# Periodic Fractional Discrete Nonlinear Schrödinger Equation and Modulational
Instability

Brian Choi  United States Military Academy<EMAIL_ADDRESS><EMAIL_ADDRESS>
Keywords. NLS, Fractional calculus, Continuum limit, Modulational instability,
Dispersive estimate
MSC 2020 35Q55, 81T27, 65M06, 34A34, 37K60

(January, 2024)

###### Abstract

The fractional discrete nonlinear Schrödinger equation (fDNLS) is studied on a
periodic lattice from the analytic and dynamic perspective by varying the mesh
size $h>0$ and the nonlocal Lévy index $\alpha\in(0,2]$. We show that the
discrete system converges to the fractional NLS as $h\rightarrow 0$ below the
energy space by directly estimating the difference between the discrete and
continuum solutions in $L^{2}(\mathbb{T})$ using the periodic Strichartz
estimates. The sharp convergence rate via the finite-difference method is
shown to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. On the other
hand for a fixed $h>0$, the linear stability analysis on a family of
continuous wave (CW) solutions reveals a rich dynamical structure of CW waves
due to the interplay between nonlinearity, nonlocal dispersion, and
discreteness. The gain spectrum is derived to understand the role of $h$ and
$\alpha$ in triggering higher mode excitations. The transition from the
quadratic dependence of maximum gain on the amplitude of CW solutions to the
linear dependence, due to the lattice structure, is shown analytically and
numerically.

##  1 Introduction.

In this paper, the fractional discrete nonlinear Schrödinger equation (fDNLS)

$\begin{split}i\dot{u}_{h}&=(-\Delta_{h})^{\frac{\alpha}{2}}u_{h}+\mu|u_{h}|^{2}u_{h},\
(x,t)\in\mathbb{T}_{h}\times\mathbb{R},\\\ u_{h}(x,0)&=u_{h,0}(x)\end{split}$
(1.1)

on a periodic lattice is studied featuring the continuum limit at low
regularity and the modulational instability of continuous wave (CW) solutions
governed by nonlocal long-range interactions described by the Lévy index
$\alpha\in(1,2)$. The formal continuum limit of (1.1) as $h\rightarrow 0$
yields the fractional nonlinear Schrödinger equation (fNLS) (A.1) where the
model is defocusing/focusing for $\mu=\pm 1$, respectively. It is immediately
observed that (1.1) is a finite-difference model of (A.1) where the time
variable is not discretized. For notations, see Section 2.

For $\alpha=2$, (A.1) recovers the well-studied NLS whose well-posedness
theory with the periodic boundary condition goes back to [3]. The method used
in this reference based on the Bourgain space motivated the rigorous study of
nonlocal (A.1) by [7, 11] where the local well-posedness in
$H^{s}(\mathbb{T})$ for $s\geq\frac{2-\alpha}{4}$ was shown. On the non-
compact Euclidean space, the well-posedness theory using the Strichartz
estimates was shown in [18, 12]. Motivated from nonlinear optics, the mixed-
fractional variant of fNLS on $\mathbb{R}^{2}$ was studied in [8] where the
coupling strengths of nonlocal interaction was assumed to be non-homogeneous
in the two transverse directions with respect to the axis of propagation.

The NLS is a ubiquitous model in nonlinear wave phenomena that arises as the
homogenized equation in various physical applications including the pulse
propagation of intense laser beam in nonlinear media and the Bose-Einstein
condensates, or the collective behavior of bosons in an ultra-cold
temperature, via the Gross-Pitaevskii hierarchy. A recent generalization of
NLS, the fractional NLS, introduces nonlocality as a parameter that measures
strong correlations between distant lattice sites. One of the motivations to
study fNLS comes from fractional quantum mechanics [30] where the Feynman path
integral formalism based on the Brownian-like paths was extended to the
$\alpha$-stable Lévy-like paths. Meanwhile our interest extends to the
relationship between fNLS and its discrete analog. The long-range variant of
DNLS is not only relevant in numerical analysis but also in physical phenomena
that are inherently discrete. On a fixed anharmonic lattice, soliton dynamics
with the coupling strengths decaying algebraically, as opposed to the nearest-
neighbor interaction, was studied in [32, 24, 15]. When $\alpha=2$, DNLS,
among many others, describes pulse propagation in discrete waveguide arrays
whose experimental validity was verified in [14]. DNLS is a well-established
model where various results, both theoretical and numerical, can be found in
[26]. Another important feature of discreteness is the Peierls-Nabarro
barrier, studied in [28] applied to DNLS, where the lattice structure yields
an effective energy barrier that eventually pins the transport of a pulse. For
an extension of this work to a nonlocal setting, see [10]. As of now, a
concrete experimental realization of fDNLS based on photonics array is
lacking; however see [31] that proposed an experiment under an optical
framework.

A further motivation to introduce nonlocal operators stems from the convergent
behavior of lattice dynamics under long-range interactions to a homogenized
nonlocal partial differential equation on a smooth domain. While the analysis
on continuum limit for fDNLS on $\mathbb{R}$ has been studied in [19], an
analogous study on $\mathbb{T}$ is absent in the literature, and it is our
intention to fill in this gap.

######  Theorem 1.1.

Let $\alpha\in(1,2]$, and define $s_{0}(\alpha)=\frac{3-\alpha}{4}$ if
$\alpha\in(1,\frac{5}{3}]$ and $s_{0}(\alpha)=\frac{1}{3}$ if
$\alpha\in(\frac{5}{3},2]$. For any $s\in(s_{0}(\alpha),\frac{\alpha}{2}]$ and
$u_{0}\in H^{s}(\mathbb{T})$, let $S(t)u_{0}$ and $S_{h}(t)d_{h}u_{0}$ denote
the well-posed solutions constructed in Proposition A.1 and Proposition 4.2,
respectively. Then there exists $C(\|u_{0}\|_{H^{s}},\alpha)>0$ such that the
error estimate

$\|p_{h}S_{h}(t)d_{h}u_{0}-S(t)u_{0}\|_{L^{2}(\mathbb{T})}\leq
Ch^{\frac{s}{1+s}},$ (1.2)

holds for all $t\in[0,T]$ where $T=T(\|u_{0}\|_{H^{s}},\alpha)>0$. If
$s=\frac{\alpha}{2}$, then $T>0$ can be taken arbitrarily large and the order
of convergence $\frac{\alpha}{2+\alpha}$ is sharp.

One of the first rigorous (weak) convergence results of fDNLS to fNLS on
$h\mathbb{Z}$ as $h\rightarrow 0$ under a general interaction kernel was shown
in [27], to be strengthened to strong convergence [19] in $L^{2}(\mathbb{R})$
under certain hypotheses when $\alpha\in(0,2)\setminus\\{1\\}$. The strong
convergence in $L^{2}(\mathbb{R}^{2})$ was shown in [9] for energy-subcritical
data corresponding to $\alpha\in(1,2)$. Our current work contrasts with those
of Ignat and Zuazua [21, 22, 23], which are based on preconditioning the
numerical scheme, via the Fourier filtering or the two-grid algorithm, that
avoids the effect of weak dispersion whose weaker dispersive decay properties
were studied in [33]. Instead our approach does not modify the finite-
difference scheme, and therefore, the weak dispersive effects rising from the
degenerate phase of the discrete Laplacian needs to be addressed. Note that
this degeneracy is a purely discrete phenomenon, which leads to a derivative-
loss in the Strichartz estimates (see Corollary 4.1).

The main result shows that the method of [17, 35] based on Lemma 4.1 is
sufficient to derive the continuum limit below the energy space. However the
method does not apply when $s\in[\frac{2-\alpha}{4},s_{0}(\alpha)]$ where
$\frac{2-\alpha}{4}$ is the Sobolev regularity threshold proved by [7].
Moreover while previous references do not comment on the sharpness of
convergence rate, we show that the order $\frac{s}{1+s}$ is sharp in the
energy space.

The limitation of applying Lemma 4.1 to our periodic nonlocal problem is
manifested in the periodic discrete dispersive estimate (4.2) far from being
sharp, caused by the non-sharp $O(1)$ difference in the integral-approximation
of the oscillatory sum in the lemma. Therefore our convergence result could be
improved, potentially by modifying the number-theoretic argument in [3] that
counts the cardinality of resonances of frequency-mixing due to nonlinearity.
However this is an interesting challenge since the Fourier symbol of the
discrete Laplacian is trigonometric instead of that of the Laplacian on the
Euclidean domain being a power-type monomial.

On the other hand, fractional modulational instability (MI) is treated
analytically and numerically with nonlocality and discreteness as parameters.
Localization of nonlinear waves where a breather-like excitation rises due to
a small perturbation in its spectrum has been an active area of research. MI
was studied in the context of Stokes wave [36], soliton dynamics [16], and
mixed-fractional NLS and fNLS [38, 1, 37, 13] just to name a few. Regions of
linear stability and instability are given analytically. The MI gain spectrum,
maximum gain, and the corresponding fastest-growth frequencies are explicitly
computed. Numerical simulations that support our theoretical results are
given. These results on $\mathbb{T}_{h}$ show convergent behavior under the
continuum limit.

In Section 2, mathematical background and notation are introduced. In Section
3, theoretical and numerical studies on fractional MI are presented that
emphasize the role of discreteness and the Lévy index. In Section 4, the
dispersive estimates for (1.1) are developed. The proof of Theorem 1.1 is
given in Section 5. The uniform well-posedness theory is given in Appendix A.

##  2 Mathematical Background.

Let $h=\frac{\pi}{M}$ where $M\geq 1$ is an integer. The periodic lattice of
uniform mesh is defined as

$\mathbb{T}_{h}=\\{x=hj:j=-M,\dots,M-1\\}\cong\mathbb{Z}/(2M\mathbb{Z}),$

which is a finite abelian group. Hence there exists a unique Haar measure
$d\mu_{h}$, up to a multiplicative constant, defined by

$L^{1}_{h}:=L^{1}((\mathbb{T}_{h},d\mu_{h});\mathbb{C})\ni
f\mapsto\int_{\mathbb{T}_{h}}fd\mu_{h}:=h\sum_{x\in\mathbb{T}_{h}}f(x),$

where the family of discrete Lebesgue spaces $L^{p}_{h}$ is defined similarly
for $p\in[1,\infty]$. The dual space $\mathbb{T}_{h}^{*}$ is defined as the
homomorphism into the circle group $S^{1}\subseteq\mathbb{C}$ given by
$Hom(\mathbb{T}_{h},S^{1})\cong\\{-M,\dots,M-1\\}\cong\mathbb{Z}/(2M\mathbb{Z})$
where each $k\in\mathbb{T}_{h}^{*}$ acts on $x\in\mathbb{T}_{h}$ by
$(k,x)\mapsto e^{ikx}$. The Plancherel’s Thoerem gives that the spaces of
$L^{2}$ functions on the lattice and its dual are isomorphic under the
discrete (inverse) Fourier transform defined by

$\mathcal{F}_{h}[f](k)=h\sum_{x\in\mathbb{T}_{h}}f(x)e^{-ikx},\
\mathcal{F}_{h}^{-1}[g](x)=(2\pi)^{-1}\sum_{k\in\mathbb{T}_{h}^{*}}g(k)e^{ikx}.$

Note the formal convergence as $h\rightarrow 0$ where $\mathbb{T}_{h}$ tends
to $\mathbb{T}=[-\pi,\pi)$ and $\mathcal{F}_{h}$ tends to $\mathcal{F}$, the
Fourier transform on $\mathbb{T}$.

The linear time evolution is governed by the integro-differential operator
$(-\Delta_{h})^{\frac{\alpha}{2}}:=\mathcal{F}_{h}^{-1}\sigma_{h}(k)\mathcal{F}_{h}$
where
$\sigma_{h}(\xi)=\left|\frac{2}{h}\sin\left(\frac{h\xi}{2}\right)\right|^{\alpha}$
for $\xi\in[-\frac{\pi}{h},\frac{\pi}{h})$. When $\alpha$ is an even integer,
note that $(-\Delta_{h})^{\frac{\alpha}{2}}$ is a local operator, exemplified
in the simplest case of $\alpha=2$ where
$\Delta_{h}f(x)=\frac{f(x+h)+f(x-h)-2f(x)}{h^{2}}$ is the center-difference
discrete Laplacian. Globally, the linear propagator is given by the unitary
operator $U_{h}(t):=e^{-it(-\Delta_{h})^{\frac{\alpha}{2}}}$ defined by the
multiplier $e^{-it\sigma_{h}(k)}$. By convention when $h=0$, denote
$U_{0}(t)=U(t):=e^{-it(-\Delta)^{\frac{\alpha}{2}}}$ where
$(-\Delta)^{\frac{\alpha}{2}}$ is defined by the symbol
$\sigma_{0}(k)=|k|^{\alpha}$ on the Fourier side. Recall that $U(t)$ is
unitary on the Sobolev space $H^{s}(\mathbb{R})$ for any $s\in\mathbb{R}$. On
the other hand, the discrete Sobolev space is defined with the norm

$\|f\|_{H^{s}_{h}}^{2}=\|\langle\nabla_{h}\rangle^{s}f\|_{L^{2}(\mathbb{T})}^{2}:=\frac{1}{2\pi}\sum_{k\in\mathbb{T}_{h}^{*}}\langle
k\rangle^{2s}|\mathcal{F}_{h}[f](k)|^{2},$

where $\langle\xi\rangle:=(1+|\xi|^{2})^{\frac{1}{2}}$, and similarly for
$\|f\|_{H^{s}(\mathbb{T})}$.

To study dispersive smoothing, it is often useful to analyze the linear
evolution of dyadic frequency components and sum each contribution utilizing
the orthogonality properties of the Littlewood-Paley operators. Throughout
this paper, let $N\in 2^{\mathbb{Z}}$ satisfy $N_{*}\leq N\leq 1$ where
$N_{*}=2^{\lceil\log_{2}(\frac{h}{\pi})\rceil}-1$. Define

$P_{N}=\begin{cases}\mathcal{F}_{h}^{-1}\chi_{\\{|k|\in(\frac{\pi
N}{2h},\frac{\pi N}{h}]\\}}\mathcal{F}_{h},\ &\text{if\ }2N_{*}\leq N\leq
1,\\\ Id-\sum\limits_{2N_{*}\leq N\leq 1}P_{N},&\text{if\
}N=N_{*},\end{cases}$

where $Id$ is the identity operator and $\chi_{E}$ is the characteristic
function on $E\subseteq[-\frac{\pi}{h},\frac{\pi}{h})$. As a shorthand, let
$P_{\leq N}:=\sum\limits_{N_{*}\leq M\leq N}P_{M}$.

To relate continuum and discrete data, the operators $d_{h}$ (discretization)
and $p_{h}$ (linear interpolation) are used in our approach where, given
$f:\mathbb{T}\rightarrow\mathbb{C}$ and
$g:\mathbb{T}_{h}\rightarrow\mathbb{C}$,

$\begin{split}d_{h}f(x)&=\frac{1}{h}\int_{x}^{x+h}f(x^{\prime})dx^{\prime},\
x\in\mathbb{T}_{h},\\\
p_{h}g(x)&=g(x_{0})+\frac{g(x_{0}+h)-g(x_{0})}{h}(x-x_{0}),\
x_{0}\in\mathbb{T}_{h},\ x\in[x_{0},x_{0}+h).\end{split}$ (2.1)

See [19, 17] for more details on the properties of $d_{h},p_{h}$ on Sobolev
spaces.

We use the notation $f\lesssim g$ (or similarly $f\gtrsim g$) if $f\leq Cg$
for some universal constant $C>0$ and denote $f\simeq g$ by $f\lesssim g$ and
$f\gtrsim g$. For more details on the harmonic analysis and numerical analysis
on discrete spaces, see [20, 17, 34].

##  3 Modulational Instability of CW Solutions.

An analysis on MI of CW solution under (1.1) is given. An explicit derivation
of the gain spectrum is presented, followed by a discussion of possible
corollaries and numerical simulations. Figures 2, 3 and 4 are generated using
the method in [4, Chapter 2] based on FFT while Figure 1 is due to
Mathematica.

(a) $A=1$

(b) $A=\frac{1}{\sqrt{2}}$

(c) $A=0.5$

(d) $\alpha=2$

(e) $\alpha=1$

(f) $\alpha=0.5$

Figure 1: The region of linear instability given by (3.3) (in blue) is plotted
in $(\xi,A)$ and $(\xi,\alpha)$ for $h=\frac{\pi}{5}$.

Define $u_{h}^{cw}=Ae^{-i\mu|A|^{2}t}$ and let $u_{h}=(A+\epsilon
v_{h}(x,t))e^{-i\mu|A|^{2}t}$ where $v_{h}(x,t)\in\mathbb{C},\ |\epsilon|\ll
1$, and $A\in\mathbb{R}$ without loss of generality. The $O(\epsilon)$ term
yields

$i\frac{dv_{h}}{dt}=(-\Delta_{h})^{\frac{\alpha}{2}}v_{h}+\mu
A^{2}(v_{h}+\overline{v}_{h}).$

Taking the real and imaginary parts, i.e. $v_{h}=f_{h}+ig_{h}$, we have

$\frac{d}{dt}\begin{pmatrix}f_{h}\\\
g_{h}\end{pmatrix}=\begin{pmatrix}0&(-\Delta_{h})^{\frac{\alpha}{2}}\\\
-(-\Delta_{h})^{\frac{\alpha}{2}}-2\mu
A^{2}&0\end{pmatrix}\begin{pmatrix}f_{h}\\\ g_{h}\end{pmatrix}.$ (3.1)

Taking the discrete Fourier transform both sides and the ansatz
$\mathcal{F}_{h}[f_{h}]=P_{h}(k)e^{-i\Omega t},\
\mathcal{F}_{h}[g_{h}]=G_{h}(k)e^{-i\Omega t}$, (3.1) becomes an eigenvalue
problem whose nontrivial solution $(P_{h},G_{h})$ exists if and only if
$\Omega$ satisfies the dispersion relation given by

$\Omega^{2}(k)=\left|\frac{2}{h}\sin(\frac{hk}{2})\right|^{\alpha}\left(\left|\frac{2}{h}\sin(\frac{hk}{2})\right|^{\alpha}+2\mu
A^{2}\right).$ (3.2)

When $\mu=1$, the system is linearly stable since $\Omega^{2}\geq 0$ and
henceforth assume $\mu=-1$. The region of linear instability and the
corresponding gain spectrum are given by

$\displaystyle\left|\frac{2}{h}\sin(\frac{hk}{2})\right|^{\alpha}$
$\displaystyle<2A^{2},\ |k|\leq\frac{\pi}{h},$ (3.3) $\displaystyle
G(\xi,A,\alpha,h)$
$\displaystyle:=\sqrt{\left|\frac{2}{h}\sin(\frac{h\xi}{2})\right|^{\alpha}\left(2A^{2}-\left|\frac{2}{h}\sin(\frac{h\xi}{2})\right|^{\alpha}\right)},$

where we denote $\xi\in\mathbb{R},\ k\in\mathbb{Z}$.

See Figure 1 that illustrates (3.3). Top row: when $A=\frac{1}{\sqrt{2}}$, the
region is independent of $\alpha$. If $A\ll 1$ and $\alpha\ll 1$, then there
exist no $k\in\mathbb{Z}\setminus\\{0\\}$ that satisfies (3.3), i.e., linear
stability. On the other hand, for $A>\frac{1}{\sqrt{2}}$ and $\alpha\ll 1$,
any $k\in\mathbb{T}_{h}$ satisfies (3.3), i.e., linear instability. Botton
row: when $\alpha=2$, $A(\xi)$ behaves as a kink and when $\alpha<2$, $A(\xi)$
behaves as a cusp near $\xi=0$. More precisely,
$A\sim_{h}\frac{|\xi|^{\frac{\alpha}{2}}}{\sqrt{2}}$ as $\xi\rightarrow 0$.
Note that the region approaches $\\{|\xi|^{\alpha}<2A^{2}\\}$ as $h\rightarrow
0$. Hence for a fixed $h>0$, the system is linearly unstable on the entire
bandwidth if $|A|$ is sufficiently large, i.e., if
$2A^{2}>(\frac{2}{h})^{\alpha}$.

Figure 2: The log plots used $(\alpha,h,k)=(0.25,\frac{\pi}{50},3)$ and
$u_{0}(x)=A+10^{-5}e^{ikx}$ where the right plot is the contour of the left
plot.

If (3.3) holds, then the maximum exponential gain $\Omega_{m}$ that occurs at
$k_{m}$, the fastest-growth frequency, can be computed explicitly by computing
the derivative of (3.2) treating $k$ as real. By direct computation, for
$\xi\in(0,\frac{\pi}{h})$,

$\frac{d}{d\xi}\Omega^{2}(\xi)=\alpha
h\cot(\frac{h\xi}{2})\left(\frac{2}{h}\sin(\frac{h\xi}{2})\right)^{\alpha}\left(\left(\frac{2}{h}\sin(\frac{h\xi}{2})\right)^{\alpha}-A^{2}\right).$

If $(\frac{2}{h})^{\alpha}\leq A^{2}$, then $k_{m}=\pm M$ and

$\Omega_{m}=\sqrt{\left(2A^{2}-\left(\frac{2}{h}\right)^{\alpha}\right)\left(\frac{2}{h}\right)^{\alpha}}.$
(3.4)

If $(\frac{2}{h})^{\alpha}>A^{2}$, let $\xi_{m}\in(0,\frac{\pi}{h})$ be real
such that $\left(\frac{2}{h}\sin(\frac{h\xi_{m}}{2})\right)^{\alpha}=A^{2}$,
or equivalently,
$\xi_{m}=\frac{2}{h}\sin^{-1}\left(\frac{h|A|^{\frac{2}{\alpha}}}{2}\right)$.
It can be verified directly that $\pm\xi_{m}$ is the unique frequency that
maximizes $-\Omega^{2}$. Therefore
$|k_{m}|\in\\{\lfloor\xi_{m}\rfloor,\lceil\xi_{m}\rceil\\}$ and
$\Omega_{m}=\sqrt{-\Omega^{2}(k_{m})}$. Observe that
$\Omega_{m}^{\prime}:=\sqrt{-\Omega^{2}(\xi_{m})}=A^{2}$, independent of
$h,\alpha$. A couple of remarks follows.

  * •

In the continuum limit, the region of instability is
$\\{|k|^{\alpha}<2A^{2}\\}$. Since $|\sin(z)|<|z|$ when
$0<|z|\leq\frac{\pi}{2}$, the region of linear instability for fDNLS strictly
contains that of fNLS.

  * •

The system is linearly stable if $|A|\ll 1,\ \alpha\ll 1$, which is in stark
contrast to the system posed on $h\mathbb{Z}$ where given any $A>0$, there
exists a real $k$ sufficiently small that satisfies (3.3). In Figure 2, the
solution is linearly stable when $|A|\ll 1$. However nonlinearity begins to
dominate from $t=10$ with the emergence of $k$ troughs where $k=3$ was used in
Figure 2. Indeed numerical experiments suggest the perturbation of $\epsilon
e^{ikx}$, with $|\epsilon|\ll 1$ and $k\in\mathbb{T}_{h}^{*}$, triggers the
emergence of $k$ troughs as the linear stability is supplanted by highly
nonlinear wave evolution. For high $A$ value, the system is linearly unstable.

Figure 3: Plots of $|u_{h}(x,t)|^{2}$ with $h=\frac{\pi}{50}$ and
$u_{0}(x)=1+10^{-5}(e^{ix}+e^{-ix})$.

  * •

A transition into chaos as $\alpha$ decreases is illustrated in Figure 3,
consistent with [29]. For $\alpha=2$, the recurrence of localization was
observed as expected; see [13] for a detailed numerical study on the nonlinear
evolution of fNLS using the split-step Fourier spectral method. As $\alpha$
decreases to $1$, such clear recurrence was not observed with the development
of irregular amplitudes. The time of first localization was observed to be
delayed as $\alpha\rightarrow 1+$.

  * •

By (3.4), $\Omega_{m}$ grows linearly in $|A|$ asymptotically as
$|A|\rightarrow\infty$ when $h^{-\alpha}\ll A^{2}$. The transition occurs when
$h^{-\alpha}\simeq A^{2}$. When $h^{-\alpha}\gg A^{2}$, we have
$\Omega_{m}^{\prime}=\sqrt{-\Omega^{2}(\xi_{m})}=A^{2}$; recall that $\xi_{m}$
may not be an integer and that $k_{m}=\lfloor\xi_{m}\rfloor$ or
$\lceil\xi_{m}\rceil$. The top plot of Figure 4 reports, for multiple Lévy
indices, the initial quadratic growth of $\Omega_{m}$ for sufficiently small
$A$, followed by a non-quadratic behavior. The linear stability analysis
suggests that the linear growth should follow, consistent with our numerical
experiments. However for larger values of $A$, the spectrum of instability for
higher harmonics is larger, and therefore the nonlinear evolution seems to be
non-negligible.

Figure 4: Parameters used for the top plot: $h=\frac{\pi}{50},\
u_{0}(x)=A+10^{-5}e^{50ix}$

##  4 Strichartz Estimates.

The non-zero curvature of the dispersion relation yields dispersive smoothing
estimates, or the Strichartz estimates, manifested as the boundedness of
evolution time-dependent operators in various norms of Lebesgue spaces
consisting of space-time functions. In [19], the proof of the nonlocal
continuum limit in the energy space $H^{\frac{\alpha}{2}}(\mathbb{R})$ when
$\alpha\in(1,2)$ is based on the sharp Strichartz estimate

$\|e^{-it(-\Delta_{h})^{\frac{\alpha}{2}}}f\|_{L^{q}_{t}(\mathbb{R};L^{r}_{h})}\lesssim\||\nabla_{h}|^{\frac{3-\alpha}{q}}f\|_{L^{2}_{h}},$
(4.1)

for $2\leq q,r\leq\infty$ satisfying $\frac{3}{q}+\frac{1}{r}=\frac{1}{2}$.
Since the approach taken to obtain (4.1) does not translate directly to a
compact domain, an alternative approach based on approximating an oscillatory
sum with an oscillatory integral (Lemma 4.1) was used in [17, 35]. Here we
adopt their method and show (Corollary 4.1)

$\|e^{-it(-\Delta_{h})^{\frac{\alpha}{2}}}f\|_{L^{q}([0,1];L^{r}_{h})}\lesssim_{\epsilon}\|f\|_{H^{\frac{2}{q}+\epsilon}_{h}}.$
(4.2)

For $f\in L^{1}_{h}\setminus\\{0\\}$, $\|U_{h}(t)f\|_{L^{\infty}_{h}}$ cannot
decay to zero as $t\rightarrow\infty$ due to the conservation of $L^{2}_{h}$
norm; see Proposition 4.2. However the discrete dispersive estimates hold
locally in time.

######  Proposition 4.1.

Let $\alpha\in(1,2]$ and
$|t|\leq\frac{\pi^{2-\alpha}}{2\alpha}\left(\frac{h}{N}\right)^{\alpha-1}$.
Then

$\|U_{h}(t)P_{\leq
N}f\|_{L^{\infty}_{h}}\lesssim|\alpha-1|^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{3}}|t|^{-\frac{1}{3}}\|f\|_{L^{1}_{h}}.$
(4.3)

Our proof of Proposition 4.1 is motivated from [17, 35] where a discrete
oscillatory sum (see (4.4)) is approximated by an oscillatory integral, after
which the Van der Corput Lemma is applied.

######  Lemma 4.1 ([39, Chapter 5, Lemma 4.4]).

Let $a<b$ and $0<\epsilon<1$. Assume
$\sup\limits_{\xi\in(a,b)}|\phi^{\prime}(\xi)|\leq 2\pi(1-\epsilon)$ and
$\phi$, monotonic in $(a,b)$. Then there exists $A_{\epsilon}>0$ independent
of $a,b,\phi$ such that

$\left|\int_{a}^{b}e^{i\phi(\xi)}d\xi-\sum_{a<k\leq b}e^{i\phi(k)}\right|\leq
A_{\epsilon}.$

###### Proof of Proposition 4.1.

Consider the identity

$\begin{split}U_{h}(t)P_{\leq N}f(x)&=\frac{1}{2\pi}\sum_{|k|\leq\frac{\pi
N}{h}}e^{i\left(-t\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}+kx\right)}\mathcal{F}_{h}f(k)\\\
&=h\sum_{x^{\prime}\in\mathbb{T}_{h}}f(x^{\prime})\sum_{|k|\leq\frac{\pi
N}{h}}\frac{1}{2\pi}e^{i\left(-t\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}+k(x-x^{\prime})\right)}=K_{t}\ast
f,\end{split}$

where $K_{t}(x):=\sum\limits_{|k|\leq\frac{\pi
N}{h}}\frac{1}{2\pi}e^{i\left(-t\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}+kx\right)}$
and $\ast$ denotes the discrete convolution defined by the measure $d\mu_{h}$.
As a shorthand, let
$\phi(\xi)=-t\left|\frac{2}{h}\sin\frac{h\xi}{2}\right|^{\alpha}+\xi x$. It
suffices to show

$\|K_{t}\|_{L^{\infty}_{h}}\lesssim|\alpha-1|^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{3}}|t|^{-\frac{1}{3}}$
(4.4)

by the Young’s inequality. By the triangle inequality,

$|K_{t}(x)|\leq\left|K_{t}(x)-\frac{1}{2\pi}\int_{-\frac{\pi N}{h}}^{\frac{\pi
N}{h}}e^{i\phi(\xi)}d\xi\right|+\left|\frac{1}{2\pi}\int_{-\frac{\pi
N}{h}}^{\frac{\pi N}{h}}e^{i\phi(\xi)}d\xi\right|=:I+II.$ (4.5)

To show that $I=O(1)$ and $II$ is consistent with (4.4) by Lemma 4.1 and the
Van der Corput Lemma, respectively, the higher order derivatives of $\phi$
need to estimated. Let $sgn(\xi)=1$ if $\xi>0$ and $sgn(\xi)=-1$ if $\xi<0$.
Then,

$\begin{split}\phi^{\prime}(\xi)&=-\alpha t\cdot
sgn(\xi)\cos\left(\frac{h\xi}{2}\right)\left|\frac{\sin(\frac{h\xi}{2})}{h/2}\right|^{\alpha-1}+x,\\\
\phi^{\prime\prime}(\xi)&=\alpha
t\left(\frac{h}{2}\right)^{2-\alpha}\frac{1-\alpha\cos^{2}\left(\frac{h\xi}{2}\right)}{\left|\sin\left(\frac{h\xi}{2}\right)\right|^{2-\alpha}},\\\
\phi^{\prime\prime\prime}(\xi)&=-\alpha
t\left(\frac{h}{2}\right)^{3-\alpha}sgn(\xi)\frac{\left(\alpha^{2}\cos^{2}\left(\frac{h\xi}{2}\right)-3\alpha+2\right)\cos\left(\frac{h\xi}{2}\right)}{\left|\sin\left(\frac{h\xi}{2}\right)\right|^{3-\alpha}}.\end{split}$
(4.6)

By direct computation, $\phi^{\prime}$ is monotonic on
$E_{1}:=(-\xi_{0},\xi_{0})$ and
$E_{2}:=[-\frac{\pi}{h},\frac{\pi}{h}]\setminus(-\xi_{0},\xi_{0})$ separately
where $\xi_{0}:=\frac{2}{h}\arccos(\alpha^{-\frac{1}{2}})$ is the unique
positive root of $\phi^{\prime\prime}$ in $(0,\frac{\pi}{h})$. By choosing
$\epsilon=\frac{1}{4}$ in Lemma 4.1, it can further be verified that
$\sup\limits_{\xi\in[-\frac{\pi N}{h},\frac{\pi
N}{h}]}|\phi^{\prime}(\xi)|\leq\frac{3\pi}{2}$ given the restriction on $|t|$,
which shows $I=O(1)$ estimating the difference on $E_{1},E_{2}$ separately.

To estimate the integral in $II$, the lower bounds of
$|\phi^{\prime\prime}|,|\phi^{\prime\prime\prime}|$ are estimated. Let

$S=\Bigl{\\{}|\xi|\leq\frac{\pi
N}{h}:\left|1-\alpha\cos^{2}\left(\frac{h\xi}{2}\right)\right|\geq\frac{|\alpha-1|}{2}\Bigr{\\}}.$

On $S$, the bound
$\left|\sin\left(\frac{h\xi}{2}\right)\right|\leq\frac{h|\xi|}{2}$ is used to
obtain

$|\phi^{\prime\prime}(\xi)|\gtrsim(\alpha-1)|t||\xi|^{-(2-\alpha)}.$ (4.7)

On $[-\frac{\pi N}{h},\frac{\pi N}{h}]\setminus S$, we have

$|\phi^{\prime\prime\prime}(\xi)|\gtrsim(\alpha-1)|t||\xi|^{-(3-\alpha)},$

since

$\left|\alpha^{2}\cos^{2}\left(\frac{h\xi}{2}\right)-3\alpha+2\right|\geq
2|\alpha-1|-\left|\alpha^{2}\cos^{2}\left(\frac{h\xi}{2}\right)-\alpha\right|\geq
2|\alpha-1|-\frac{\alpha}{2}|\alpha-1|\geq\alpha-1,$

and

$\left|\alpha\cos^{2}\left(\frac{h\xi}{2}\right)\right|\geq
1-\left|\alpha\cos^{2}\left(\frac{h\xi}{2}\right)-1\right|\geq
1-\frac{|\alpha-1|}{2}\geq\frac{1}{2}.$

Hence

$\begin{split}\left|\int_{-\frac{\pi N}{h}}^{\frac{\pi
N}{h}}e^{i\phi(\xi)}d\xi\right|&\leq\left|\int_{S}e^{i\phi(\xi)}d\xi\right|+\left|\int_{[-\frac{\pi
N}{h},\frac{\pi N}{h}]\setminus S}e^{i\phi(\xi)}d\xi\right|\\\
&\lesssim\max\left((\alpha-1)^{-\frac{1}{2}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{2}}|t|^{-\frac{1}{2}},(\alpha-1)^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{3}}|t|^{-\frac{1}{3}}\right)\\\
&\lesssim(\alpha-1)^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{3}}|t|^{-\frac{1}{3}},\end{split}$
(4.8)

where the last inequality follows from interpolating $\left|\int_{-\frac{\pi
N}{h}}^{\frac{\pi N}{h}}e^{i\phi(\xi)}d\xi\right|\lesssim\frac{N}{h}$ and the
Van der Corput estimate obtained from (4.7). Substituting $I=O(1)$ and (4.8)
into (4.5), the desired estimate (4.4) is shown since for
$|t|\lesssim(\frac{h}{N})^{\alpha-1}$ and $N\geq N_{*}$,

$(\alpha-1)^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{1-\frac{\alpha}{3}}|t|^{-\frac{1}{3}}\gtrsim(\alpha-1)^{-\frac{1}{3}}\left(\frac{N}{h}\right)^{\frac{2}{3}}\gtrsim(\alpha-1)^{-\frac{1}{3}}.$

∎

######  Corollary 4.1.

Let $(q,r)\in[2,\infty]^{2}$ be lattice-admissible and $\alpha\in(1,2]$. Then
for all $\epsilon>0$,

$\displaystyle\|U_{h}(t)f\|_{L^{q}([0,1];L^{r}_{h})}$
$\displaystyle\lesssim_{\epsilon}|\alpha-1|^{-\frac{1}{6}(1-\frac{2}{r})}\|f\|_{H^{\frac{2}{q}+\epsilon}_{h}},$
(4.9)
$\displaystyle\left|\left|\int_{0}^{t}U_{h}(t-t^{\prime})F(t^{\prime})dt^{\prime}\right|\right|_{L^{q}([0,1];L^{r}_{h})}$
$\displaystyle\lesssim_{\epsilon}|\alpha-1|^{-\frac{1}{3}(1-\frac{2}{r})}\|F\|_{L^{1}([0,1];H^{\frac{2}{q}+\epsilon}_{h})}.$
(4.10)

###### Proof.

Observe that $\tilde{U}_{h}(t)=P_{\leq
N}U_{h}(t(\frac{N}{h})^{3-\alpha})P_{\leq N}$ satisfies the hypothesis of [25,
Theorem 1.2], and therefore

$\|U_{h}(t)P_{\leq
N}f\|_{L^{q}([0,\tau];L^{r}_{h})}\lesssim|\alpha-1|^{-\frac{1}{6}(1-\frac{2}{r})}\left(\frac{N}{h}\right)^{\frac{3-\alpha}{q}}\|f\|_{L^{2}_{h}},$
(4.11)

where
$\tau=\frac{\pi^{2-\alpha}}{2\alpha}\left(\frac{h}{N}\right)^{\alpha-1}$. By
iterating (4.11) on $[0,1]$ using the unitarity of $U_{h}(t)$, (4.9) is shown.
Another application of [25, Theorem 1.2] yields the inhomogeneous estimate
(4.10). Note that the implicit constant of (4.10) is that of (4.9) squared by
the TT∗ argument based on the complex interpolation of dispersive estimates
between $\|U_{h}(t)P_{\leq N}f\|_{L^{2}_{h}}\leq\|f\|_{L^{2}_{h}}$ and (4.3)
given by

$\|U_{h}(t)P_{\leq
N}f\|_{L^{r}_{h}}\lesssim|\alpha-1|^{-\frac{1}{3}(1-\frac{2}{r})}\left(\frac{N}{h}\right)^{(1-\frac{\alpha}{3})(1-\frac{2}{r})}|t|^{-\frac{1}{3}(1-\frac{2}{r})}\|f\|_{L^{\frac{r}{r-1}}_{h}}.$

∎

######  Remark 4.1.

Our proof of Proposition 4.1 and Corollary 4.1 is adapted from [17]. The proof
of Proposition 4.1 differs from that of [17] in that we use $\epsilon>0$ when
applying Lemma 4.1 whereas [17] used $\epsilon=0$, which may lead to the error
bound $A_{\epsilon}$ to blow up. This subtlety can be easily circumvented by
choosing $0<\epsilon<\frac{1}{2}$. Furthermore note that the right-hand side
(RHS) of (4.9) is measured in $H^{\frac{2}{q}+\epsilon}_{h}$ whose Sobolev
regularity is independent of $\alpha$. The $\epsilon>0$ is a result of the
non-endpoint Sobolev embedding.

The time interval $[0,T]$ of the local well-posedness of (1.1) in the
Strichartz space $X_{T}:=C([0,T];L^{2}_{h})\cap L^{6}([0,T];L^{\infty}_{h})$
cannot be determined uniformly in $h>0$ solely from $L^{p}_{h}\hookrightarrow
L^{q}_{h}$, for $p<q$, since the embedding is not uniform in $h$ as
$\|f\|_{L^{q}_{h}}\leq h^{\frac{1}{p}-\frac{1}{q}}\|f\|_{L^{p}_{h}}$. An
application of Corollary 4.1 yields a uniform estimate in $h$.

######  Proposition 4.2.

Let $s>\frac{1}{3}$ and $\alpha\in(1,2]$. For every $u_{h,0}\in H^{s}_{h}$,
there exists a unique $u_{h}\in X_{T}$ such that

$u_{h}(t)=U_{h}(t)u_{h,0}-i\mu\int_{0}^{t}U_{h}(t-\tau)\left(|u_{h}(\tau)|^{2}u_{h}(\tau)\right)d\tau$
(4.12)

where $T_{h}\sim_{\alpha}\|u_{h,0}\|_{H^{s}_{h}}^{-3}$ is independent of
$h>0$. Furthermore discrete mass ($M_{h}$) and discrete energy ($H_{h}$) are
conserved where

$M_{h}[u_{h}(t)]=\|u_{h}(t)\|_{L^{2}_{h}}^{2};\
H_{h}[u_{h}(t)]=\frac{1}{2}\||\nabla_{h}|^{\frac{\alpha}{2}}u_{h}\|_{L^{2}_{h}}^{2}+\frac{\mu}{4}\|u_{h}\|_{L^{4}_{h}}^{4}.$

###### Proof.

Let $\Gamma u_{h}$ be the RHS of (4.12). By Corollary 4.1 and the a priori
estimate,

$\begin{split}\|\Gamma u_{h}\|_{X_{T}}&:=\|\Gamma
u_{h}\|_{C_{T}H^{s}_{h}}+\|\Gamma
u_{h}\|_{L^{6}_{T}L^{\infty}_{h}}\lesssim\|u_{h,0}\|_{H^{s}_{h}}+\||u_{h}|^{2}u_{h}\|_{L^{1}_{T}H^{s}_{h}}\\\
&\lesssim\|u_{h,0}\|_{H^{s}_{h}}+\left|\left|\|u_{h}\|_{L^{\infty}_{h}}^{2}\|u_{h}\|_{H^{s}_{h}}\right|\right|_{L^{1}_{T}}\leq\|u_{h,0}\|_{H^{s}_{h}}+T^{\frac{2}{3}}\|u_{h}\|_{L^{6}_{T}L^{\infty}_{h}}^{2}\|u_{h}\|_{C_{T}H^{s}_{h}},\\\
\|\Gamma u_{h}-\Gamma v_{h}\|_{X_{T}}&\lesssim
T^{\frac{2}{3}}(\|u_{h}\|_{X_{T}}^{2}+\|v_{h}\|_{X_{T}}^{2})\|u_{h}-v_{h}\|_{X_{T}},\end{split}$

there exists a unique fixed point $u_{h}$ in a small closed ball of $X_{T}$
such that $u_{h}=\Gamma u_{h}$. The domain of $\Gamma$ is extended from the
neighborhood of the origin to the entire $X_{T}$ by the continuity argument. ∎

##  5 Convergence as $h\rightarrow 0$.

The proof of (1.2) is presented. Our strategy is to directly estimate the
difference, or the error, between the solutions on $\mathbb{T}$ and
$\mathbb{T}_{h}$. To address the subtlety that $u(t)$ and $u_{h}(t)$ are
defined on different spaces, we lift $u_{h}(t)$ via linear interpolation.
Since there is no canonical way to interpolate discrete data into continuum
data or vice versa via discretization, there is flexibility in how the error
is defined and computed. For example, the exact solution whose spatiotemporal
frequency is concentrated at a single site admits linear convergence (B) in
$L^{2}(\mathbb{T})$ and quadratic convergence (5.3) in $L^{2}_{h}$. Whether or
not other methods of error estimation yield similar results as Theorem 1.1 is
left for further research.

The technical lemmas used to prove the theorem are adapted from [19, 17]
either directly or with minimal modifications.

######  Lemma 5.1.

Let $0\leq s\leq 1$. Then

$\|p_{h}U_{h}(t)u_{h,0}-U(t)u_{0}\|_{L^{2}(\mathbb{T})}\lesssim|t|h^{\frac{s}{1+s}}(\|u_{h,0}\|_{H^{s}_{h}}+\|u_{0}\|_{H^{s}(\mathbb{T})})+\|p_{h}u_{h,0}-u_{0}\|_{L^{2}(\mathbb{T})}.$

######  Lemma 5.2.

For $0\leq s\leq 1,\ 0\leq\tau\leq t$,

$\|(p_{h}U_{h}(t-\tau)-U(t-\tau)p_{h})F(\tau)\|_{L^{2}(\mathbb{T})}\lesssim|t-\tau|h^{\frac{s}{1+s}}\|F(\tau)\|_{H^{s}_{h}(\mathbb{T})}.$

######  Lemma 5.3.

For $p>1,\ 0<s\leq 1$,

$\|p_{h}(|u_{h}|^{p-1}u_{h})-|p_{h}u_{h}|^{p-1}p_{h}u_{h}\|_{L^{2}(\mathbb{T})}\lesssim
h^{s}\|u_{h}\|_{L^{\infty}_{h}}^{p-1}\|u_{h}\|_{H^{s}_{h}}.$

######  Lemma 5.4.

For $0\leq s\leq 1$,

$\|p_{h}d_{h}f-f\|_{L^{2}(\mathbb{T})}\lesssim
h^{s}\|f\|_{H^{s}(\mathbb{T})}.$

###### Proof of (1.2).

In this proof, $\|\cdot\|$ denotes the norm in $L^{2}(\mathbb{T})$. Given
$u_{0}\in H^{s}(\mathbb{T})$, let $u(t)=S(t)u_{0}$ and
$u_{h}(t)=S_{h}(t)d_{h}u_{0}$. It can be directly verified that $d_{h}$
defines a bounded linear operator from $L^{2}(\mathbb{T})$ to $L^{2}_{h}$ and
$H^{1}(\mathbb{T})$ to $H^{1}_{h}$, and consequently it follows from
interpolating the two estimates that
$d_{h}:H^{s^{\prime}}(\mathbb{T})\rightarrow H^{s^{\prime}}_{h}$ is bounded
for any $s^{\prime}\in[0,1]$. Therefore the time of existence in Proposition
4.2 is bounded from below since
$T_{h}\sim_{\alpha}\|d_{h}u_{0}\|_{H^{s}_{h}}^{-3}\gtrsim\|u_{0}\|_{H^{s}}^{-3}$.
Let $T^{\prime}(\|u_{0}\|_{H^{s}})>0$ be the time of existence stated in
Proposition A.1 and take

$T:=\min\\{T^{\prime},\inf\limits_{h}T_{h}\\}>0.$ (5.1)

By the triangle inequality and Lemma 5.1, Lemma 5.2, and Lemma 5.3, we have

$\displaystyle\|(p_{h}S_{h}(t)d_{h}-S(t))u_{0}\|$
$\displaystyle\leq\|(p_{h}U_{h}(t)d_{h}-U(t))u_{0}\|+\int_{0}^{t}\|(p_{h}U_{h}(t-\tau)-U(t-\tau)p_{h})(|u_{h}|^{2}u_{h})(\tau)\|d\tau$
$\displaystyle+\int_{0}^{t}\|p_{h}(|u_{h}|^{2}u_{h})-|p_{h}u_{h}|^{2}p_{h}u_{h}\|d\tau+\int_{0}^{t}\||p_{h}u_{h}|^{2}p_{h}u_{h}-|u|^{2}u\|d\tau$
$\displaystyle=:I_{1}+I_{2}+I_{3}+I_{4}$ (5.2)
$\displaystyle\lesssim(1+|T|)h^{\frac{s}{1+s}}\|u_{0}\|_{H^{s}_{h}}$
$\displaystyle+\int_{0}^{t}(|t-\tau|h^{\frac{s}{1+s}}+h^{s})\|u_{h}\|_{L^{\infty}_{h}}^{2}\|u_{h}\|_{H^{s}_{h}}d\tau+\int_{0}^{t}(\|p_{h}u_{h}\|_{L^{\infty}}^{2}+\|u\|_{L^{\infty}}^{2})\|p_{h}u_{h}-u\|d\tau,$
(5.3)

having used the discrete Sobolev estimate
$\||u_{h}|^{2}u_{h}\|_{H^{s}_{h}}\lesssim\|u_{h}\|_{L^{\infty}}^{2}\|u_{h}\|_{H^{s}_{h}}$.
Similar to the proof of Proposition 4.2 where
$X_{T}=C([0,T_{h}];L^{2}_{h})\cap L^{6}([0,T_{h}];L^{\infty}_{h})$, the
nonlinear terms are estimated as

$\left|\left|\|u_{h}\|_{L^{\infty}_{h}}^{2}\|u_{h}\|_{H^{s}_{h}}\right|\right|_{L^{1}_{T}}\leq
T_{h}^{\frac{2}{3}}\|u_{h}\|_{X_{T}}^{3}\lesssim\|d_{h}u_{0}\|_{H^{s}_{h}}\lesssim\|u_{0}\|_{H^{s}}.$

Furthermore a direct estimation on the linear interpolation yields
$\|p_{h}u_{h}\|_{L^{\infty}}\lesssim\|u_{h}\|_{L^{\infty}_{h}}$ since for
$x_{0}\in\mathbb{T}_{h}$ and $x\in[x_{0},x_{0}+h)$,

$|p_{h}u_{h}(x)|\leq|u_{h}(x_{0})|+\frac{|u_{h}(x_{0}+h)|+|u_{h}(x_{0})|}{h}|x-x_{0}|\leq
3\|u_{h}\|_{L^{\infty}_{h}}.$

Altogether

$\eqref{estimate1}\lesssim(1+|T|)h^{\frac{s}{1+s}}\|u_{0}\|_{H^{s}_{h}}+\int_{0}^{t}(\|u_{h}\|_{L^{\infty}_{h}}^{2}+\|u\|_{L^{\infty}}^{2})\|p_{h}u_{h}-u\|d\tau,$

and the Gronwall’s inequality for $t\in[0,T]$ yields

$\|p_{h}u_{h}(t)-u(t)\|\lesssim(1+|T|)h^{\frac{s}{1+s}}\|u_{0}\|_{H^{s}_{h}}e^{\int_{0}^{T}\|u_{h}(\tau)\|_{L^{\infty}_{h}}^{2}+\|u(\tau)\|_{L^{\infty}}^{2}d\tau}.$

It suffices to show that there exists $C(\|u_{0}\|_{H^{s}},\alpha)>0$
independent of $h$ such that
$\|u_{h}\|_{L^{2}_{T}L^{\infty}_{h}}^{2}+\|u\|_{L^{2}_{T}L^{\infty}}^{2}\leq
C$. By Proposition 4.2,

$\|u_{h}\|_{L^{2}_{T}L^{\infty}_{h}}^{2}\leq
T_{h}^{\frac{2}{3}}\|u_{h}\|_{X_{T}}^{2}=O(1)$

is independent of $h$. To show $\|u\|_{L^{2}_{T}L^{\infty}}^{2}=O(1)$, note
that

$\|u\|_{L^{2}_{T}L^{\infty}}^{2}\leq
T^{\frac{1}{2}}\|u\|_{L^{4}_{T}L^{\infty}}^{2}\lesssim
T^{\frac{1}{2}}\|u\|_{L^{4}_{T}W^{\frac{1}{4}+\epsilon^{\prime},4}}^{2},$
(5.4)

by the Hölder’s inequality and the Sobolev embedding
$W^{\frac{1}{4}+\epsilon^{\prime},4}(\mathbb{T})\hookrightarrow
L^{\infty}(\mathbb{T})$ where $0<\epsilon^{\prime}<s-\frac{3-\alpha}{4}$. Then
Proposition A.2 is applied at the minimum regularity $\frac{2-\alpha}{4}$ and
$0<\epsilon\ll 1$ to obtain

$\eqref{estimate3}\lesssim
T^{\frac{1}{2}}\|u\|_{X_{T}^{\frac{1}{4}+\epsilon^{\prime},\frac{1}{2}-\epsilon}}\|u\|_{X_{T}^{\frac{3-\alpha}{4}+\epsilon^{\prime},\frac{1}{2}+\epsilon}}\lesssim
T^{\frac{1}{2}}\|u\|_{X_{T}^{\frac{3-\alpha}{4}+\epsilon^{\prime},\frac{1}{2}+\epsilon}}^{2}\lesssim
T^{\frac{1}{2}}\|u\|_{C_{T}H^{s}}^{2}\lesssim
T^{\frac{1}{2}}\|u_{0}\|_{H^{s}}^{2},$

where the second and the third estimates reflect the embeddings
$X_{T}^{s_{1},b_{1}}\hookrightarrow X_{T}^{s_{2},b_{2}}$, for $s_{1}\geq
s_{2},\ b_{1}\geq b_{2}$, and $X_{T}^{s,\frac{1}{2}+\epsilon}\hookrightarrow
C([0,T];H^{s}(\mathbb{T}))$, respectively, and the last inequality follows
from the proof of Proposition A.1 based on the fixed point argument. ∎

######  Remark 5.1.

Since there is no canonical way to define a numerical error, the convergence
rate depends on the method of discretization and interpolation. As (2.1), we
discretize data on a smooth domain by averaging over an interval of length $h$
and linearly interpolate discrete data. Though we do not take the following
approach, [2, 6] considered the pointwise projection of continuous functions
onto $h\mathbb{Z}$ and the Shannon interpolation that takes the discrete
convolution of discrete data against the sinc function. It is commented in [6]
that the Shannon interpolation is better suited to show convergence in higher
Sobolev norms. It is of interest to show the continuum limit of our model in
higher Sobolev norms, which would require a uniform-in-$h$ control of the
Sobolev norms of discrete solutions. However the method of modified energy
used in the previous references to obtain bounds on higher Sobolev norms is
not directly applicable in our nonlocal case due to the complexity of the
Leibniz rule for fractional derivatives.

Now we show the sharpness of the convergence rate of the continuum limit at
the energy space $H^{\frac{\alpha}{2}}(\mathbb{T})$. The mass and energy
conservation for sufficiently regular data admits the global result.

######  Proposition 5.1.

Let $\alpha\in(1,2]$ and $u_{0}\in H^{\frac{\alpha}{2}}(\mathbb{T})$. Then
there exists $C_{1},C_{2}>0$ depending only on
$\|u_{0}\|_{H^{\frac{\alpha}{2}}}$ and $\alpha$ such that the error estimate

$\|p_{h}S_{h}(t)d_{h}u_{0}-S(t)u_{0}\|_{L^{2}(\mathbb{T})}\leq
C_{1}h^{\frac{\alpha}{2+\alpha}}e^{C_{2}|t|}(1+\|u_{0}\|_{H^{\frac{\alpha}{2}}})^{3},$
(5.5)

holds for all $t\in\mathbb{R}$.

###### Proof.

With initial data of finite energy, the Sobolev embedding
$H^{\frac{\alpha}{2}}(\mathbb{T})\hookrightarrow L^{\infty}(\mathbb{T})$
allows a more straightforward proof, without resorting to the Strichartz
estimates, than the one presented in Section 5. ∎

It is of interest to show the existence of $u_{0}\in
H^{\frac{\alpha}{2}}(\mathbb{T})$ such that

$C_{1}(t,\|u_{0}\|_{H^{\frac{\alpha}{2}}})h^{\frac{2}{2+\alpha}}\leq\|p_{h}S_{h}(t)d_{h}u_{0}-S(t)u_{0}\|_{L^{2}}\leq
C_{2}(t,\|u_{0}\|_{H^{\frac{\alpha}{2}}})h^{\frac{2}{2+\alpha}},$

for any $t\in\mathbb{R}$. Instead we derive a partial result that is local in
time, which shows the sharpness of the convergence rate
$\frac{\alpha}{2+\alpha}$ in (5.5).

######  Proposition 5.2.

Let $\alpha\in(1,2]$. There exists $0<T\ll 1$ such that for any $0<h\ll 1$, we
have $u_{0}^{h}\in H^{\frac{\alpha}{2}}(\mathbb{T})$ with
$\|u_{0}^{h}\|_{H^{\frac{\alpha}{2}}}=O(1)$, independent of $h$, satisfying

$\|p_{h}S_{h}(t)d_{h}u_{0}^{h}-S(t)u_{0}^{h}\|_{L^{\infty}_{T}L^{2}}\geq
c(T,\alpha)h^{\frac{2}{2+\alpha}},$

where $c>0$ is independent of $h>0$.

###### Proof.

We fix $\alpha$ once and for all and all constants resulting from the Sobolev
embedding or the Strichartz estimates are considered as implicit constants.
Fix $T>0$ as (5.1) corresponding to
$\\{u_{0}:\|u_{0}\|_{H^{\frac{\alpha}{2}}}=O(1)\\}$. Recall from the proof of
Proposition 4.2 that
$\|u_{h}\|_{L^{\infty}_{T}H^{\frac{\alpha}{2}}}\lesssim\|d_{h}u_{0}\|_{H_{h}^{\frac{\alpha}{2}}}$
where an explicit description of $u_{0}$ is given in (5.12). Let $t\in[0,T]$
and $h\ll_{\alpha}1$. From (5.2),

$\begin{split}I_{2}(t)+I_{3}(t)&\lesssim\int_{0}^{t}(|t-\tau|h^{\frac{\alpha}{2+\alpha}}+h^{\frac{\alpha}{2}})\|u_{h}\|_{L^{\infty}_{h}}^{2}\|u_{h}\|_{H_{h}^{\frac{\alpha}{2}}}d\tau\lesssim\|u_{h}\|^{3}_{L^{\infty}_{T}H_{h}^{\frac{\alpha}{2}}}\int_{0}^{t}(|t-\tau|h^{\frac{\alpha}{2+\alpha}}+h^{\frac{\alpha}{2}})d\tau\\\
&\lesssim\|d_{h}u_{0}\|_{H^{\frac{\alpha}{2}}_{h}}^{3}(t^{2}h^{\frac{\alpha}{2+\alpha}}+|t|h^{\frac{\alpha}{2}})\lesssim\|u_{0}\|_{H^{\frac{\alpha}{2}}}^{3}Th^{\frac{\alpha}{2+\alpha}},\end{split}$

where we shrink $T$, if necessary, to admit the last inequality. On the other,

$\begin{split}\int_{0}^{t}(\|p_{h}u_{h}\|_{L^{\infty}}^{2}+\|u\|_{L^{\infty}}^{2})\|p_{h}u_{h}-u\|d\tau&\leq(\|p_{h}u_{h}\|_{L^{2}([0,t];L^{\infty})}^{2}+\|u\|_{L^{2}([0,t];L^{\infty})}^{2})\|p_{h}u_{h}-u\|_{L^{\infty}([0,t];L^{2})}\\\
&\lesssim\|p_{h}u_{h}-u\|_{L^{\infty}_{T}L^{2}},\end{split}$

since
$\|p_{h}u_{h}\|_{L^{2}([0,t];L^{\infty})}^{2}+\|u\|_{L^{2}([0,t];L^{\infty})}^{2}=O(1)$
as in the proof of Theorem 1.1. Hence by the triangle inequality, we have

$\|p_{h}S_{h}(\cdot)d_{h}u_{0}-S(\cdot)u_{0}\|_{L^{\infty}_{T}L^{2}}\gtrsim\|p_{h}U_{h}(t)d_{h}u_{0}-U(t)u_{0}\|_{L^{2}}-\|u_{0}\|_{H^{\frac{\alpha}{2}}}^{3}Th^{\frac{\alpha}{2+\alpha}}.$
(5.6)

Observe that

$\|p_{h}U_{h}(t)d_{h}u_{0}-U(t)u_{0}\|_{L^{2}}\geq\|p_{h}U_{h}(t)d_{h}u_{0}-U(t)p_{h}d_{h}u_{0}\|_{L^{2}}-\|U(t)(p_{h}d_{h}u_{0}-u_{0})\|_{L^{2}}=:A_{1}-A_{2},$
(5.7)

where $A_{2}\lesssim h^{\frac{\alpha}{2}}\|u_{0}\|_{H^{\frac{\alpha}{2}}}$ by
the unitarity of $U(t)$ and Lemma 5.4. By the Plancherel’s Theorem,

$\displaystyle\sqrt{2\pi}A_{1}$
$\displaystyle=\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})\mathcal{F}[p_{h}d_{h}u_{0}]\|_{l^{2}}$
$\displaystyle\geq\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})\widehat{u}_{0}\|_{l^{2}}-\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})(\mathcal{F}[p_{h}d_{h}u_{0}]-\widehat{u}_{0})\|_{l^{2}}$
$\displaystyle\geq\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})\widehat{u}_{0}\|_{l^{2}}-Ch^{\frac{\alpha}{2}}\|u_{0}\|_{H^{\frac{\alpha}{2}}},$
(5.8)

where $C>0$ is by Lemma 5.4. The lower bound of the phase difference is
estimated. By direct computation,

$\left|e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}}\right|=2\left|\sin\left(\frac{2^{\alpha}t}{h^{\alpha}}\left(\left|\sin\frac{hk}{2}\right|^{\alpha}-\left|\frac{hk}{2}\right|^{\alpha}\right)\right)\right|.$

For $|k|\leq(10h)^{-1}$, the Taylor expansion yields the alternating series

$\left|\sin\frac{hk}{2}\right|^{\alpha}-\left|\frac{hk}{2}\right|^{\alpha}=\left|\frac{hk}{2}\right|^{\alpha}\left(-\frac{\alpha}{24}(hk)^{2}+\frac{\alpha(5\alpha-2)}{5760}(hk)^{4}-O_{\alpha}((hk)^{4}))\right),$
(5.9)

and therefore

$\frac{2^{\alpha}|t|}{h^{\alpha}}\cdot\left|\left|\sin\frac{hk}{2}\right|^{\alpha}-\left|\frac{hk}{2}\right|^{\alpha}\right|\leq\frac{\alpha|t|}{24}h^{2}|k|^{2+\alpha}.$
(5.10)

To apply the lower bound estimate

$|\sin\theta|\geq\frac{2}{\pi}|\theta|,\ \text{for any\
}|\theta|\leq\frac{\pi}{2},$

let
$\theta=\frac{2^{\alpha}t}{h^{\alpha}}\left(\left|\sin\frac{hk}{2}\right|^{\alpha}-\left|\frac{hk}{2}\right|^{\alpha}\right)$
and by (5.10), assume that $E$ holds where

$E=\\{k\in\mathbb{Z}:|k|\leq\left(\frac{12\pi}{\alpha}\right)^{\frac{1}{2+\alpha}}|t|^{-\frac{1}{2+\alpha}}h^{-\frac{2}{2+\alpha}}\\}.$

Then by the trigonometric lower bound estimate and (5.9),

$\left|e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}}\right|\geq\frac{2^{2+\alpha}|t|}{\pi
h^{\alpha}}\cdot\left|\left|\sin\frac{hk}{2}\right|^{\alpha}-\left|\frac{hk}{2}\right|^{\alpha}\right|\geq\frac{\alpha|t|}{12\pi}h^{2}|k|^{2+\alpha},$

and using this lower bound estimate, we have

$\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})\widehat{u}_{0}\|_{l^{2}}\geq\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})\widehat{u}_{0}\|_{l^{2}(E)}\gtrsim|t|h^{2}\||k|^{2+\alpha}\widehat{u}_{0}\|_{l^{2}(E)}.$
(5.11)

Let $k_{0}=|t|^{-\frac{1}{2+\alpha}}h^{-\frac{2}{2+\alpha}}$. For $\epsilon>0$
to be determined below, define

$\widehat{u}_{0}^{h}(k)=\epsilon k_{0}^{-\frac{\alpha}{2}}\delta_{k_{0}}(k)$
(5.12)

where $\delta_{k_{0}}$ is the Kronecker delta function supported at $k=k_{0}$.
Hence by (5.6), (5.7), (5), and (5.11),

$\begin{split}\|p_{h}S_{h}(\cdot)d_{h}u_{0}^{h}-S(\cdot)u_{0}^{h}\|_{L^{\infty}_{T}L^{2}}&\gtrsim
Th^{2}\||k|^{2+\alpha}\widehat{u}^{h}_{0}\|_{l^{2}(E)}-\|u_{0}^{h}\|_{H^{\frac{\alpha}{2}}}^{3}Th^{\frac{\alpha}{2+\alpha}}-\|u_{0}^{h}\|_{H^{\frac{\alpha}{2}}}h^{\frac{\alpha}{2}}\\\
&\simeq(\epsilon
T^{\frac{\alpha/2}{2+\alpha}}-\epsilon^{3}T)h^{\frac{\alpha}{2+\alpha}}-\epsilon
h^{\frac{\alpha}{2}}\\\
&\gtrsim(\frac{\epsilon}{2}-\epsilon^{3})T^{\frac{\alpha/2}{2+\alpha}}h^{\frac{\alpha}{2+\alpha}},\end{split}$

where the last inequality assumes $T\leq 1$ and $h$ sufficiently small
depending on $\alpha$ and $T$; for example,
$h\leq\left(\frac{T^{\frac{\alpha/2}{2+\alpha}}}{2}\right)^{\frac{4+2\alpha}{\alpha^{2}}}$
would do. The proof is complete by taking $0<\epsilon<\frac{1}{\sqrt{2}}$. ∎

######  Corollary 5.1.

Let $T>0$ be as Proposition 5.2 and suppose

$\sup_{\|u_{0}\|_{H^{\frac{\alpha}{2}}(\mathbb{T})}<R}\|p_{h}S_{h}(\cdot)d_{h}u_{0}-S(\cdot)u_{0}\|_{L^{\infty}_{T}L^{2}}\lesssim_{T,R,\alpha}h^{p},$
(5.13)

for some $R>0,\ p>0$. Then $\max\\{p:\eqref{sharpness_cor}\ \text{holds for
all\ }h>0\\}=\frac{2}{2+\alpha}$.

###### Proof.

By Proposition 5.1, $\frac{2}{2+\alpha}$ satisfies $\eqref{sharpness_cor}$ for
any $R>0$. Any $p>\frac{2}{2+\alpha}$ is not in the desired set by an explicit
construction in Proposition 5.2. ∎

######  Remark 5.2.

The sharpness of the convergence rate is expected to be $O(h^{\frac{s}{1+s}})$
in $H^{s}(\mathbb{T})$ for $s<\frac{\alpha}{2}$, i.e., for data of infinite
energy. The proof in this regime, given the technical difficulty due to the
absence of the Sobolev embedding, is left for further research.

We have shown that the convergence rate of the numerical scheme given by (1.2)
or (5.5) is sublinear at worst for general Sobolev data. In numerical
computations using softwares, the high frequency components of $u_{0}$ are
often truncated, and therefore the Fourier support of $\widehat{u}_{0}$ is
assumed to be compact. To motivate further discussion on the relationship
between numerical convergence and the compactness of Fourier support, see
Appendix B for concrete examples of exact solutions also considered in [5]. In
the following proposition, the sharp linear convergence illustrated by the
example (B.2) is generalized. More remains to be studied on the nonlinear
evolution of Fourier modes on the lattice.

######  Proposition 5.3.

For $\alpha\in(1,2]$, assume $u_{0}\in H^{\frac{\alpha}{2}}(\mathbb{T})$ and
$k_{m}:=\max\\{|k|:k\in supp(\widehat{u}_{0})\\}<\infty$. Suppose there exist
$T>0,\ k_{c}>0$, and $h_{0}>0$ such that
$supp(\mathcal{F}_{h}[u_{h}(t)])\subseteq[-k_{c},k_{c}]$ for all $|t|\leq T,\
h<h_{0}$. Then,

$\|p_{h}S_{h}(t)d_{h}u_{0}-S(t)u_{0}\|_{L^{2}(\mathbb{T})}\leq
Ch(k_{max})^{1-\frac{\alpha}{2}},$ (5.14)

where $k_{max}=\max(k_{m},k_{c})$ and
$C=C(\|u_{0}\|_{H^{\frac{\alpha}{2}}},\alpha)>0$, for $t\in[0,T]$ and $h>0$
sufficiently small given by (5.15).

###### Proof.

The argument proceeds as in the proof of Theorem 1.1 where we may assume $T>0$
is at most the time of existence stated in Theorem 1.1. Take

$h<C_{0}\min(h_{0},(3k_{max})^{-1},(Tk_{max}^{1+\alpha})^{-1},(T^{2}k_{max}^{1+\alpha})^{-1}),$
(5.15)

where $C_{0}(\|u_{0}\|_{H^{\frac{\alpha}{2}}},\alpha)>0$ to be determined.
Since $2k_{m}<\frac{2\pi}{h}$, the period of $\mathbb{T}_{h}^{*}$, we have
$supp(\mathcal{F}_{h}[d_{h}u_{0}])=supp(\widehat{u}_{0})$ by (B.1). Hence by
the triangle inequality and Lemma 5.4,

$\displaystyle\|p_{h}U_{h}(t)d_{h}u_{0}-U(t)u_{0}\|_{L^{2}}$
$\displaystyle\leq\|p_{h}U_{h}(t)d_{h}u_{0}-U(t)p_{h}d_{h}u_{0}\|_{L^{2}}+\|U(t)(p_{h}d_{h}u_{0}-u_{0})\|_{L^{2}}$
$\displaystyle\lesssim\|(e^{-it\left|\frac{2}{h}\sin\frac{hk}{2}\right|^{\alpha}}-e^{-it|k|^{\alpha}})P_{h}(k)\mathcal{F}_{h}[d_{h}u_{0}]\|_{l^{2}_{\\{|k|\leq
h^{-1}\\}}}+h\|u_{0}\|_{H^{1}}$
$\displaystyle\lesssim\alpha|t|h^{2}\||k|^{2+\alpha}\mathcal{F}_{h}[d_{h}u_{0}]\|_{l^{2}_{\\{|k|\leq
h^{-1}\\}}}+h\|u_{0}\|_{H^{1}}$ (5.16)
$\displaystyle\lesssim\|u_{0}\|_{H^{\frac{\alpha}{2}}}\left(\alpha
Tk_{m}^{2+\frac{\alpha}{2}}h^{2}+k_{m}^{1-\frac{\alpha}{2}}h\right)\lesssim\|u_{0}\|_{H^{\frac{\alpha}{2}}}k_{m}^{1-\frac{\alpha}{2}}h,$
(5.17)

where the third inequality estimates the phase difference as (5.10) for
$|k|\leq h^{-1}$ and the last inequality follows from $h<(\alpha
Tk_{m}^{1+\alpha})^{-1}$. Then the nonlinear terms are estimated. Since
$supp(\mathcal{F}_{h}[|u_{h}|^{2}u_{h}])\subseteq[-3k_{c},3k_{c}]$,

$\displaystyle\int_{0}^{t}\|(p_{h}U_{h}(t-\tau)-U(t-\tau)p_{h})(|u_{h}|^{2}u_{h})(\tau)\|_{L^{2}}d\tau$
$\displaystyle\lesssim
h^{2}\int_{0}^{t}|t-\tau|\cdot\||k|^{2+\alpha}P_{h}(k)\mathcal{F}_{h}[|u_{h}|^{2}u_{h}]\|_{l^{2}_{\\{|k|\leq
h^{-1}\\}}}d\tau$ $\displaystyle\lesssim
h^{2}(3k_{c})^{2+\frac{\alpha}{2}}\int_{0}^{t}|t-\tau|\cdot\||u_{h}(\tau)|^{2}u_{h}(\tau)\|_{H_{h}^{\frac{\alpha}{2}}}d\tau$
$\displaystyle\lesssim CT^{2}h^{2}(3k_{c})^{2+\frac{\alpha}{2}},$ (5.18)

where the first inequality is estimated as (5.16) and the last inequality of
$H_{h}^{\frac{\alpha}{2}}$ is by the Sobolev algebra property with $C$
depending only on the mass and energy of $u_{0}$ by the discrete Gagliardo-
Nirenberg inequality. Another nonlinear term is estimated by Lemma 5.3.

$\int_{0}^{t}\|p_{h}(|u_{h}|^{2}u_{h})-|p_{h}u_{h}|^{2}p_{h}u_{h}\|_{L^{2}}d\tau\lesssim
h\int_{0}^{t}\|u_{h}\|_{L^{\infty}_{h}}^{2}\|u_{h}\|_{H^{1}_{h}}d\tau\lesssim\|u_{h}\|_{H_{h}^{\frac{\alpha}{2}}}^{3}Thk_{c}^{1-\frac{\alpha}{2}}.$
(5.19)

Combining (5.17), (5.18), (5.19), we have

$\begin{split}\|p_{h}u_{h}(t)-u(t)\|_{L^{2}}&\lesssim
hk_{m}^{1-\frac{\alpha}{2}}+T^{2}h^{2}(3k_{c})^{2+\frac{\alpha}{2}}+Thk_{c}^{1-\frac{\alpha}{2}}+\int_{0}^{t}(\|p_{h}u_{h}\|_{L^{\infty}}^{2}+\|u\|_{L^{\infty}}^{2})\|p_{h}u_{h}-u\|_{L^{2}}d\tau\\\
&\lesssim(1+T)h(k_{max})^{1-\frac{\alpha}{2}}+\int_{0}^{t}(\|p_{h}u_{h}\|_{L^{\infty}}^{2}+\|u\|_{L^{\infty}}^{2})\|p_{h}u_{h}-u\|_{L^{2}}d\tau.\end{split}$

That
$\|p_{h}u_{h}\|_{L^{2}_{T}L^{\infty}}^{2}+\|u\|_{L^{2}_{T}L^{\infty}}^{2}=O(1)$
can be shown as in the proof of Theorem 1.1, and therefore (5.14) holds by the
Gronwall’s inequality. ∎

######  Remark 5.3.

The computation of numerical error could take place in different function
spaces with different interpolation methods. While Proposition 5.3 gives
linear convergence in $L^{2}(\mathbb{T})$ for solutions with compact Fourier
support, the faster quadratic convergence is not expected for linearly
interpolated solutions. Indeed the second derivative acting on $p_{h}$ yields
the Dirac delta functions, which are not square-integrable. Observe, however,
that the exact solution (B.2) with the initial datum
$u_{0}(x)=A|n|^{-s}e^{inx}$ converges quadratically in $L^{2}_{h}$ as can be
shown explicitly as

$\displaystyle\|u_{h}(t)-d_{h}u(t)\|_{L^{2}_{h}}$
$\displaystyle=\sqrt{2\pi}|A||n|^{-s}\left|\frac{e^{ihn}-1}{ihn}\right|\cdot\left|e^{-it(|\frac{2}{h}\sin\frac{hn}{2}|^{\alpha}+\mu|A|^{2}|n|^{-2s}|\frac{e^{ihn}-1}{ihn}|^{2}}-e^{-it(|n|^{\alpha}+\mu|A|^{2}|n|^{-2s}}\right|$
$\displaystyle=\left(\frac{\sqrt{2\pi}}{24}|Atn|^{2-3s}\cdot\left|\alpha|n|^{\alpha+2s}+2\mu|A|^{2}\right|\right)h^{2}+O(h^{3}).$
(5.20)

##  6 Conclusion.

Motivated by recent trends in fractional calculus and nonlocal dynamics, we
investigated fDNLS on a periodic lattice. The continuum limit for data below
the energy space was shown, thereby extending [19, 17, 9]. However the method
of periodic discrete Strichartz estimates was insufficient to establish the
desired convergence up to $s=\frac{2-\alpha}{4}$, the known lowest Sobolev
regularity at which the local well-posedness of (A.1) was established in [7].
In the discrete regime, we studied the modulational instability of CW
solutions, thereby extending [1, 38]. It was shown that the nonlocal parameter
$\alpha$ triggers a broader spectrum of higher mode excitations if
$|A|>\frac{1}{\sqrt{2}}$ while the spectrum shrinks if
$|A|<\frac{1}{\sqrt{2}}$. The dependence of the maximum gain on $A,h,\alpha$
was shown analytically and numerically, consistent with the emergence of chaos
[29] as $\alpha$ departs from $\alpha=2$ where the long-range coupling yields
strong correlation between two distant lattice sites. The nonlinear patterns
revealed by our numerical simulations, such as the nonlinear instability for
highly nonlocal systems and the nonlinear dependence of the maximum gain on
the wave amplititude, call for further research.

##  Appendix A Appendix: Well-posedness and Uniform Estimates.

The well-posedness results of (1.1), (A.1) are given, followed by the uniform
estimates needed to establish the continuum limit.

The quantitative measure of dispersive smoothing can be obtained by averaging
over space and time variables under the unitary evolution. Recall that the
Bourgain norm measures the $L^{2}$ norm of the space-time Fourier transform
weighted by the deviation from the hypersurface defined as the zero-set of the
dispersion relation. Let $s,b\in\mathbb{R}$ and
$\widehat{f}(k,\tau)=\int_{\mathbb{R}}\int_{\mathbb{T}}f(x,t)e^{-i(kx+\tau
t)}dxdt$. Define

$\|f\|_{X^{s,b}}^{2}=\int_{\mathbb{R}}\sum\limits_{k\in\mathbb{Z}}\langle
k\rangle^{2s}\langle\tau-|k|^{\alpha}\rangle^{2b}|\widehat{f}(k,\tau)|^{2}d\tau.$

To establish local well-posedness in $[0,T]$, consider
$\mathcal{C}_{f}=\\{g\in X^{s,b}:g=f\ \text{on\ }[0,T]\\}$, and define the
quotient space whose norm is defined by
$\|f\|_{X^{s,b}_{T}}=\inf\limits_{g\in\mathcal{C}_{f}}\|g\|_{X^{s,b}}$.

######  Proposition A.1 ([7, Theorem 1.1]).

Given $\alpha\in(1,2)$ and $s\geq\frac{2-\alpha}{4}$, the fNLS

$i\partial_{t}u=(-\Delta)^{\frac{\alpha}{2}}u+\mu|u|^{2}u,\
(x,t)\in\mathbb{T}\times\mathbb{R}$ (A.1)

is locally well-posed in $H^{s}(\mathbb{T})$. More precisely, for any initial
datum $u_{0}\in H^{s}(\mathbb{T})$, there exists a unique $u\in
X^{s,\frac{1}{2}+\epsilon}_{T}\subseteq C([0,T];H^{s}(\mathbb{T}))$, for every
$0<\epsilon\ll 1$, such that the integral representation of (A.1) given by

$u(t)=U(t)u_{0}-i\mu\int_{0}^{t}U(t-\tau)\left(|u(\tau)|^{2}u(\tau)\right)d\tau$

holds for all $t\in[0,T]$ where $T=T(\|u_{0}\|_{H^{s}})>0$. Furthermore mass
($M$) and energy ($H$) are conserved where

$M[u(t)]=\|u(t)\|_{L^{2}}^{2};\
H[u(t)]=\frac{1}{2}\||\nabla|^{\frac{\alpha}{2}}u\|_{L^{2}}^{2}+\frac{\mu}{4}\|u\|_{L^{4}}^{4}.$

A crucial estimate used in the proof of Proposition A.1 is the following
bilinear estimate.

######  Proposition A.2 ([7, Proposition 3.2]).

For $s\geq\frac{2-\alpha}{4}$ and $0<\epsilon\ll 1$, we have

$\|uv\|_{L^{2}(\mathbb{R}\times\mathbb{T})}\lesssim_{\epsilon}\|u\|_{X^{0,\frac{1}{2}-\epsilon}}\|v\|_{X^{s,\frac{1}{2}+\epsilon}}.$

##  Appendix B Appendix: Exact Solutions.

To illustrate the trivial case, consider $u_{0}(x)=A\in\mathbb{C}$. By direct
computation,

$p_{h}S_{h}(t)d_{h}u_{0}(x)-S(t)u_{0}(x)=Ae^{-i\mu|A|^{2}t}-Ae^{-i\mu|A|^{2}t}=0.$

Another example of exact solution is a family of sinusoids that oscillate at a
single spatiotemporal frequency. Let $A\in\mathbb{C}\setminus\\{0\\},\
n\in\mathbb{Z}\setminus\\{0\\},\ s\in\mathbb{R}$, and consider
$u_{0}(x)=A|n|^{-s}e^{inx}$ for $x\in\mathbb{T}$. By definition of $d_{h}$ and
$\mathcal{F}_{h}$, and given the Fourier expansion
$f(x)=\frac{1}{2\pi}\sum\limits_{k\in\mathbb{Z}}\widehat{f}(k)e^{ikx}$, we
have

$d_{h}[e^{ik\cdot}](x^{\prime})=\begin{cases}\frac{e^{ihk}-1}{ihk}e^{ikx^{\prime}},&k\neq
0\\\ 1,&k=0,\end{cases};\
\mathcal{F}_{h}[d_{h}f](k^{\prime})=\begin{cases}\sum\limits_{q\in\mathbb{Z}}\frac{\widehat{f}(pq+k^{\prime})}{pq+k^{\prime}}\frac{e^{ihk^{\prime}}-1}{ih},&k^{\prime}\neq
0\\\ \widehat{f}(0),&k^{\prime}=0,\end{cases}$ (B.1)

where $x^{\prime}\in\mathbb{T}_{h},\ k^{\prime}\in\mathbb{T}_{h}^{*}$. Note
that the domain of $\mathcal{F}_{h}[d_{h}f]$ can be extended from
$\mathbb{T}_{h}^{*}$ to $\mathbb{Z}$ periodically since the summation over
$q\in\mathbb{Z}$ is over all periods with the period $2M=\frac{2\pi}{h}$. By
direct computation,

$\begin{split}S(t)u_{0}(x)&=A|n|^{-s}e^{-it(|n|^{\alpha}+\mu|A|^{2}|n|^{-2s}}e^{inx},\
x\in\mathbb{T}\\\
S_{h}(t)d_{h}u_{0}(x)&=A|n|^{-s}\frac{e^{ihn}-1}{ihn}e^{-it(|\frac{2}{h}\sin\frac{hn}{2}|^{\alpha}+\mu|A|^{2}|n|^{-2s}|\frac{e^{ihn}-1}{ihn}|^{2}}e^{inx},\
x\in\mathbb{T}_{h},\end{split}$ (B.2)

i.e., the exact solutions to (A.1) and (1.1), respectively. Recalling from
[19, Lemma 5.5] that $p_{h}$ is a Fourier multiplier with the symbol
$P_{h}(k)=\left(\frac{\sin(\frac{hk}{2})}{\frac{hk}{2}}\right)^{2}$, or
equivalently $\mathcal{F}[p_{h}f](k)=P_{h}(k)\mathcal{F}_{h}[f](k)$, we have

$\displaystyle\|p_{h}u_{h}(t)$
$\displaystyle-u(t)\|_{L^{2}(\mathbb{T})}=(2\pi)^{-\frac{1}{2}}\|P_{h}(k)\mathcal{F}_{h}[u_{h}(t)](k)-\widehat{u(t)}(k)\|_{l^{2}_{k}}$
$\displaystyle=(2\pi)^{\frac{1}{2}}|A||n|^{-s}\left|P_{h}(n)\frac{e^{ihn}-1}{ihn}e^{-it(|\frac{2}{h}\sin\frac{hn}{2}|^{\alpha}+\mu|A|^{2}|n|^{-2s}|\frac{e^{ihn}-1}{ihn}|^{2}}-e^{-it(|n|^{\alpha}+\mu|A|^{2}|n|^{-2s}}\right|$
$\displaystyle=\left(\sqrt{\frac{\pi}{2}}|A||n|^{1-s}\right)h+O(h^{2}),$ (B.3)

which yields sharp linear convergence, where the last equality is by the
Taylor’s Theorem.

## References

  * [1] Alejandro Aceves and Austin Copeland.  Spatiotemporal dynamics in the fractional nonlinear Schrödinger equation.  Frontiers, Nonlinear Photonics, 3, 2022.
  * [2] Joackim Bernier.  Bounds on the growth of high discrete Sobolev norms for the cubic discrete nonlinear Schrödinger equations on $h\mathbb{Z}$.  Discrete and Continuous Dynamical Systems-Series A, 39(6):3179–3195, 2019.
  * [3] J. Bourgain.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, part I: Schrödinger equations.  Geometric and functional analysis, 3(2):107–156, 1993.
  * [4] Steven L Brunton and J Nathan Kutz.  Data-driven science and engineering: Machine learning, dynamical systems, and control.  Cambridge University Press, 2022.
  * [5] Nicolas Burq, Pierre Gérard, and Nikolay Tzvetkov.  An instability property of the nonlinear Schrödinger equation on ${S}^{d}$.  Mathematical Research Letters, 9(3):323–335, 2002.
  * [6] Quentin Chauleur.  Growth of Sobolev norms and strong convergence for the discrete nonlinear Schrödinger equation.  working paper or preprint, June 2023.
  * [7] Yonggeun Cho, Gyeongha Hwang, Soonsik Kwon, and Sanghyuk Lee.  Well-posedness and ill-posedness for the cubic fractional Schrödinger equations.  Discrete and Continuous Dynamical Systems, 35(7):2863–2880, 2015.
  * [8] Brian Choi and Alejandro Aceves.  Well-posedness of the mixed-fractional nonlinear Schrödinger equation on $\mathbb{R}^{2}$.  Partial Differential Equations in Applied Mathematics, 6:100406, 2022.
  * [9] Brian Choi and Alejandro Aceves.  Continuum limit of 2D fractional nonlinear Schrödinger equation.  Journal of Evolution Equations, 23(2):30, 2023.
  * [10] Brian Choi, Austin Marstaller, and Alejandro Aceves.  On localization of the fractional discrete nonlinear Schrödinger equation.  arXiv preprint arXiv:2309.11395, 2023.
  * [11] Seckin Demirbas, M. Burak Erdoğan, and Nikolaos Tzirakis.  Existence and uniqueness theory for the fractional Schrödinger equation on the torus, volume 34 of Adv. Lect. Math. (ALM), pages 145–162.  Int. Press, Somerville, MA, 2016.
  * [12] Van Duong Dinh.  Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces.  International Journal of Apllied Mathematics, 31(4):483–525, September 2018.
  * [13] Siwei Duo, Taras I Lakoba, and Yanzhi Zhang.  Dynamics of plane waves in the fractional nonlinear Schrödinger equation with long-range dispersion.  Symmetry, 13(8):1394, 2021.
  * [14] HS Eisenberg, Yaron Silberberg, R Morandotti, AR Boyd, and JS Aitchison.  Discrete spatial optical solitons in waveguide arrays.  Physical Review Letters, 81(16):3383, 1998.
  * [15] Yu. B. Gaididei, S. F. Mingaleev, P. L. Christiansen, and K. Ø. Rasmussen.  Effects of nonlocal dispersive interactions on self-trapping excitations.  Phys. Rev. E, 55:6141–6150, May 1997.
  * [16] Andrey Gelash, Dmitry Agafontsev, Vladimir Zakharov, Gennady El, Stéphane Randoux, and Pierre Suret.  Bound state soliton gas dynamics underlying the spontaneous modulational instability.  Physical review letters, 123(23):234102, 2019.
  * [17] Younghun Hong, Chulkwang Kwak, Shohei Nakamura, and Changhun Yang.  Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations.  Journal of Evolution Equations, 21(1):391–418, 2021.
  * [18] Younghun Hong and Yannick Sire.  On fractional Schrödinger equations in Sobolev spaces.  Communications on Pure & Applied Analysis, 14(6):2265–2282, 2015.
  * [19] Younghun Hong and Changhun Yang.  Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit.  SIAM Journal on Mathematical Analysis, 51(2):1297–1320, 2019.
  * [20] Younghun Hong and Changhun Yang.  Uniform Strichartz estimates on the lattice.  Discrete and Continuous Dynamical Systems, 39(6):3239–3264, 2019.
  * [21] Liviu I Ignat and Enrique Zuazua.  A two-grid approximation scheme for nonlinear Schrödinger equations: dispersive properties and convergence.  Comptes Rendus Mathematique, 341(6):381–386, 2005.
  * [22] Liviu I Ignat and Enrique Zuazua.  Numerical dispersive schemes for the nonlinear Schrödinger equation.  SIAM journal on numerical analysis, 47(2):1366–1390, 2009.
  * [23] Liviu I Ignat and Enrique Zuazua.  Convergence rates for dispersive approximation schemes to nonlinear Schrödinger equations.  Journal de mathématiques pures et appliquées, 98(5):479–517, 2012.
  * [24] Magnus Johansson, Yuri B Gaididei, Peter L Christiansen, and KØ Rasmussen.  Switching between bistable states in a discrete nonlinear model with long-range dispersion.  Physical Review E, 57(4):4739, 1998.
  * [25] Markus Keel and Terence Tao.  Endpoint Strichartz estimates.  American Journal of Mathematics, 120(5):955–980, 1998.
  * [26] Panayotis G Kevrekidis.  The discrete nonlinear Schrödinger equation: mathematical analysis, numerical computations and physical perspectives, volume 232.  Springer Science & Business Media, 2009.
  * [27] K Kirkpatrick, E Lenzmann, and G Staffilani.  On the continuum limit for discrete NLS with long-range lattice interactions.  Commun. Math. Phys., 317:563––591, 2013.
  * [28] Yuri S. Kivshar and David K. Campbell.  Peierls-Nabarro potential barrier for highly localized nonlinear modes.  Phys. Rev. E, 48:3077–3081, Oct 1993.
  * [29] Nickolay Korabel and George M Zaslavsky.  Transition to chaos in discrete nonlinear Schrödinger equation with long-range interaction.  Physica A: Statistical Mechanics and its Applications, 378(2):223–237, 2007.
  * [30] Nikolai Laskin.  Fractional quantum mechanics and Lévy path integrals.  Physics Letters A, 268(4-6):298–305, 2000.
  * [31] Stefano Longhi.  Fractional Schrödinger equation in optics.  Optics Letters, 40(6):1117, mar 2015.
  * [32] Serge F Mingaleev, Yuri B Gaididei, and Franz G Mertens.  Solitons in anharmonic chains with power-law long-range interactions.  Physical Review E, 58(3):3833, 1998.
  * [33] Atanas Stefanov and Panayotis G Kevrekidis.  Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and klein–gordon equations.  Nonlinearity, 18(4):1841, 2005.
  * [34] Lloyd Nicholas Trefethen.  Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations.  Unpublished text, available at http://people.maths.ox.ac.uk/trefethen/pdetext.html, 1996.
  * [35] Luis Vega.  Restriction theorems and the Schrödinger multiplier on the torus.  In Partial differential equations with minimal smoothness and applications, pages 199–211. Springer, 1992.
  * [36] Vladimir E Zakharov, Alexander I Dyachenko, and Alexander O Prokofiev.  Freak waves as nonlinear stage of Stokes wave modulation instability.  European Journal of Mechanics-B/Fluids, 25(5):677–692, 2006.
  * [37] Jinggui Zhang.  Modulation instability in fractional Schrödinger equation with cubic–quintic nonlinearity.  Journal of Nonlinear Optical Physics & Materials, 31(04):2250019, 2022.
  * [38] Lifu Zhang, Zenghui He, Claudio Conti, Zhiteng Wang, Yonghua Hu, Dajun Lei, Ying Li, and Dianyuan Fan.  Modulational instability in fractional nonlinear Schrödinger equation.  Communications in Nonlinear Science and Numerical Simulation, 48:531–540, 2017.
  * [39] Antoni Zygmund.  Trigonometric series, volume 1.  Cambridge university press, 2002.