\alpha_{k}\\}_{k\in\mathbb{Z}}\in l^{2,s}$ with
$0<s$ and let $M:=\sum_{k}|\alpha_{k}|^{2}$ and construct solutions for $t\geq
T(\\{\alpha_{k}\\})$ to

$i\partial_{t}B_{k}=\frac{1}{t}\sum_{NR_{k}}e^{it\omega_{k,j_{1},j_{2}}}(\alpha_{j_{1}}+B_{j_{1}})\overline{(\alpha_{j_{2}}+B_{j_{2}})}(\alpha_{j_{3}}+B_{j_{3}})(t)-\frac{1}{t}|(\alpha_{k}+B_{k})|^{2}(\alpha_{k}+B_{k})(t),$

with $\|B_{k}(t)\|_{l^{2,s}}\overset{t\rightarrow\infty}{\longrightarrow}0$ by
performing a logarithmic phase change and by doing a fixed point argument
for222To avoid smallness hypothesis on $\\{\alpha_{k}\\}_{k\in\mathbb{Z}}$
that would come from the linear term in the last integral term of the fixed
point operator, this last integrant can be replaced by an oscilatory one in
the same spirit as in §2.1 of [5].

$\Phi(\\{B_{j}\\}_{j\in\mathbb{Z}})_{k}(t)=i\int_{t}^{\infty}\sum_{NR_{k}}e^{i\tau\omega_{k}+i(|\alpha_{j_{1}}|^{2}-|\alpha_{j_{2}}|^{2}+|\alpha_{j_{3}}|^{2}-|\alpha_{k}|^{2})\log\tau}(\alpha_{j_{1}}+B_{j_{1}})\overline{(\alpha_{j_{2}}+B_{j_{2}})}(\alpha_{j_{3}}+B_{j_{3}})(\tau)\frac{d\tau}{\tau},$
$-i\int_{t}^{\infty}(|(\alpha_{k}+B_{k})|^{2}-|\alpha_{k}|^{2})(\alpha_{k}+B_{k})(\tau)\frac{d\tau}{\tau},$

in the space of sequences $\\{B_{k}\\}_{k\in\mathbb{Z}}$ having the property
that for all truncations $\eta_{\nu}$ of support of size one, at distance of
size $\nu$ from the origin,

$\sum_{\lambda,k}\langle\lambda\rangle^{2b}\langle
k\rangle^{2s}|\widehat{\eta_{\nu}B_{k}}(\lambda)|^{2}\leq c(\nu),$

with $c(\nu)$ summable in $\nu$ and $b>\frac{1}{2}$.

##  3\. Proof of Theorem 1.6

As explained in the Introduction, we shall prove that
$\|\\{B_{j}(t)\\}\|_{l^{2,\frac{3}{2}^{+}}}$ is uniformly bounded in time. To
do so we shall first prove a uniform in time bound on the $l^{2,1}$-norms,
then we shall revisit Lemma 2.2 at higher regularity, and gathering all
together we shall estimate the evolution in time of the
$l^{2,\frac{3}{2}^{+}}$-norm similarly as for Lemma 2.4.

We start by noting that from (11) we get the energy evolution law: If

(39) $E(v)(t):=\frac{1}{2}\|\nabla
v(t)\|_{L^{2}}^{2}-\frac{1}{4t}\|v(t)\|_{L^{4}}^{4},$

then

(40) $\partial_{t}E(v)(t)=\frac{1}{4t^{2}}\|v(t)\|_{L^{4}}^{4}.$

######  Lemma 3.1.

For $0<s<\frac{1}{2}$ we have the following control:

$\sum_{k}\langle k\rangle^{2}|B_{k}(t)|^{2}\leq\sum_{k}\langle
k\rangle^{2}|B_{k}(1)|^{2}+C(s,\|\\{B_{j}(1)\\}\|_{l^{2}})\|\\{B_{j}(1)\\}\|_{l^{2,s}}^{\theta},$

where $\theta(s)>2$ and $\theta(\frac{1}{2}^{-})=2^{+}$.

###### Proof.

By using the energy law (39)-(40) and Sobolev embeddings we have

$\|\nabla v(t)\|_{L^{2}}^{2}-\|\nabla v(1)\|_{L^{2}}^{2}\leq
C\sup_{1\leq\tau\leq t}\|v(\tau)\|_{L^{4}}^{4}\leq C\sup_{1\leq\tau\leq
t}\|v(\tau)\|_{H^{\frac{1}{4}}}^{4}=C\sup_{1\leq\tau\leq
t}\|\\{B_{j}(\tau)\\}\|_{H^{\frac{1}{4}}}^{4}.$

Therefore by using Lemma 2.4 with $r=\frac{1}{4}$ we end up with
$\|\\{B_{j}(1)\\}\|_{l^{2,\frac{1}{4}}}^{4}$ that gives by interpolation
$\|\\{B_{j}(1)\\}\|_{l^{2,s}}^{\theta}(s)$ with $\theta(s)>2$ as
$s<\frac{1}{2}$, and $\theta(\frac{1}{2}^{-})=2^{+}$.

∎

######  Lemma 3.2.

We have:

$\|\\{B_{j}(t)\\}\|_{l^{2,\frac{3}{2}^{+}}}\leq
C(r,\|\\{B_{j}(1)\\}\|_{l^{2,\frac{3}{2}^{+}}}).$

###### Proof.

The proof follows the one of Lemma 2.4 with $r=\frac{3}{2}^{+}$, which gives
in particular the weaker estimate (see Lemma 2.5 in [14]):

$\Big{|}\frac{\langle k\rangle^{2r}-\langle j_{1}\rangle^{2r}+\langle
j_{2}\rangle^{2r}-\langle
j_{3}\rangle^{2r}}{k^{2}-j_{1}^{2}+j_{2}^{2}-j_{3}^{2}}\Big{|}\leq
C\max\\{\langle k\rangle,\langle j_{1}\rangle,\langle j_{2}\rangle,\langle
j_{3}\rangle\\}^{1^{+}},\quad\forall k,(j_{1},j_{2},j_{3})\in NR_{k}.$

On the first three terms in (23) we use this estimate and Lemma 3.1 to get

$\sum_{k;NR_{k}}|\varphi_{k,j_{1},j_{2},j_{3}}B_{j_{1}}\overline{B_{j_{2}}}B_{j_{3}}\overline{B_{k}(\tau)}|\leq
C\|B(\tau)\|_{l^{2,1}}^{4}\leq C(\|\\{B_{j}(1)\\}\|_{l^{2,1}}).$

The last integral term in (23) involving $\partial_{\tau}B_{k}$ can be treated
similarly as in the beginning of the proof of Lemma 2.4, distributing the loss
$\max\\{\langle j_{4}-j_{5}+j_{6}\rangle,\langle j_{1}\rangle,\langle
j_{2}\rangle,\langle j_{3}\rangle\\}^{1^{+}}$ coming from
$\varphi_{j_{4}-j_{5}+j_{6},j_{1},j_{2},j_{3}}$ on the terms corresponding to
the highest frequencies: in the case (31) on $F_{4}$ and $F_{5}$, and in the
remaining case on $F_{1}$. ∎

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# Influence of Spatial Coherence on Phonon Transmission across Aperiodically
Arranged Interfaces

Theodore Maranets  ,  Milad Nasiri  and  Yan Wang  Department of Mechanical
Engineering, University of Nevada, Reno, Reno, NV, 89557, USA
<EMAIL_ADDRESS>
###### Abstract.

In this study, we employ the atomistic wave-packet method to directly simulate
coherent phonon transport and scattering dynamics in an aperiodic superlattice
structure with aperiodically arranged interfaces. Our investigation reveals
that interference dynamics, contingent upon the relative magnitudes of phonon
wavelength, spatial coherence length, and average interface spacing,
profoundly influence transmission spectra and mode-conversion behavior.
Particularly, longer-wavelength phonons possessing larger coherence lengths
exhibit more pronounced destructive interference and mode-conversion, leading
to generally lower transmission. The insights gained into coherent phonon wave
physics from this study can significantly augment the analysis and design of
phononic devices.

Keywords: Coherence, Coherent phonon, Superlattice, Interface, Interference

The study of phonon transport, scattering, and localization within
metamaterials characterized by secondary periodicity, such as superlattices
(SLs) and nanomeshes, has garnered substantial research attention [1, 2]. The
purposeful manipulation of these processes holds substantial promise for
advancing a diverse range of applications. In SLs, for instance, interfaces
can efficiently scatter phonons, and with optimization of the SL structure,
achieving ultralow lattice thermal conductivity ($\kappa_{L}$) that approach
or even break the random-alloy-limit $\kappa_{L}$ [3, 4, 5, 6]. Consequently,
SLs have garnered attention as potential thermoelectric materials due to their
ultralow $\kappa_{L}$ [4, 7]. On the other hand, it is crucial to explore
strategies for enhancing heat dissipation in devices incorporating SL-like
structures, such as transistor arrays and quantum cascade lasers.

It has been theorized that thermal transport in systems featuring a secondary
periodicity is dictated by phonon coherence, particularly at low temperatures.
Notably, in SLs, the emergence of a minimum $\kappa_{L}$ as the period length
decreases to a few atomic layers is attributed to a transition from interface
scattering to coherent wave transport [4, 8, 9, 10]. The existence of two
transport regimes is explained by the presence of two distinct types of phonon
modes [8, 11, 12]. The first type represents phonon modes belonging to the
constituent base materials of the SL, e.g., the phonon modes of bulk silicon
(Si) and germanium (Ge) within a Si/Ge SL. The second type includes phonon
modes aligning with the secondary periodicity of the SL, following the
dispersion relations of the SL rather than those of the base materials. The
former is referred to as incoherent phonons, while the latter is denoted as
coherent phonons, a terminology adopted in this letter. These two phonon types
induce different length dependencies (i.e., the number of periods) of
$\kappa_{L}$ in SLs [12, 13]. When coherent phonons dominate, $\kappa_{L}$ can
almost linearly increase with the number of SL periods due to the ballistic
transport of coherent phonons. Conversely, if incoherent phonons dominate,
$\kappa_{L}$ remains nearly constant, constrained by interface scattering
effects. Moreover, the two phonon types impact the temperature dependence of
$\kappa_{L}$ [14] and interfacial transmission is affected by the
nonequilibrium of coherent and incoherent phonons [15].

Recent advancements have been made in understanding coherent phonon transport
in SLs, specifically, the conceptual introduction of spatial coherence [16,
17]. Spatial coherence length $l_{c}$, defined as the real-space width of a
phonon wave-packet, serves as a measure of the spatial correlations of atomic
motion within the phonon wave. $l_{c}$ has been demonstrated to influence the
interference states, thereby effectively characterizing the transition from
incoherent to coherent transport in SLs [16]. Due to the relative infancy of
this theory, there is a present scarcity of research [12, 13, 18, 19, 20, 21]
elucidating the propagation, scattering, and localization of incoherent and
coherent modes in SLs or nanomeshes, particularly those in aperiodic forms,
within the context of spatial coherence. A rigorous and quantitative analysis
of these behaviors is imperative for a comprehensive understanding of their
unique characteristics, which is pivotal for developing effective strategies
to tailor the $\kappa_{L}$ of SL(-like) structures for diverse applications,
either enhancing or reducing it as needed.

In this letter, we conduct, to our knowledge, the first application of the
atomistic phonon wave-packet method to study coherent phonon scattering in a
modified SL structure in contrast to previous works that have solely explored
incoherent phonon transport in SLs [22, 23]. As demonstrated with the method
by Latour and Chalopin [17], coherent phonons travel through the periodic SL
perfectly with zero interface scattering. Coherent phonons are scattered,
however, when the periodic structure is modified, such as the case of
aperiodic layering [24, 25]. The character of coherent phonons in aperiodic
SLs has mostly been indirectly analyzed through macroscopic coefficients such
as $\kappa_{L}$ and/or thermal boundary conductance in the existing literature
[6, 12, 14, 20, 21, 26]. Leveraging the wave-packet method, it becomes
feasible to directly simulate the transport of phonons with well-defined
polarization and wavelength $\lambda$ [22, 27]. Furthermore, we can uniquely
specify the value of $l_{c}$ along with the size $L$ of the aperiodic
structure. Harnessing the method’s unique strength in controlling these three
degrees of freedom, we investigate the variations in transmission spectra of
coherent phonons with spatial coherence, ultimately shedding light on the wave
dynamics within the aperiodic SL structure that impacts heat conduction.

Figure 1. Phonon dispersion relations along the cross-plane or [100] direction
for incoherent phonons (a) and coherent phonons (b) of the material system
employed in this work with the longitudinal-acoustic (LA) and transverse-
acoustic (TA) branches indicated. (c) Schematic illustration of the simulation
domain. An aperiodic SL device of thickness $L=250$ nm is sandwiched between
two periodic SL contacts. The incident LA-mode coherent phonon wave-packet of
wavevector $k_{0}$ and spatial coherence length $l_{c}$ is generated in the
left-contact and allowed to propagate into the device. The total energies of
the left and right-contacts, as well as the device are monitored throughout
the simulation to compute the transmission per Eq. 4. (d) Schematic
illustration of the coherent phonon wave-packet propagating perfectly without
scattering in the periodic SL. Note: the illustrated wave-packets are not
physically accurate as the minimum wavelength for a coherent phonon is two
periods.

The atomistic method applied in this work is adapted from the technique
developed by Schelling et al. [27] We chose a Lennard-Jones (LJ) solid argon
structure as our model material system for reduced computational cost. The LJ
model for interatomic interactions is given as [28]:

(1)
$\displaystyle\phi_{ij}(r_{ij})=4\epsilon\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right]\qquad.$

The molecular weights of the two layered materials are 40 $g/mol$ and 90
$g/mol$. The thickness of each layer in the periodic SL is 2 unit-cells (UC).
The parameter set of $\epsilon_{Ar}=0.0104$ $eV$, $\sigma_{Ar}=0.34$ nm, and a
cutoff radius of $2.5\sigma_{Ar}$ has been used in previous studies of LJ-
modeled solid argon which possesses van der Waals bonding in nature [5, 29].
In this work, we use $\epsilon=16\epsilon_{Ar}$ for the interactions of all
atom types to mimic stronger bonded materials such as Si/Ge and GaAs/AlAs as
done in previous LJ-based atomistic investigations [6, 12, 14, 30]. The
relaxed lattice constant is $a=5.269$ Å, making the period thickness $\sim 2$
nm. The cross-plane incoherent and coherent phonon dispersion relations for
this material system are presented in Figs. 1a and 1b, respectively.

The wave-packet simulations are conducted in the following process. First, a
wavevector $k_{0}$ is chosen from the longitudinal-acoustic (LA) branch of the
cross-plane coherent phonon dispersion in Fig. 1b. The generated wave-packet
will thus be centered at $k_{0}$ in reciprocal-space. For this $k_{0}$, there
is an associated frequency $\omega_{0}=\omega(k_{0})$ and group velocity
$v_{g0}=v_{g}(k_{0})$ from the LA branch. The wave-packet is generated at the
beginning of the simulation by applying the following equations for atom
displacement and velocity:

(2) $\displaystyle
u_{i,n}=\frac{A_{i}}{\sqrt{m_{i}}}\varepsilon_{k_{0},i}\exp{(i[k_{0}\cdot(x_{n}-x_{0})-\omega_{0}t]})\exp{(-4(x_{n}-x_{0}-v_{g0}t)^{2}/l_{c}^{2})}\qquad.$
(3) $\displaystyle v_{i,n}=\frac{\partial u_{i,n}}{\partial t}\qquad.$

where the real parts of $u_{i,n}(t=0)$ and $v_{i,n}(t=0)$ are the initial
displacement and velocity for the $i$th atom in the $n$th unit cell,
respectively. $A_{i}$ is the amplitude of the wave-packet, $m_{i}$ is the mass
of atom $i$, $\varepsilon_{k_{0},i}$ is the eigenvector of the $i$th atom,
$x_{n}$ is the position of the $n$th unit cell, and $x_{0}$ is the initial
position of the wave-packet center. Notably, $l_{c}$ in Eq. 2 corresponds to
the full width at half maximum of the Gaussian-shaped wave-packet. Phonon
properties are determined from lattice dynamics calculations.

Fig. 1c depicts the setup of the simulation domain. An aperiodic SL device of
length $L=250$ nm is sandwiched between two periodic SL contacts each of
length $4,875$ nm, making the total size of the system 10 $\mu$m. The device
is constructed per the methodology described in Ref. [6]. The cross-section of
the simulation domain is $4\times 4$ UC2. Periodic boundary conditions are
applied to all three dimensions. The initial wave-packet is generated
sufficiently away from the device and the duration of the simulation is such
that by the end, nearly all of the incident energy has been either reflected
or transmitted through the device. Following the wave-packet initialization,
the simulation is performed in a microcanonical ensemble. The LAMMPS software
[31] is used to implement the atomistic simulations.

The transmission spectra $\mathcal{T}(k)$ across the aperiodic SL is computed
by

(4) $\displaystyle\mathcal{T}=\frac{E_{rc,final}}{E_{lc,initial}}\qquad.$

. where $E_{rc,final}$ is the total energy transmitted into the right-contact
(rc) by the end of the simulation and $E_{lc,initial}$ is the total energy in
the left-contact (lc) at the beginning of the simulation. If the device were a
periodic SL, then the entire domain would be periodic and $\mathcal{T}=1$ as
the coherent phonon travels perfectly with zero interface scattering. This
scenario is illustrated in Fig. 1d.

Figure 2. Transmission $\mathcal{T}$ versus wavevector $k$ (or inverse of
wavelength $\lambda$) for the LA-mode coherent phonon wave-packet passing
through the aperiodic SL device illustrated in Fig. 1c. The spectra curves
correspond to wave-packets of different spatial coherence lengths. Key
wavelengths $\lambda_{1}=30.7$ nm, $\lambda_{2}=17.9$ nm, $\lambda_{3}=9.3$
nm, and $\lambda_{4}=5.4$ nm analyzed in this letter are indicated in the
figure. Very low wavevector data points are omitted as these wavelengths are
larger than the small-$l_{c}$ and thus their ballistic transport cannot be
simulated.

In Fig. 2, we present $\mathcal{T}(k)$ for the LA-mode coherent phonon wave-
packet with $l_{c}=0.25L$ and $l_{c}=4L$, representing two scenarios where the
spatial coherence length is significantly shorter or longer than the aperiodic
SL device length. Evidently, the transmission of the large-$l_{c}$ wave-packet
is generally lower than that of the smaller counterpart ($l_{c}=0.25L$) at
long wavelengths. As the wavelength decreases, the curves converge, indicating
a reduced influence of spatial coherence on the transmission of shorter-
wavelength coherent phonons in the aperiodic SL device.

Remarkably, the observed increase in transmission with decreasing coherence
length introduces new perspectives into the understanding of phonon transport
in aperiodic SL devices. As previously reported by Chakraborty et al. [14],
the thermal conductivity $\kappa_{L}$ of aperiodic SLs increases with
temperature, attributed to intensified inelastic phonon transmission across
interfaces within the aperiodic SL, akin to the behavior observed in single
interfaces [32]. Our coherence-length-dependent transmission results in Fig. 2
unveil a distinct mechanism for the temperature-dependent $\kappa_{L}$ of
aperiodic SLs, absent in the case of a single interface. Specifically, Latour
et al. [16] noted a reduction in spatial coherence length at higher
temperatures. Accordingly, we attribute the increased $\kappa_{L}$ of
aperiodic SLs at elevated temperatures to the shortened coherence lengths at
higher temperatures, which increases coherent phonon transmission as
demonstrated by Fig. 2. Recognizing the monotonic correlation between
temperature and coherence length, our work underscores the potential of the
wave-packet method as a robust tool for exploring phonon behaviors,
particularly those associated with coherence length, at finite temperatures,
despite the simulation being conducted at a background temperature of 0 K.

The influence of spatial coherence on phonon transmission across the aperiodic
SL device arises from two major contributing factors, as illustrated in Fig.
S1 of Supplemental Materials. Firstly, it governs the spectrum of phonon
wavelengths that a wave-packet can accommodate in reciprocal space.
Specifically, a wave-packet with larger $l_{c}$ encompasses a narrower
spectrum of phonon wavelengths centered around $k_{0}$, thereby increasing the
likelihood of robust interference among the phonons within the wave-packet,
particularly destructive interference within the aperiodic SL structure.
Secondly, it controls the extent to which a wave-packet can span multiple
phonon wave periods in real-space. As $l_{c}/\lambda$ increases, the number of
phonon wave periods encompassed by the wave-packet increases proportionally.
This creates more opportunities for the transmitted and (multiple-)reflected
components to destructively interfere with each other in the aperiodic SL
device.

Table 1. Snapshots of the atom velocity versus real-space position when the wave-packet scatters through the aperiodic SL device for wavelengths $\lambda_{1}=30.7$ nm, $\lambda_{2}=17.9$ nm, and $\lambda_{4}=5.4$ nm at spatial coherence lengths $l_{c}=0.25L$ and $l_{c}=4L$, where $L=250$ nm. Movies of these scenarios are located in the Supplemental Materials. The dotted vertical lines denote the position and thickness of the device. Fig. 3a: $\lambda_{1}=30.7$ nm, $l_{c}=0.25L$, $t=460$ ps |  Fig. 3b: $\lambda_{2}=5.4$ nm, $l_{c}=0.25L$, $t=460$ ps |  Fig. 3c: $\lambda_{4}=5.4$ nm, $l_{c}=0.25L$, $t=560$ ps
---|---|---
|  |
Fig. 3d: $\lambda_{1}=30.7$ nm, $l_{c}=4L$, $t=460$ ps |  Fig. 3e $\lambda_{2}=5.4$ nm, $l_{c}=4L$, $t=460$ ps |  Fig. 3f: $\lambda_{4}=5.4$ nm, $l_{c}=4L$, $t=560$ ps
|  |

The amplified destructive interference leads to a reduction in the
transmission of the coherent phonon, as evidenced by Fig. 2. To further
elucidate this phenomenon, we present snapshots (Table 1) and accompanying
movies (see Supplemental Materials) showcasing the atomic velocity profile of
wave-packets characterized by three representative wavelengths (as marked in
Fig. 2): $\lambda_{1}$ (long), $\lambda_{2}$, and $\lambda_{4}$ (short), under
the scenarios of $l_{c}=0.25L$ and $l_{c}=4L$. For long-$\lambda$ phonons, the
small-$l_{c}$ wave-packet (Figs. 3a and 3b), encompassing a wide range of
wavelengths in reciprocal space and thereby being less monochromatic,
traverses the aperiodic SL device with notably less amplitude modulation,
indicative of minimal interference. This stands in contrast to the scenarios
involving large-$l_{c}$ wave-packet, as illustrated in Figs. 3d and 3e, where
significant amplitude modulation is observed.

The convergence of the spectra curves, corresponding to various coherence
lengths $l_{c}$, at short wavelengths $\lambda$ in Fig. 2, can be attributed
to pronounced destructive interference in the short-$\lambda$ limit. With
identical $l_{c}$, the number of phonon wave periods in a wave-packet
increases for shorter-$\lambda$ phonons, facilitating destructive
interference. Moreover, shorter-$\lambda$ phonons are more susceptible to
interface scattering in SL structures [33, 34]. Both these factors suppress
phonon transmission, thus diminishing differences in transmission at the
short-$\lambda$ limit.

Examining Figs. 3c and 3f for the short-wavelength case ($\lambda_{4}$) and
the corresponding movies, it becomes evident that the initial wave-packet
splits into smaller packets with a discernible phase lag between each,
attributed to partial reflection or transmission at various interfaces.
Noteworthy is the emergence of a standing wave within the aperiodic SL device
in both Fig. 3c and Fig. 3f, a consequence of the interference between
counter-propagating modes. Notably, the wave-packet with a larger $l_{c}$
(Fig. 3f) exhibits a larger amplitude, highlighting the significance of
spatial coherence on interference.

Finally, we explore the occurrence of phonon mode conversion when a coherent
wave-packet interacts with the interfaces in the aperiodic SL device,
considering the pronounced interference and interface scattering events
revealed by velocity plots in Table 1 and corresponding movies. The wavelet
transform method allows us to directly probe reciprocal-space dynamics for
different real-space positions [35, 36, 37]. Here, we apply the wavelet
transform to the time-dependent atomic velocities to investigate coherent
phonon scattering within the aperiodic SL device. Table 2 presents snapshots
of the wavelet transform for wavelengths $\lambda_{1}$ and $\lambda_{3}$ and
for coherence lengths $l_{c}=0.25L$ and $l_{c}=4L$. Full movies of these
scenarios can be found in Supplemental Materials. Figs. 4a and 4b reveal that,
for small-$l_{c}$ wave-packets, the reflected and transmitted components are
exclusively centered around the incident wavevector $k_{0}$, indicating an
absence of mode conversion. In contrast, the scattering of large-$l_{c}$ wave-
packets (Figs. 4c and 4d) exhibits significant mode-conversion at long
wavelengths ($\lambda_{1})$ but not at shorter wavelengths ($\lambda_{3}$).

Table 2. Snapshots of the reciprocal-space wavelet transform when the wave-packet scatters through the aperiodic SL device for wavelength $\lambda_{1}=30.7$ nm and $\lambda_{3}=9.3$ nm at spatial coherence lengths $l_{c}=0.25L$ and $l_{c}=4L$, where $L=250$ nm. Movies of these scenarios are located in the Supplemental Materials. The dotted vertical lines denote the position and thickness of the device. The dashed horizontal line denotes the central wavevector $k_{0}$ of the incident wave-packet. Illuminated regions in the heat map not centered about the $k_{0}$ line indicates excitation of additional vibrational modes. |  Fig. 4a: $\lambda_{1}=30.7$ nm, $l_{c}=0.25L$, $t=424$ ps |  Fig. 4b: $\lambda_{3}=9.3$ nm, $l_{c}=0.25L$, $t=428$ ps |
---|---|---|---
|  |  |
|  Fig. 4c: $\lambda_{1}=30.7$ nm, $l_{c}=4L$, $t=424$ ps |  Fig. 4d: $\lambda_{3}=9.3$ nm, $l_{c}=4L$, $t=428$ ps |
|  |  |

In conclusion, we have investigated the propagation and scattering of coherent
phonons in contrast to previously studied incoherent phonons across
aperiodically arranged interfaces within the aperiodic SL structure, employing
the atomistic wave-packet method. The distinctive capability of this technique
to delineate spatial coherence length unveils interference dynamics in
coherent phonon transmission, which depends on the relative magnitudes of
phonon wavelength $\lambda$, the spatial coherence of the wave-packet $l_{c}$,
and the spatial extent of the aperiodic SL device $L$. Notably, our findings
demonstrate that a larger coherence length enhances destructive interference
within the aperiodic SL, resulting in suppressed transmission. This introduces
a novel mechanism, hitherto unreported, wherein an elevated temperature can
increase the thermal conductivity of aperiodic SLs by mitigating destructive
interference through the reduction of phonon coherence length. Additionally,
our study reveals the excitation of new vibrational modes, i.e., mode
conversion, when wave-packets with large coherence lengths traverse the
aperiodic SL. This work provides valuable insights into the pivotal roles
played by spatial coherence and wavelength in coherent phonon transport,
advancing our comprehension of thermal transport in metamaterials or modern
devices featuring a secondary periodicity.

## Acknowledgements

The authors thank the support from the Nuclear Regulatory Commission (award
number: 31310021M0004) and the National Science Foundation (award number:
2047109). The authors also would like to acknowledge the support of Research
and Innovation and the Cyberinfrastructure Team in the Office of Information
Technology at the University of Nevada, Reno for facilitation and access to
the Pronghorn High-Performance Computing Cluster.

## Conflict of Interests

The authors have no conflicts to disclose.

## Author Contributions

Theodore Maranets: Conceptualization (equal), Formal analysis (equal),
Methodology (lead), Software (lead), Writing—Original Draft (lead),
Writing—Review and Editing (equal). Milad Nasiri: Conceptualization
(supporting), Formal analysis (supporting), Methodology (supporting), Software
(supporting), Writing—Original Draft (supporting), Writing—Review and Editing
(equal). Yan Wang: Conceptualization (equal), Formal analysis (equal),
Supervision (lead), Funding Acquisition (lead), Writing—Original Draft
(supporting), Writing—Review and Editing (equal).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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