# On Localization of the Fractional Discrete Nonlinear Schrödinger Equation

Brian Choi Corresponding author. Southern Methodist University, United States
Military Academy<EMAIL_ADDRESS>Austin Marstaller Southern Methodist
University<EMAIL_ADDRESS>Alejandro Aceves Southern Methodist
University<EMAIL_ADDRESS>
###### Abstract

The continuum and discrete fractional nonlinear Schrödinger equations (fDNLS)
represent new models in nonlinear wave phenomena with unique properties. In
this paper, we focus on various aspects of localization associated to fDNLS
featuring modulational instability, asymptotic construction of onsite and
offsite solutions, and the role of Peierls-Nabarro barrier. In particular, the
localized onsite and offsite solutions are constructed using the map approach.
Under the long-range interaction characterized by the Lévy index $\alpha>0$,
the phase space of solutions is infinite-dimensional unlike that of the well-
studied nearest-neighbor interaction. We show that an orbit corresponding to
this spatial dynamics translates to an approximate solution that decays
algebraically. We also show as $\alpha\rightarrow\infty$, the discrepancy
between local and nonlocal dynamics becomes negligible on a compact time
interval, but persists on a global time scale. Moreover it is shown that data
of small mass scatter to free solutions under a sufficiently high
nonlinearity, which proves the existence of strictly positive excitation
threshold for ground state solutions of fDNLS.

Keywords. nonlinear modes, localization, stability, asymptotic approximation,
nonlocal dynamics

MSC 2020 34A08, 34A12, 34A34, 37K45, 37K60, 78M35

##  1 Introduction.

Research in arrays of coupled nonlinear oscillators remains a field of intense
research whether it aims at understanding thermalization or the emergence of
localized coherent structures, or global synchronization as exemplified by the
FPUT or the Kuramoto oscillator model, respectively. Their relevance goes
beyond theory as they model systems in a wide range of applications including
photonics, plasmonics [1, 2, 3], Bose-Einstein condensates [4, 5, 6, 7],
biological/chemical phenomena [8, 9], neuroscience [10] and the power grid
[11]. For more comprehensive surveys, see [12, 13, 14, 15]. In all these
cases, the emerging models are large systems of coupled ordinary differential
equations. Similarly for continuous fields, universal equations such as the
nonlinear Schrödinger equation (NLSE) or the sine-Gordon equation have
provided a platform to study many features in nonlinear wave phenomena
including the existence and interactions of solitons, and the formation of
coherent structures and singular blow-up dynamics.

Specific to nonlinear photonics and its applications, two universal models are
the NLSE and the discrete nonlinear Schrödinger equation (DNLS) (1.3). In the
discrete case, the model describes waveguide or resonator arrays with the
nearest-neighbor coupling, whereas the continuum version models, for example,
pulses in optical fibers or light filaments propagating in air. In this paper,
we present results for nonlocal, nonlinear discrete systems, departing from
the nearest-neighbor coupling, whose long-range interaction is described by a
nonlocal interaction kernel whose coupling strength decays algebraically. This
model is known as the fractional discrete nonlinear Schrödinger equation
(fDNLS) given by

$i\dot{u}_{n}=\epsilon\sum_{m\in\mathbb{Z}\setminus\\{n\\}}\frac{u_{n}-u_{m}}{|n-m|^{1+\alpha}}-|u_{n}|^{2}u_{n},\
(n,t)\in\mathbb{Z}\times\mathbb{R},\ \epsilon>0,\ \alpha>0,$ (1.1)

where our interest goes beyond this specific model to study a general long-
range interaction given by (2.2). Such nonlocal model was previously studied
[16, 17, 18, 19, 20] in various contexts including stability, chaos, and
higher-dimensional dynamics.

Unlike the vast amount of experimental results on waveguide arrays modeled by
the DNLS, at time there is no concrete photonics-based array for which the
fDNLS is an experimentally-verifiable model. We do believe, however, the
results could pave the way to future realizations where non-locality presents
a new degree of freedom. It should be noted that the continuum version of the
fDNLS

$i\partial_{t}U=(-\Delta)^{\frac{\alpha}{2}}U-|U|^{2}U,\
(x,t)\in\mathbb{R}^{d+1},$ (1.2)

where $U(x,t)\in\mathbb{C}$ models, for example, the envelope dynamics of an
electric field, was proposed as a model for an optical cavity [21] whereby a
proper design of lenses allows one to engineer diffraction to behave as
$|k|^{\alpha},\ 0<\alpha<2$ in the Fourier representation. The suggested
benefit of such a design is to produce laser beam outputs with unique (Airy-
like) profiles as opposed to the classical Gaussian-like outputs.

The continuum and discrete dynamics, either viewed as a discretization of a
continuum model, or in reverse, the continuum limit of an intrinsically
discrete model, exhibit distinct features, one of them being the Peierls-
Nabarro Barrier (PNB) in the discrete model [22, 23, 2]. Of central concern in
the theory of nonlinear lattices is the propagation of localized waves in
discrete media. A motion along the lattice requires the localized wave to
switch its structure from onsite to offsite, and vice versa, interacting with
the effective energy potential barrier, or the PNB, rising from the inherent
discreteness of the lattice. The continuum power-type NLSE, say (1.2) with
$\alpha=2$, satisfies the Galilean invariance: if $u(x,t)$ is a solution for
$x\in\mathbb{R}^{d}$, then so is
$e^{i\frac{v}{2}\cdot(x-\frac{tv}{2})}u(x-vt,t)$ for any $v\in\mathbb{R}^{d}$.
In particular, a ground state solution can be boosted to yield a family of
solutions whose amplitudes are traveling waves. Moreover the mass-critical
NLSE is invariant under the pseudoconformal symmetry, which could be used to
show the existence of excitation threshold of mass [24, 25] for certain NLSE
with sufficiently high nonlinearity. On the other hand, the lattice structure
lacks these symmetries rising from the smooth structure of the Euclidean
space, giving rise to, among many others, a non-zero PNB. We confirm the
presence of nonlocal PNB in our asymptotic analysis where the energy
difference between onsite and offsite solutions is computed explicitly. Our
theory is supported by numerical simulations of the dynamics of (1.1) for
varying degrees of non-locality where localized solutions with an initial
boost are eventually pinned at a lattice site.

On the other hand, the role of non-locality and discreteness on the formation
of localized states by modulational instability (MI) is investigated. A
periodic train of pulses, under a small perturbation in its spectrum that
originates from nonlinearity, localizes as a breather-like excitation driven
by MI. The literature on MI is vast, and for our purposes, readers are
directed to [26, 27, 28, 29, 30, 31] for survey results with applications in
nonlinear optics or fluid dynamics. However a detailed analysis of MI of
nonlocal dynamics is still at its infancy. See [32] for MI in relation to the
fractional NLSE. Furthermore see [33] that investigates the mixed-fractional
NLSE in the context of MI.

While this paper only discusses the discrete model, observe in (1.1) that by
considering $\epsilon=c(d,\alpha)h^{-\alpha}$ where $c(d,\alpha)$ is given by
(2.5) and $\alpha\in(0,2)$, (1.1) could be understood as a finite difference
scheme of the continuum fNLSE (1.2). The well-posedness theory of (1.2)
requires a more delicate analysis than its discrete version (see Proposition
3.1) given that blow-up phenomena could occur [34]. See [35, 36] for the
rigorous analysis of the well-posedness of fNLSE in Sobolev spaces. The
functional-analytic tools of the aforementioned references do not directly
apply when the non-locality, represented by the linear dispersion relation, is
not uniform in space (which occurs when the spatial dimension is at least
two), and thus to account for such non-homogeneity that rises in the mixed-
fractional NLSE, the non-smooth analogue of the Littlewood-Paley theory was
developed to establish the well-posedness in anisotropic Sobolev spaces [37].

Even with the continuum model being well-posed for data of sufficiently high
regularity, it is not trivial to show the continuum limit as $h\rightarrow 0$,
if it holds at all. For example, the mass-critical cubic NLSE on
$\mathbb{R}^{2}$ contains solutions that blow up in finite time whereas the
lattice dynamics is always global. It was shown rigorously in [38] that the
weak convergence of nonlocal discrete to continuum dynamics of the Schrödinger
evolution holds. A further research done by [39] (see references therein)
refined the weak to strong convergence by employing discrete Strichartz
estimates uniform in the discreteness parameter. Furthermore note [40] for the
first result in the continuum limit of fDNLS to fNLSE in dimension two. In the
context of onsite/offsite solutions, it is expected that the PNB would tend to
zero in the continuum limit since the Galilean invariance needs to be
recovered. This is indeed true, and moreover the quantitative bound that PNB
vanishes exponentially fast as $h\rightarrow 0$ was shown in [41, 42] for DNLS
and fDNLS, respectively.

This paper is organized as follows. In Section 2, the main model and its
properties are introduced. In Section 3, the global dynamics corresponding to
$\alpha\gg 1$ is investigated. In the formal limit as
$\alpha\rightarrow\infty$, (1.1) converges to the DNLS

$i\dot{u}_{n}=-\epsilon\delta^{2}u_{n}-|u_{n}|^{2}u_{n},\
(n,t)\in\mathbb{Z}^{d}\times\mathbb{R},\ \epsilon>0,$ (1.3)

with $d=1$, where $\delta^{2}u_{n}:=\sum\limits_{|j-n|=1}u_{j}-2du_{n}$ and
$|\cdot|$ defined with the $l^{1}$ norm on $\mathbb{Z}^{d}$. The nature of
this limit, whether it is regular or singular, is subtle as it relates to the
Soliton Resolution Conjecture and the long-time dynamics of Hamiltonian
systems whose solutions do not dissipate. In Section 4, linear stability
analysis on the CW solution of fDNLS is shown with explicit regions of MI. The
onset of nonlinear bound states resulting from MI is shown via numerical
simulations. In Section 5, the family of onsite and offsite solutions and the
corresponding PNB are constructed asymptotically for DNLS and fDNLS,
respectively. Although the rigorous aspects of numerical analysis
(consistency, convergence, etc) are not our main focus, we provide the
numerical discretization of the fractional Laplacian in Appendix A.

##  2 Background.

In this paper, we consider the generalized nonlocal model, also considered in
[38], where the infinitesimal generator is given by

$\mathscr{L}_{\alpha}f_{n}=\sum_{m\neq n}J_{|n-m|}(f_{n}-f_{m}),\ (f_{n})\in
l^{2}(\mathbb{Z}),\ \alpha>0,$ (2.1)

where $J=(J_{n})_{n=1}^{\infty},\ J_{n}\geq 0$ is the $\alpha$-kernel
satisfying the limit property

$\lim\limits_{n\rightarrow\infty}n^{1+\alpha}J_{n}=A_{\alpha}\in(0,\infty),$

and when $\alpha=\infty$, define $J$ as an $\infty$-kernel if
$\lim\limits_{n\rightarrow\infty}n^{1+\alpha}J_{n}=0$ for all $\alpha>0$;
assume that $(J_{n})$ is not identically zero. Note that
$\mathscr{L}_{\alpha}$ defines a family of self-adjoint, bounded linear
operators on $l^{2}(\mathbb{Z})$. In applications, our focus lies in specific
long-range interaction kernels defined by $J_{n}=|n|^{-(1+\alpha)}$, but
assume the general form unless otherwise specified.

A particular nonlinear model generated by (2.1) is given by

$i\dot{u}_{n}=\epsilon\mathscr{L}_{\alpha}u_{n}-|u_{n}|^{2}u_{n},\ u(0)=f\in
l^{2}(\mathbb{Z}),\ \epsilon>0,$ (2.2)

and the stationary model, by taking the ansatz $u_{n}(t)=e^{iwt}Q_{n}$ where
$w>0,\ Q_{n}\in\mathbb{R}$, is given by

$-wQ_{n}=\epsilon\mathscr{L}_{\alpha}Q_{n}-Q_{n}^{3}.$ (2.3)

Since the dynamics of interest is posed on $\mathbb{Z}^{d}$, it is natural to
use Fourier analysis. For $f\in l^{1}(\mathbb{Z}^{d})$, define
$\mathcal{F}[f](k)=\sum\limits_{n\in\mathbb{Z}^{d}}f_{n}e^{in\cdot k}$ for
$k\in\mathbb{T}^{d}=(-\pi,\pi]^{d}$, after which $\mathcal{F}$ extends
uniquely to an isomorphism
$l^{2}(\mathbb{Z}^{d})\xrightarrow[]{\cong}L^{2}(\mathbb{T}^{d})$ by the
standard density argument.

Under the flow (2.2), there are at least two conserved quantities given by

$N[u(t)]=\sum_{n}|u_{n}|^{2},\
E[u(t)]=\frac{\epsilon}{2}\langle\mathscr{L}_{\alpha}u_{n},u_{n}\rangle_{l^{2}}-\frac{1}{4}\sum_{n}|u_{n}|^{4},$
(2.4)

representing the particle number (or mass) and energy, respectively. The
kinetic energy terms corresponding to DNLS (1.3) and fDNLS (1.1) are given by
$\frac{\epsilon}{2}\sum\limits_{n\in\mathbb{Z}}|u_{n+1}-u_{n}|^{2}$ and
$\frac{\epsilon}{4}\sum\limits_{n,m:n\neq
m}\frac{|u_{n}-u_{m}|^{2}}{|n-m|^{1+\alpha}}$, respectively.

For $\alpha\in(0,2),\ x\in\mathbb{R}^{d}$, the singular integral
representation of the fractional Laplacian [43] is defined as

$(-\Delta)^{\frac{\alpha}{2}}u(x)=c(d,\alpha)\text{p.v.}\int\limits_{\mathbb{R}^{d}}\frac{u(x)-u(y)}{|x-y|^{d+\alpha}}dy:=\frac{2^{\alpha}\Gamma(\frac{\alpha}{2}+\frac{d}{2})}{\pi^{\frac{d}{2}}|\Gamma(-\frac{\alpha}{2})|}\text{p.v.}\int\limits_{\mathbb{R}^{d}}\frac{u(x)-u(y)}{|x-y|^{d+\alpha}}dy.$
(2.5)

In fact, there are many inequivalent definitions of the fractional Laplacian
on bounded domains, and hence a particular numerical discretization of
$(-\Delta)^{\frac{\alpha}{2}}$ needs to be defined since numerical simulations
depend on an appropriate spatial truncation. In Section 4 where the evolution
of $u_{n}(0)=A>0,\ -N\leq n\leq N$ is studied, the periodic boundary condition
is imposed whereas the zero exterior Dirichlet boundary condition is imposed
to simulate localized wave solutions in Section 5.

##  3 Global Well-posedness and Small Data Scattering.

By the contraction mapping argument and the embedding
$l^{p}(\mathbb{Z}^{d})\hookrightarrow l^{q}(\mathbb{Z}^{d})$ whenever $p\leq
q$, the well-posedness of (1.1), (1.3) is established. As long as the long-
range interaction is described by a self-adjoint operator and the nonlinear
interaction, by a local nonlinearity, the following well-posedness result is
proved similarly as [38, Proposition 4.1].

######  Proposition 3.1.

Let $L$ be a bounded, self-adjoint operator on $l^{2}(\mathbb{Z}^{d})$ and
$N:\mathbb{C}\rightarrow\mathbb{C}$ such that

$N(0)=0,\ |N(z_{1})-N(z_{2})|\leq
C\left(\max(|z_{1}|,|z_{2}|)\right)|z_{1}-z_{2}|,$

where $C:[0,\infty)\rightarrow[0,\infty)$ is increasing. Then the initial-
value problem

$i\dot{u}_{n}=Lu_{n}+N(u_{n}),\ u(0)=f\in l^{2}(\mathbb{Z}^{d}),$ (3.1)

is globally well-posed; for any $f\in l^{2}(\mathbb{Z}^{d})$, there exists a
unique solution $u\in C^{1}_{loc}(\mathbb{R};l^{2}(\mathbb{Z}^{d}))$ to (3.1)
such that $u(0)=f$ and the data-to-solution map $f\mapsto u$ is Lipschitz
continuous.

######  Proposition 3.2.

Assume the hypotheses of Proposition 3.1 hold. Let
$\\{L_{\alpha}\\}_{\alpha>0}$ and $L$ be bounded, self-adjoint operators on
$l^{2}(\mathbb{Z}^{d})$ such that
$L_{\alpha}\xrightarrow[\alpha\rightarrow\infty]{}L$ in norm. Let
$u^{(\alpha)},v\in C^{1}_{loc}(\mathbb{R};l^{2}(\mathbb{Z}^{d}))$ be the well-
posed solutions to (3.1) given by $L_{\alpha},L$, respectively, satisfying
$u^{(\alpha)}(0)=f^{(\alpha)}\in l^{2}(\mathbb{Z}^{d}),\ v(0)=g\in
l^{2}(\mathbb{Z}^{d})$. Assume
$\sup\limits_{\alpha>0}\|f^{(\alpha)}\|_{l^{2}}\leq M$. Then there exists
$C=C(M,\|g\|_{l^{2}})>0$ such that for all $t\geq 0$,

$\|u^{(\alpha)}(t)-v(t)\|_{l^{2}}\leq
e^{Ct}\Big{(}\|f^{(\alpha)}-g\|_{l^{2}}+t\|L_{\alpha}-L\|\cdot\|g\|_{l^{2}}\Big{)}.$
(3.2)

###### Proof.

For notational brevity, say $u=u^{(\alpha)},\ f=f^{(\alpha)}$. The well-
posedness of $u,v$ follows by Proposition 3.1. Setting $\phi=u-v$, we have

$i\dot{\phi}_{n}=L_{\alpha}\phi_{n}+(L_{\alpha}-L)v_{n}+N(u_{n})-N(v_{n}).$

By integrating, it follows that

$\phi_{n}(t)=e^{-itL_{\alpha}}\phi_{n}(0)-i\int_{0}^{t}e^{-i(t-t^{\prime})L_{\alpha}}\bigl{\\{}(L_{\alpha}-L)v_{n}(t^{\prime})+N(u_{n}(t^{\prime}))-N(v_{n}(t^{\prime}))\bigl{\\}}dt^{\prime},$

and by the triangle inequality, the unitarity of $e^{-itL_{\alpha}}$, the
conservation of particle numbers, and the embedding
$l^{2}(\mathbb{Z}^{d})\hookrightarrow l^{\infty}(\mathbb{Z}^{d})$, we have

$\|\phi(t)\|_{l^{2}}\leq\|\phi(0)\|_{l^{2}}+t\|L_{\alpha}-L\|\cdot\|g\|_{l^{2}}+C(\max(M,\|g\|_{l^{2}}))\int_{0}^{t}\|\phi(t^{\prime})\|_{l^{2}}dt^{\prime},$

where $C>0$ is the (local) Lipschitz constant of the nonlinearity $N$. The
proof follows from the Gronwall’s inequality. ∎

######  Remark 3.1.

In the context of long-range interaction and DNLS, the operators defined by

$L_{\alpha}f_{n}=\epsilon\sum\limits_{m\neq
n}\frac{f_{n}-f_{m}}{|n-m|^{d+\alpha}},\ L=-\epsilon\delta^{2},$

on $l^{2}(\mathbb{Z}^{d})$ are Fourier multipliers with the symbols given by

$\begin{split}\sigma_{\alpha}(k)&=2\epsilon\sum_{n\neq
0}\frac{\sin^{2}\left(\frac{n\cdot k}{2}\right)}{|n|^{d+\alpha}},\\\
\sigma(k)&=2\epsilon\sum_{|n|=1}\sin^{2}\left(\frac{n\cdot
k}{2}\right)=2d-\sum_{|n|=1}e^{in\cdot k},\end{split}$

respectively, for $k\in\mathbb{T}^{d}$ and
$|(z_{1},\dots,z_{d})|=\sum\limits_{j=1}^{d}|z_{j}|$. Then the norm-
convergence hypothesis of Proposition 3.2 is satisfied since

$|\sigma_{\alpha}(k)-\sigma(k)|=2\epsilon\left|\sum_{|n|\geq
2}\frac{\sin^{2}\left(\frac{nk}{2}\right)}{|n|^{d+\alpha}}\right|\leq\sum_{|n|\geq
2}\frac{2\epsilon}{|n|^{d+\alpha}}\lesssim_{d}\epsilon\int_{2}^{\infty}\frac{dr}{r^{1+\alpha}}=\frac{\epsilon}{\alpha
2^{\alpha}}\xrightarrow[\alpha\to\infty]{}0,$

uniformly in $k$. Hence the short-time dynamics of fDNLS is well-approximated
by that of the DNLS for large $\alpha$ on $\mathbb{Z}^{d}$.

The previous remark on the approximation of DNLS via fDNLS for large $\alpha$
need not hold for the long-time dynamics, and the exponential bound (3.2)
provides no coercive growth-in-time estimates. Moreover the emerging pattern
in Figure 3 with different wavenumbers corresponding to various $\alpha$ is a
numerical evidence that $u^{(\alpha)}\nrightarrow v$ in
$C(\mathbb{R};l^{2}(\mathbb{Z}^{d}))$ as $\alpha\rightarrow\infty$ where
$u^{(\alpha)},v$ are solutions in
$C^{1}_{loc}(\mathbb{R};l^{2}(\mathbb{Z}^{d}))$, that follow from Proposition
3.2, with $u^{(\alpha)}(0)=v(0)\in l^{2}(\mathbb{Z}^{d})\setminus\\{0\\}$. The
long-time dynamics of an extended Hamiltonian system exhibits a rich structure
featuring multi-breathers, transition into chaos, and small data scattering,
just to name a few, where such variety of features rises from conservation
laws in stark contrast to dissipative systems. The following proposition is an
example of small data scattering uniform in $\alpha$ for sufficiently high
nonlinearity. As a corollary, this proves the existence of strictly positive
excitation threshold of ground state solutions for fDNLS. Note that [44]
showed that $p\geq 5$ for DNLS is the sufficient and necessary condition for
the positive excitation threshold. By the following result, we leave it as a
future work to investigate the case $5\leq p<7$ for fDNLS.

######  Proposition 3.3.

Let $\mathscr{L}_{\infty}:=\mathscr{L}=-\delta^{2}$ and $\mathscr{L}_{\alpha}$
be defined by $J_{n}=|n|^{-(1+\alpha)}$, and define $U(t)=e^{-it\mathscr{L}}$
and $U^{(\alpha)}(t)=e^{-it\mathscr{L}_{\alpha}}$. By Proposition 3.1, let
$u^{(\alpha)},v:=u^{(\infty)}\in C^{1}_{loc}(\mathbb{R};l^{2}(\mathbb{Z}))$ be
the well-posed solutions to

$\begin{split}i\dot{u}_{n}^{(\alpha)}&=\mathscr{L}_{\alpha}u_{n}^{(\alpha)}+\mu|u_{n}^{(\alpha)}|^{p-1}u_{n}^{(\alpha)},\
u^{(\alpha)}(0)=f\in l^{2}(\mathbb{Z}),\\\
i\dot{v}_{n}&=\mathscr{L}v_{n}+\mu|v_{n}|^{p-1}v_{n},\ v(0)=f\in
l^{2}(\mathbb{Z}),\end{split}$

where $\mu=\pm 1,\ p\geq 7$. Then there exists $\tilde{\alpha}>0$ such that
for all $\tilde{\alpha}<\alpha<\infty$, the data-to-asymptotic-state map is
well-defined as an isometric homeomorphism in a small neighborhood of
$l^{2}(\mathbb{Z})$ uniformly in $\tilde{\alpha}<\alpha\leq\infty$; more
precisely, there exists $\delta_{1}(\tilde{\alpha})>0$ such that whenever
$\|f\|_{l^{2}}<\delta_{1}$, there exists unique $u_{+}^{(\alpha)}\in
l^{2}(\mathbb{Z})$ such that

$\|u^{(\alpha)}(t)-U^{(\alpha)}(t)u_{+}^{(\alpha)}\|_{l^{2}}\xrightarrow[t\rightarrow\infty]{}0.$
(3.3)

Furthermore there exists $\delta_{2}(\tilde{\alpha})>0$ such that whenever
$0<\|f\|_{l^{\frac{5}{4}}}<\delta_{2}$, we have

$\|u^{(\alpha)}(t)\|_{l^{5}}\leq
C(\tilde{\alpha})(1+|t|)^{-\frac{1}{5}}\|f\|_{l^{\frac{5}{4}}},$ (3.4)

and

$\varlimsup_{\alpha\rightarrow\infty}\sup_{t\in[0,\infty)}\|u^{(\alpha)}(t)-v(t)\|_{l^{2}}>0.$
(3.5)

######  Lemma 3.1.

There exists $\alpha_{0}^{*}>0$ such that for all $\alpha>\alpha_{0}^{*}$,
there exists $t_{\alpha}>2^{\alpha}$ and

$\|(U^{(\alpha)}(t_{\alpha})-U(t_{\alpha}))f\|_{l^{2}}\geq c>0,$

for all $f\in l^{2}(\mathbb{Z})\setminus\\{0\\}$ where $c>0$ is independent of
$\alpha$ and $f$.

###### Proof.

By the Plancherel Theorem and the reverse Hölder inequality,

$\displaystyle\|(U^{(\alpha)}(t/4)-U(t/4))f\|_{l^{2}}^{2}$
$\displaystyle=\frac{1}{2\pi}\bigintsss_{-\pi}^{\pi}\left|\exp\left(-it\sum_{m=1}^{\infty}\frac{\sin^{2}\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right)-\exp\left(-it\sin^{2}\left(\frac{k}{2}\right)\right)\right|^{2}\cdot|\widehat{f}(k)|^{2}dk$
$\displaystyle\gtrsim\|\widehat{f}\|_{L^{1/3}}^{2}\left(\bigintsss_{-\pi}^{\pi}\left|\exp\left(-it\sum_{m=2}^{\infty}\frac{\sin^{2}\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right)-1\right|^{-\frac{2}{5}}dk\right)^{-5}$
$\displaystyle\gtrsim\left(\bigintsss_{0}^{\pi}\left(1-\cos\left(t\sum\limits_{m=2}^{\infty}\frac{\sin^{2}\left(\frac{mk}{2}\right)}{m^{1+\alpha}}\right)\right)^{-\frac{1}{5}}dk\right)^{-5},$
(3.6)

since $0<\|\widehat{f}\|_{L^{1/3}}\lesssim\|\widehat{f}\|_{L^{2}}<\infty$. For
the convenience of notation, let

$X(k)=t\sum\limits_{m=2}^{\infty}\frac{\sin^{2}\left(\frac{mk}{2}\right)}{m^{1+\alpha}}.$

Then the integral above on $k\in[0,\pi]$ is split into

$\int_{0}^{\pi}(1-\cos X)^{-\frac{1}{5}}dk=\int_{\\{1-\cos X<c\\}}(1-\cos
X)^{-\frac{1}{5}}dk+\int_{\\{1-\cos X\geq c\\}}(1-\cos
X)^{-\frac{1}{5}}dk=:I+II,$

where $c>0$ is sufficiently small. Then $II$ is $O(1)$ depending only on $c$,
and therefore it suffices to estimate $I$.

First consider $0\leq X<X_{0}$ where $X_{0}$ is the smallest positive real
such that $1-\cos X_{0}=c$, and suppose $t=2^{1+\alpha}(2\pi+X_{0})$. Then

$0\leq\sin^{2}k<\frac{X_{0}}{2\pi+X_{0}}-2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}},$ (3.7)

and $k\in[0,\pi]$ that satisfies (3.7) are near $k=0,\pi$ since the series
term is negligible for all $\alpha>0$ sufficiently large due to the uniform
estimate

$2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}<2\left(\frac{1}{3}+\frac{1}{\alpha}\right)\left(\frac{2}{3}\right)^{\alpha},$
(3.8)

that follows from majorizing the series into an appropriate integral. The
arguments for the estimation near $k=0,\pi$ are similar, and therefore assume
$k$ is near $0$. Note that (3.7) is satisfied for $0\leq k<k_{0}$ where
$k_{0}$ is the smallest positive real root of

$\sin^{2}(k_{0})=\frac{X_{0}}{2\pi+X_{0}}-2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk_{0}}{2})}{m^{1+\alpha}}.$

Then for all $\alpha>0$ sufficiently large depending on $X_{0}$, we have
$k_{0}<k_{0}^{*}$ where $k_{0}^{*}$ is the smallest positive real root of
$\sin^{2}(k_{0}^{*})=\frac{X_{0}}{2\pi+X_{0}}$. By the Taylor expansion,

$k_{0}<k_{0}^{*}\lesssim X_{0}^{\frac{1}{2}},$

and

$\begin{split}\int_{\\{0\leq X<X_{0}\\}}(1-\cos
X)^{-\frac{1}{5}}dk&\lesssim\int_{\\{0\leq
X<X_{0}\\}}X^{-\frac{2}{5}}dk\lesssim\int_{\\{0\leq
X<X_{0}\\}}(2\pi+X_{0})^{-\frac{2}{5}}\sin^{-\frac{4}{5}}(k)dk\\\
&\lesssim\int_{0}^{k_{0}}k^{-\frac{4}{5}}dk\lesssim
X_{0}^{\frac{1}{10}},\end{split}$ (3.9)

where the implicit constants depend only on $c,X_{0}$.

Now consider $|X-2\pi|<X_{0}$. Since the analysis for $0<X-2\pi<X_{0}$ and
$-X_{0}<X-2\pi<0$ are similar, we focus on the former. Then

$\displaystyle
2\pi<2^{1+\alpha}(2\pi+X_{0})\left(\frac{\sin^{2}k}{2^{1+\alpha}}+\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}\right)$
$\displaystyle<2\pi+X_{0}$
$\displaystyle\frac{2\pi}{2\pi+X_{0}}-2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}$
$\displaystyle<\sin^{2}k<1-2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}<1.$

Define $k_{1}(\alpha),k_{1}^{*}\in(0,\frac{\pi}{2})$ such that

$\begin{split}\sin^{2}k_{1}&=\frac{2\pi}{2\pi+X_{0}}-2^{1+\alpha}\sum_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}\\\
\sin^{2}k_{1}^{*}&=\frac{2\pi}{2\pi+X_{0}}.\end{split}$

The Taylor expansion of $\cos X$ near $X=2\pi$ yields

$\begin{split}\int_{\\{0<X-2\pi<X_{0}\\}}(1-\cos
X)^{-\frac{1}{5}}dk&\lesssim\int_{E}|X-2\pi|^{-\frac{2}{5}}dk\\\
&=\int_{k_{1}}^{\frac{\pi}{2}}|X-2\pi|^{-\frac{2}{5}}dk+\int_{\frac{\pi}{2}}^{\pi-
k_{1}}|X-2\pi|^{-\frac{2}{5}}dk=:III+IV\end{split}$

where

$E=\\{k\in[0,\pi]:\frac{2\pi}{2\pi+X_{0}}-2^{1+\alpha}\sum\limits_{m\geq
3}\frac{\sin^{2}(\frac{mk}{2})}{m^{1+\alpha}}<\sin^{2}k<1\\}=(k_{1},\pi-
k_{1}).$

The term $IV$ can be estimated similarly as $III$, and therefore the work for
$III$ is shown. For sufficiently large $\alpha>0$, the Taylor expansion of
$X(k)$ near $k=k_{1}$ yields

$\begin{split}X(k)&=X(k_{1})+X^{\prime}(k_{1})(k-k_{1})+O(|k-k_{1}|^{2})\\\
&=2\pi+2^{\alpha}(2\pi+X_{0})\sum_{m\geq
2}\frac{\sin(mk_{1})}{m^{\alpha}}(k-k_{1})+O(|k-k_{1}|^{2}).\end{split}$

By arguing as (3.8), the coefficient of $k-k_{1}$ is bounded above by a
constant independent of $\alpha$. Conversely, since $k_{1}<k_{1}^{*}$ and
$k_{1}\xrightarrow[\alpha\rightarrow\infty]{}k_{1}^{*}$, we have
$2k_{1}\in(2k_{1}^{*}-\delta,2k_{1}^{*})$ for some $\delta>0$ and
$\sin(2k_{1})>0$. The lower bound is given by

$2^{\alpha}\sum_{m\geq
2}\frac{\sin(mk_{1})}{m^{\alpha}}\geq\sin(2k_{1})-\sum_{m\geq
3}\left(\frac{2}{m}\right)^{\alpha}>\sin(2k_{1}^{*})-\sum_{m\geq
3}\left(\frac{2}{m}\right)^{\alpha}\gtrsim 1,$

where the implicit constant is independent of $\alpha$. Hence the linear
coefficient is bounded above and below by a positive constant independent of
$\alpha$, and similarly, the quadratic coefficient is bounded above uniformly
in $\alpha$. Then

$III\lesssim\int_{k_{1}}^{\frac{\pi}{2}}|k-k_{1}|^{-\frac{2}{5}}dk=O(1).$
(3.10)

Hence by (3.9), (3.10), and $t=2^{1+\alpha}(2\pi+X_{0})$, there exists $M>0$
independent of $\alpha$ such that

$\int_{\\{1-\cos X<c\\}}(1-\cos
X)^{-\frac{1}{5}}dk\leq\int_{\\{0<X<X_{0}\\}}(1-\cos
X)^{-\frac{1}{5}}dk+\int_{\\{|X-2\pi|<X_{0}\\}}(1-\cos X)^{-\frac{1}{5}}dk\leq
M,$

and this shows the lower bound of (3). ∎

###### Proof of Proposition 3.3.

The proof strategy is as follows. As [45] in their analysis of DNLS, derive
linear dispersive estimates (Strichartz estimates) uniform in $\alpha$ and
apply them to establish the small data scattering (3.3). By the fixed point
argument in the Strichartz space, derive (3.4). Lastly estimate
$\|U^{(\alpha)}(t)-U(t)\|$ for $t\gg 1$ to obtain (3.5).

By the discrete Fourier transform,
$U^{(\alpha)}(t)f_{n}=\left(K_{t,\alpha}\ast f\right)_{n}$ where

$K_{t,\alpha}(n):=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-it\phi(k)}dk,\
\phi(k)=4\sum_{m=1}^{\infty}\frac{\sin^{2}\left(\frac{mk}{2}\right)}{m^{1+\alpha}}+\frac{nk}{t}.$

By the Dominated Convergence Theorem,

$\phi^{\prime\prime}(k)\xrightarrow[\alpha\rightarrow\infty]{}2\cos(k),\
\phi^{\prime\prime\prime}(k)\xrightarrow[\alpha\rightarrow\infty]{}-2\sin(k),$

and therefore
$\max\limits_{k\in[-\pi,\pi]}\left(|\phi^{\prime\prime}(k)|,|\phi^{\prime\prime\prime}(k)|\right)\gtrsim
1$ for $\alpha$ sufficiently large. In fact, the bound holds as long as
$\alpha\geq\alpha_{0}>3$ under which the term-by-term differentiation of
$\phi^{\prime\prime\prime}$ holds uniformly. By the Van der Corput Lemma [46,
p.334], this implies

$\sup_{n\in\mathbb{Z}}|K_{t,\alpha}(n)|\lesssim_{\alpha_{0}}(1+|t|)^{-\frac{1}{3}},$

where the inhomogeneous bound follows from applying the triangle inequality to
$K_{t,\alpha}$. As a corollary, we obtain

$\|u^{(\alpha)}(t)\|_{l^{s}}\lesssim(1+|t|)^{-\frac{s-2}{3s}}\|f\|_{l^{s^{\prime}}},$
(3.11)

by the complex interpolation of linear operators where $s\in[2,\infty]$ and
$s^{\prime}:=\frac{s}{s-1}$. Let $(q,r)\in[2,\infty]^{2}$ satisfy
$\frac{3}{q}+\frac{1}{r}=\frac{1}{2}$, and say that such pair is DNLS-
admissible. A further corollary [47, Theorem 1.2] implies

$\begin{split}\|U^{(\alpha)}(t)f\|_{L^{q}_{t}l^{r}(I\times\mathbb{Z})}&\lesssim\|f\|_{l^{2}},\\\
\left|\left|\int_{0}^{\infty}U^{(\alpha)}(-t^{\prime})N(t^{\prime})dt^{\prime}\right|\right|_{l^{2}}&\lesssim\|N\|_{L^{\tilde{q}^{\prime}}_{t}l^{\tilde{r}^{\prime}}(\mathbb{R}\times\mathbb{Z})},\\\
\left|\left|\int_{0}^{t}U^{(\alpha)}(t-t^{\prime})N(t^{\prime})dt^{\prime}\right|\right|_{L^{q}_{t}l^{r}(I\times\mathbb{Z})}&\lesssim\|N\|_{L^{\tilde{q}^{\prime}}_{t}l^{\tilde{r}^{\prime}}(I\times\mathbb{Z})},\end{split}$
(3.12)

where $(\tilde{q},\tilde{r})$ is DNLS-admissible and $I\subseteq\mathbb{R}$.
Note that the implicit constants depend on the DNLS-admissible pairs and
$\alpha_{0}$.

The solution in the integral form satisfies

$u^{(\alpha)}(t)=U^{(\alpha)}(t)f-i\mu\int_{0}^{t}U^{(\alpha)}(t-t^{\prime})\left(|u^{(\alpha)}(t^{\prime})|^{p-1}u^{(\alpha)}(t^{\prime})\right)dt^{\prime}.$
(3.13)

Let $X=C(\mathbb{R};l^{2}(\mathbb{Z}))\cap
L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}(\mathbb{R}\times\mathbb{Z})$ and let
$\Gamma u$ be the RHS of (3.13). By the fixed point argument, $\Gamma$ has a
unique fixed point in $X$ if $\|f\|_{l^{2}}=O(1)$ where the bound depends only
on $\alpha_{0}$ since the Strichartz constant of (3.11) can be chosen
uniformly. Then

$\begin{split}\left|\left|\int_{t_{1}}^{t_{2}}U^{(\alpha)}(-t^{\prime})\left(|u^{(\alpha)}(t^{\prime})|^{p-1}u^{(\alpha)}(t^{\prime})\right)dt^{\prime}\right|\right|_{l^{2}}\lesssim\|u^{(\alpha)}\|_{L^{\frac{6p}{5}}_{t}l^{p}([t_{1},t_{2}]\times\mathbb{Z})}^{p}\leq\|u^{(\alpha)}\|_{L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}([t_{1},t_{2}]\times\mathbb{Z})}^{p}\xrightarrow[t_{1},t_{2}\rightarrow\infty]{}0,\end{split}$

where the first inequality is by (3.12), the second inequality by the Hölder’s
inequality and $p\geq 7$, and the last limit by
$\|u^{(\alpha)}\|_{L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}([t_{1},t_{2}]\times\mathbb{Z})}\leq\|u^{(\alpha)}\|_{L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}(\mathbb{R}\times\mathbb{Z})}\lesssim\|f\|_{l^{2}}<\infty$
with $t_{1}\leq t_{2}$ without loss of generality. This shows the existence of
$u_{+}^{(\alpha)}=\lim\limits_{t\rightarrow\infty}U^{(\alpha)}(-t)u^{(\alpha)}(t)\in
l^{2}(\mathbb{Z})$ and by (3.13),

$u^{(\alpha)}(t)=U^{(\alpha)}(t)u^{(\alpha)}_{+}+i\mu\int_{t}^{\infty}U^{(\alpha)}(t-t^{\prime})\left(|u^{(\alpha)}(t^{\prime})|^{p-1}u^{(\alpha)}(t^{\prime})\right)dt^{\prime}.$
(3.14)

To show that the inverse of data-to-asymptotic-state (wave operator) is
continuous, we argue by the fixed point theorem on
$X_{T}=C([T,\infty);l^{2}(\mathbb{Z}))\cap
L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}([T,\infty)\times\mathbb{Z})$ where
$\Gamma^{\prime}u$ is defined as the RHS of (3.14) and $T>0$ is chosen such
that
$\|U^{(\alpha)}(t)u_{+}^{(\alpha)}\|_{L^{\frac{6p}{5}}_{t}l^{\frac{2p}{p-5}}([T,\infty)\times\mathbb{Z})}$
is small; this proof is standard in the literature and can be consulted in,
say, [48, Chapter 3]. Let $u^{(\alpha)}\in X_{T}$ be the unique fixed point.
Flowing $u^{(\alpha)}$ backward in time, one obtains $f:=u^{(\alpha)}(0)$.
Hence the desired homeomorphism follows where the isometry is due to the
conservation of $l^{2}$ norm.

Here it is shown that the asymptotic-state map $\alpha\mapsto
u_{+}^{(\alpha)}\in l^{2}(\mathbb{Z})$, with a fixed initial datum, is
continuous. From (3.14),

$u_{+}^{(\alpha)}=f-i\mu\int_{0}^{\infty}U^{(\alpha)}(-t^{\prime})\left(|u^{(\alpha)}(t^{\prime})|^{p-1}u^{(\alpha)}(t^{\prime})\right)dt^{\prime},$

and hence it suffices to show the continuity of the integral. For a fixed
$t^{\prime}\in(0,\infty)$, the integrand converges to
$U(-t^{\prime})\left(|v(t^{\prime})|^{p-1}v(t^{\prime})\right)\in
l^{2}(\mathbb{Z})$ as $\alpha\rightarrow\infty$; the argument of Proposition
3.2 may be modified to show pointwise (in $t^{\prime}$) continuity in
$\alpha$. One may follow the argument of [45, Theorem 4] verbatim to derive
the decay estimate (3.4). The fDNLS obeys the same Strichartz estimates as
DNLS, and therefore the nonlinear decay for
$\|f\|_{l^{\frac{5}{4}}}<\delta_{2}$, for some $\delta_{2}(\alpha_{0})>0$,
follows where $\alpha\geq\alpha_{0}$ guarantees the uniformity in $\alpha$.
Applying (3.4) and the unitarity, we have

$\int_{0}^{\infty}\||u^{(\alpha)}|^{p-1}u^{(\alpha)}\|_{l^{2}}dt^{\prime}=\int_{0}^{\infty}\|u^{(\alpha)}\|_{l^{2p}}^{p}dt^{\prime}\leq\int_{0}^{\infty}\|u^{(\alpha)}\|_{l^{5}}^{p}dt^{\prime}\lesssim_{\alpha_{0}}\int_{0}^{\infty}(1+|t^{\prime}|)^{-\frac{p}{5}}\|f\|_{l^{\frac{5}{4}}}^{p}dt^{\prime}<\infty,$

and therefore $\alpha\mapsto u_{+}^{\alpha}$ is continuous by the Dominated
Convergence Theorem.

Lastly (3.5) is proved by contradiction. Let
$0<\|f\|_{l^{\frac{5}{4}}}<\delta_{2}$. Let $v_{+}\in
l^{2}(\mathbb{Z})\setminus\\{0\\}$ be the asymptotic state corresponding to
$f$ under the DNLS flow. For any $\epsilon>0$, there exists $\alpha_{1}>0$
such that $\|u^{(\alpha)}(t)-v(t)\|_{l^{2}}<\epsilon$ for all $t\in[0,\infty)$
for $\alpha\geq\alpha_{1}$. By continuity, we have
$\|u_{+}^{(\alpha)}-v_{+}\|_{l^{2}}<\epsilon$ for $\alpha\geq\alpha_{2}>0$,
and let $\alpha>\max(\alpha_{0}^{*},\alpha_{0},\alpha_{1},\alpha_{2})$ where
$\alpha_{0}^{*}>0$ is from Lemma 3.1. The scattering results imply that there
exists $T>0$ such that if $t\geq T$, then

$\|u^{(\alpha)}(t)-U^{(\alpha)}(t)u_{+}^{(\alpha)}\|_{l^{2}},\|v(t)-U(t)v_{+}\|_{l^{2}}<\epsilon.$

The triangle inequality yields

$\|(U^{(\alpha)}(t)-U(t))v_{+}\|_{l^{2}}-\|u_{+}^{(\alpha)}-v_{+}\|_{l^{2}}-\|u^{(\alpha)}(t)-U^{(\alpha)}(t)u_{+}^{(\alpha)}\|_{l^{2}}-\|v(t)-U(t)v_{+}\|_{l^{2}}\leq\|u^{(\alpha)}(t)-v(t)\|_{l^{2}},$

and therefore

$\sup_{t\in[T,\infty)}\|(U^{(\alpha)}(t)-U(t))v_{+}\|_{l^{2}}<4\epsilon,$

a contradition by Lemma 3.1. ∎

##  4 Modulational Instability of CW Solutions.

Here we focus on periodic solutions in the direction of propagation (in $t$)
that do not decay in space. Hence there is no localization, initially, but MI
triggers the formation of coherent states, a process that requires a further
study. For $A>0$, define $u_{n}^{cw}=Ae^{iA^{2}t}$ as the continuous wave (CW)
solution and consider $u_{n}=(A+\mu v_{n}(t))e^{iA^{2}t}$ where
$v_{n}(t)\in\mathbb{C}$ and $|\mu|\ll 1$.

######  Proposition 4.1.

The CW solution is linearly unstable if and only if $\Phi:=$
$\epsilon\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))-A^{2}<0$. The zero set
$\\{\Phi=0\\}$ is the graph of $A=A(k,\alpha,\epsilon)$ where $A$ is analytic
on $\mathbb{T}\setminus\\{0\\}\times(0,\infty)\times(0,\infty)$. Furthermore
there exists $C=C(\alpha)>0$ such that

$A\sim\left(\epsilon C\delta(k)\right)^{\frac{1}{2}}\text{as}\ k\rightarrow
0,$

where

$\delta(k)=\begin{cases}|k|^{\alpha},&\alpha\in(0,2),\\\
(-\log|k|)|k|^{2},&\alpha=2,\\\ |k|^{2},&\alpha\in(2,\infty].\end{cases}$

(a) $\alpha=0.5$

(b) $\alpha=50$

(c) $A=0.5$

(d) $A=1.5$

Figure 1: Instability regions (blue) for $J_{n}=n^{-(1+\alpha)}$ in the
$(k,A)$-plane and $(k,\alpha)$-plane where $k\in[-\pi,\pi]$ and $\epsilon=1$.

###### Proof.

The $\mathcal{O}(\mu)$ expansion yields

$i\partial_{t}v_{n}=\epsilon\mathscr{L}_{\alpha}v_{n}-A^{2}(v_{n}+\overline{v_{n}}),$
(4.1)

and setting $v_{n}=f_{n}+ig_{n}:=\Re v_{n}+i\Im v_{n}$, (4.1) is equivalent to

$\frac{d}{dt}\begin{pmatrix}f_{n}\\\
g_{n}\end{pmatrix}=\begin{pmatrix}0&\epsilon\mathscr{L}_{\alpha}\\\
-\epsilon\mathscr{L}_{\alpha}+2A^{2}&0\end{pmatrix}\begin{pmatrix}f_{n}\\\
g_{n}\end{pmatrix}.$ (4.2)

Taking the Fourier transform $F(k,t)=\mathcal{F}[f_{n}(t)](k),\
G(k,t)=\mathcal{F}[g_{n}(t)](k)$ and the ansatz

$\begin{pmatrix}F\\\ G\end{pmatrix}=\begin{pmatrix}P(k)\\\
Q(k)\end{pmatrix}e^{-i\Omega t},$

(4.2) reduces to

$\begin{pmatrix}i\Omega&2\epsilon\sum\limits_{m=1}J_{m}(1-\cos km)\\\
-\left(2\epsilon\sum\limits_{m=1}J_{m}(1-\cos
km)-2A^{2}\right)&i\Omega\end{pmatrix}\begin{pmatrix}P\\\
Q\end{pmatrix}=\begin{pmatrix}0\\\ 0\end{pmatrix},$ (4.3)

where a nontrivial solution to (4.3) exists if and only if $\Omega$ satisfies
the dispersion relation

$\Omega^{2}=4\left(\epsilon\sum_{m=1}^{\infty}J_{m}(1-\cos(km))\right)\left(\epsilon\sum_{m=1}^{\infty}J_{m}(1-\cos(km))-A^{2}\right).$
(4.4)

The frequency $\Omega$ is purely imaginary if and only if
$\epsilon\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))<A^{2}$, leading to
modulational instability.

Suppose $\Phi=0$ and note that $\partial_{A}\Phi=-2A$. Since $A=0$ if and only
if $k=0$, $A(k,\alpha,\epsilon)$ is analytic whenever $k\neq 0$ by the
analytic Implicit Function Theorem. In the neighborhood of $k=0$, [38, Lemma
A.1] implies that there exists $C(\alpha)>0$ such that

$\frac{A^{2}}{\epsilon C\delta(k)}=\frac{A^{2}}{\epsilon
C\delta(k)}\cdot\frac{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))}{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))}=\frac{\sum\limits_{m=1}^{\infty}J_{m}(1-\cos(km))}{C\delta(k)}\xrightarrow[k\rightarrow
0]{}1.$

∎

When $J_{n}=n^{-(1+\alpha)}$, Figure 1 shows the region of instability. Note
the non-analytic dependence of $A,\alpha$ on $k$ near the zero wavenumber. The
instability region expands for bigger values of $A$, which is consistent with
the $-A^{2}$ term in (4.4).

To compute the value(s) of $k\in[-\pi,\pi]$ that maximizes the exponential
gain of the modulational instability, the RHS of (4.4) needs to be minimized
under the condition $\Phi<0$; that a minimum exists follows from the Extreme
Value Theorem since $\Omega(k)^{2}$ is continuous by the definition of
$(J_{n})_{n=1}^{\infty}$. An explicit computation that minimizes
$\Omega(k)^{2}$ is shown for a specific interaction kernel.

Figure 2: $k_{max}$ as a function of $\alpha$ when $J_{n}=n^{-(1+\alpha)}$.
The values $A=1,\epsilon=1$ are used to generate this plot. For such
parameters, $A_{0}\geq 2>A$ where $A_{0}$ is defined in Corollary 4.1, and
therefore (4.5) is used to solve $k_{max}$ in $\alpha$. Observe that small
$\alpha$ gives small $k_{max}$, manifested in the top left plot of Figure 3.
That $k_{max}$ has an upper bound is numerically verified in Figure 3 for
increasing values of $\alpha$.

######  Corollary 4.1.

For $\alpha,\epsilon>0$, let $J_{n}=n^{-(1+\alpha)},\
\tilde{w}(k)=\sum\limits_{m=1}^{\infty}\frac{1-\cos km}{m^{1+\alpha}},$ and
$A_{0}:=\left(4\epsilon(1-2^{-(1+\alpha)})\zeta(1+\alpha)\right)^{\frac{1}{2}}$.
Then for any $A\geq A_{0}$,

$\min_{k\in[-\pi,\pi]}\Omega(k)^{2}=\Omega(\pm\pi)^{2}=-A_{0}^{2}(2A^{2}-A_{0}^{2}),$

and if $0<A<A_{0}$, then there exists a unique $k_{max}\in(0,\pi)$ that
minimizes $\Omega^{2}$ where $k_{max}$ satisfies

$\tilde{w}(k_{max})=\frac{A^{2}}{2\epsilon}$ (4.5)

and

$\min_{k\in[-\pi,\pi]}\Omega(k)^{2}=\Omega(\pm k_{max})^{2}=-A^{4}.$

###### Proof.

Since $\Omega(k)^{2}$ is even and $\Omega(0)=0$, let $k\in(0,\pi]$ without
loss of generality. By direct computation, it can be shown that
$\tilde{w}(0)=0,\ \tilde{w}\in C^{1}(0,\pi)$, and $\tilde{w}$ is increasing on
$(0,\pi)$. From the derivative, we have

$\begin{split}\frac{d}{dk}\left(\Omega(k)^{2}\right)&=4\epsilon\frac{d}{dk}\tilde{w}(k)\left(2\epsilon\overset{\sim}{w}(k)-A^{2}\right),\end{split}$

and therefore $\Omega^{2}$ is decreasing if and only if
$2\epsilon\tilde{w}(k)-A^{2}\leq 0$. Since

$\max_{k\in[-\pi,\pi]}\tilde{w}(k)=\tilde{w}(\pm\pi)=\sum_{m\geq 1,\ m\
\text{odd}}\frac{2}{m^{1+\alpha}}=2(1-2^{-(1+\alpha)})\zeta(1+\alpha),$ (4.6)

if $A\geq A_{0}$, then $2\epsilon\tilde{w}(k)-A^{2}\leq 0$ for all
$k\in(0,\pi]$. Hence $\Omega^{2}$ is minimized at $k=\pm\pi$ and the value of
$\Omega(\pm\pi)^{2}$ follows from (4.6). If $0<A<A_{0}$, then there exists a
unique $k_{max}\in(0,\pi)$ satisfying
$\tilde{w}(k_{max})=\frac{A^{2}}{2\epsilon}$ by the Intermediate Value Theorem
and

$\Omega(k_{max})^{2}=4\epsilon\tilde{w}(k_{max})\left(\epsilon\tilde{w}(k_{max})-A^{2}\right)=-A^{4}.$

∎

What dynamics emerges after the initial linear growth due to MI is a question
that has been posed since the studies of the FPUT chain. Based on many studies
in discrete and continuous models, it is expected that the discrete
modulational instability is the potential mechanism for the formation of
nonlinear localized modes, such as discrete breathers and envelope solitons.
What is not known is which type emerges in the fNLSE and what is the role of
the parameter $\alpha$. Figure 3 shows a sample of emerging patterns triggered
by numerical noise. While at this time we will not examine these emerging
patterns, it is clear that $\alpha$-dependent coherent and robust patterns
develop.

Figure 3: Emerging patterns triggered by numerical noise. Background intensity
is the same in all cases, $A=1$ . The nonlinear state should be quasiperiodic,
with an $\alpha$ dependence on the separation between peaks, likely to be
close to the most unstable MI wavenumber $k_{max}$.

##  5 Asymptotic Construction of Stationary Solutions.

An important class of solutions for nonlinear coupled oscillators, are
localized states carrying finite energy. It is expected as shown in the
previous section, that stable localized modes are indeed coherent structures
the system selects to allocate energy. Generically, there are no known
analytic solutions, so typically one relies on numerical or asymptotic
approximations. In order to highlight similarities and differences between the
local and nonlocal coupling, we include known results from the DNLS. In one
dimensional arrays, solutions are onsite if their intensity is centered at a
grid site, and offsite if centered between two consecutive grid sites. To
avoid confusions in notation, we denote $q_{n}$ by the onsite solution (mode
$A$) satisfying $q_{n}=q_{-n}$ and $g_{n}$, the offsite solution (mode $B$)
satisfying $g_{n}=g_{-(n-1)}$.

In Section 5.1, the stationary equation of DNLS is recasted as a discrete
recurrence relation similar to [12, Chapter 11]. In Section 5.3 where the
interaction kernel is of long range, the recurrence relation does not define a
Markov chain; instead the forward map depends on all previous iterates and the
phase space of solutions is infinite dimensional. Our asymptotic method has a
disadvantage that the orbit of the recurrence relation does not give an exact
solution, but has the advantage that it yields a localized sequence that
satisfies the stationary equation in an appropriate limit.

###  5.1 Localized Solutions for DNLS.

We revisit the derivation of stationary solutions in [22, Section II] while
relaxing the hypotheses that $\mathrm{supp}\\{q_{n}\\}$ and
$\mathrm{supp}\\{g_{n}\\}$ are compact. To motivate Proposition 5.1, consider
(2.3) with $J_{n}=1$, if $n=1$, and $J_{n}=0$ otherwise (nearest-neighbor
coupling). Assume $q_{n}\geq 0$ and $q_{n+1}\ll q_{n}$ for all $n\geq 0$; see
Proposition 5.1 for a precise statement. Neglecting $q_{\pm 1}$ yields

$wq_{0}=-2\epsilon q_{0}+q_{0}^{3},$

and therefore $q_{0}=\sqrt{w+2\epsilon}$. For $n\geq 1$, further assume
$q_{n}\ll 1$, after which (2.3) neglecting $q_{n+1}$ and the nonlinear term
yields

$wq_{n}=\epsilon(q_{n-1}-2q_{n}),$

and therefore $q_{n}=\frac{\epsilon}{(w+2\epsilon)}q_{n-1}$. By this method, a
forward map is defined where $q_{n}$ depends on $q_{n-1}$. A similar
computation can be done with $\\{g_{n}\\}$. This explicit construction yields
a sequence in $l^{2}(\mathbb{Z})$ that satisfies (1.3) asymptotically as
$\rho:=\frac{\epsilon}{w}\rightarrow 0$.

######  Proposition 5.1.

For $w,\epsilon>0$, let $\\{q_{n}\\}$ satisfy

$q_{n}=\left(\frac{\epsilon}{w+2\epsilon}\right)^{|n|}\sqrt{w+2\epsilon},\
n\in\mathbb{Z}.$ (5.1)

Then $q_{n+1}=o(q_{n})$ as $\rho:=\frac{\epsilon}{w}\rightarrow 0$. Moreover
$\\{q_{n}\\}$ satisfies (2.3), with the nearest-neighbor coupling,
asymptotically as $\rho\rightarrow 0$ uniformly in $n\in\mathbb{Z}$, or more
precisely,

$\lim_{\rho\rightarrow
0}\sup_{n\in\mathbb{Z}}\left|\frac{wq_{n}}{\epsilon(q_{n+1}+q_{n-1}-2q_{n})+q_{n}^{3}}-1\right|=0.$
(5.2)

###### Proof.

By symmetry, let $n\geq 0$. An explicit computation yields,

$\frac{q_{n+1}}{q_{n}}=\frac{\rho}{1+2\rho}\xrightarrow[\rho\rightarrow
0]{}0,$

and

$\frac{wq_{n}}{\epsilon(q_{n+1}+q_{n-1}-2q_{n})+q_{n}^{3}}=\begin{cases}\frac{1}{1+\rho^{2}(1+2\rho)^{-1}+\rho^{2n}(1+2\rho)^{1-2n}},\
n>0,\\\ \frac{1}{1+2\rho^{2}(1+2\rho)^{-1}},\ n=0.\end{cases}$ (5.3)

Taking the limit of the RHS of (5.3) as $\rho\rightarrow 0$, (5.2) is shown. ∎

######  Remark 5.1.

For $n\geq 1$, define $\\{g_{n}\\}$ as

$g_{n}=\left(\frac{\epsilon}{w+2\epsilon}\right)^{n-1}\sqrt{w+\epsilon},$
(5.4)

and extend to $\mathbb{Z}$ by $g_{n}=g_{-(n-1)}$. Then $g_{n+1}=o(g_{n})$ as
$\rho\rightarrow 0$ for all $n\geq 1$ and satisfies (5.2) with $q_{n}$
replaced by $g_{n}$.

###  5.2 Peierls-Nabarro Barrier for DNLS.

A direct computation via geometric series and (2.4) yields an explicit
expression for $E_{DNLS}$.

######  Proposition 5.2.

Let $q_{n},g_{n}$ be given by (5.1), (5.4), respectively. Then,

$\begin{split}E_{A}(\epsilon,w)&=\frac{\epsilon(w+\epsilon)(w+2\epsilon)}{w+3\epsilon}-\frac{(w+2\epsilon)^{2}((w+2\epsilon)^{4}+\epsilon^{4})}{4((w+2\epsilon)^{4}-\epsilon^{4})},\\\
E_{B}(\epsilon,w)&=\frac{\epsilon(w+\epsilon)^{2}}{w+3\epsilon}-\frac{(w+\epsilon)^{2}}{2\left(1-\frac{\epsilon^{4}}{(w+2\epsilon)^{4}}\right)}.\end{split}$
(5.5)

Figure 4: PNB for DNLS as a function of $\epsilon,w_{A}$. The numerical plot
is consistent with Corollary 5.1 that there exists a unique local (and global)
maximum at $(\epsilon^{*},w_{A}^{*})=(0,0)$.

Suppose $q_{n}$ oscillates in time at $w_{A}$ and $g_{n}$ at $w_{B}$. If
$q_{n},g_{n}$ are two modes of the same traveling wave solution, then
$N_{A}=N_{B}$. Since $N_{A}\sim w_{A},\ N_{B}\sim 2w_{B}$ for $w_{A},w_{B}\gg
1$, assume $w_{A}=2w_{B}$. As [22, Equation 9], the Peierls-Nabarro barrier is
defined as the energy difference of the two modes at $w_{A},w_{B}$,
respectively, which can be computed explicitly by (5.5). See Figure 4.

######  Corollary 5.1.

Let $\Delta E_{AB}=E_{A}(\epsilon,w_{A})-E_{B}(\epsilon,\frac{w_{A}}{2})$.
Setting $\epsilon=kw_{A}$ for $k>0$, we have

$\Delta E_{AB}=-\gamma(k)w_{A}^{2},$

where $\gamma(k)$ is a strictly positive rational function satisfying
$\gamma(k)\xrightarrow[k\rightarrow 0+]{}\frac{1}{8}$ and $\inf\limits_{k\geq
0}\gamma(k)>0$. Therefore $\Delta E_{AB}<0$ for any $\epsilon,w_{A}>0$ and
$\lim\limits_{(\epsilon,w_{A})\rightarrow(\epsilon^{*},w_{A}^{*})}\Delta
E_{AB}=0$ if and only if $(\epsilon^{*},w_{A}^{*})=(0,0)$. Furthermore as
$w_{A}\rightarrow\infty$,

$\Delta E_{AB}\sim_{\epsilon}-\frac{w_{A}^{2}}{8},\
\frac{E_{A}(\epsilon,w_{A})}{E_{B}(\epsilon,w_{B})}\sim_{\epsilon}2.$

######  Remark 5.2.

As $h=\epsilon^{-\frac{1}{2}}\rightarrow 0$, we have $\Delta
E_{AB}\sim-\frac{4}{15h^{4}}$ by direct computation via (5.5), which seems
inconsistent with the exponential smallness of PNB

$|\Delta E_{AB}|\lesssim(\sqrt{w}h)^{2-d}e^{-\frac{C}{\sqrt{w}h}},\ d=1,2,3,$

that recovers the Galilean invariance of NLS as $h\rightarrow 0$; see [41,
Theorem 3.3]. However, since $q_{n},g_{n}$ do not satisfy the DNLS
asymptotically as $\epsilon\rightarrow\infty$, $\Delta E_{AB}$ for
$\epsilon\gg 1$ given in Corollary 5.1, or equivalently $h\ll 1$ on
$h\mathbb{Z}$, does not accurately describe the PNB.

###  5.3 Localized Solutions for fDNLS.

Localized on and offsite solutions for the fDNLS were studied in [9]. The
authors correctly point out that the long-distance behavior of the
intrinsically localized states depends on the rate of decay $1+\alpha$ and
that as $\alpha$ gets larger, localization resembles that of the DNLS. As it
relates to the asymptotic behavior ($n$ large), for the stationary solutions
of

$-\omega{q}_{n}=\epsilon\sum_{m\in\mathbb{Z}\setminus\\{n\\}}\frac{q_{n}-q_{m}}{|n-m|^{1+\alpha}}-q_{n}^{3},\
(n,t)\in\mathbb{Z}\times\mathbb{R},\ \epsilon>0,\ \alpha>0,$

and using the Green’s function ($G_{n-m}$) formalism so that
$q_{n}=\sum_{m}G_{n-m}q_{m}^{3}$, the authors in [9] conclude that the tail of
the mode decays algebraically if $2<1+\alpha<3$ and exponentially in $n$ for
$1+\alpha>3$. Our approach is different and instead we show below that in fact
in all instances the tail decays algebraically.

Define

$\displaystyle q_{0}=\sqrt{w+2\epsilon J},\
q_{n}=\frac{\epsilon\left(J_{n}q_{0}+\sum\limits_{m=1}^{n-1}\left(J_{n-m}+J_{n+m}\right)q_{m}\right)}{\epsilon\left(2J-J_{2n}\right)+w},$
$n\geq 1$ (5.6a) $\displaystyle g_{0}=\sqrt{w+\epsilon(2J-J_{1})},\
g_{n}=\frac{\epsilon\sum\limits_{m=1}^{n-1}\left(J_{n-m}+J_{n+m-1}\right)g_{m}}{\epsilon\left(2J-J_{2n-1}\right)+w},$
$n\geq 2$. (5.6b)

To provide an insight for the definitions above, the first two terms of
$q_{n}$ are shown explicitly. For $q_{0}$, neglect $q_{n}$ for $|n|\geq 1$.
Then

$-wq_{0}=2\epsilon Jq_{0}-q_{0}^{3},$

and hence $q_{0}$. For $q_{1}$, neglect the nonlinear term $q_{1}^{3}$ and
$q_{n}$ where $|n|\geq 2$. Since $q_{1}=q_{-1}$,

$-wq_{1}=\epsilon\sum_{m\neq 1}J_{|1-m|}(q_{1}-q_{m})=2\epsilon
Jq_{1}-\epsilon\left(J_{1}q_{0}+J_{2}q_{1}\right),$

and so follows $q_{1}$. Although the analytic description of the asymptotic
sequences is not immediately tractable, they are given by the power series
expansion in the high-frequency regime.

######  Proposition 5.3.

Let $\rho=\frac{\epsilon}{w}$ and $0<\rho<\frac{1}{2J}$. Then there exist
sequences $\\{c_{nk}\\}_{k\geq 1},\\{d_{nk}\\}_{k\geq 1}\subseteq\mathbb{R}$
such that

$\begin{split}\frac{q_{n}}{q_{0}}&=\sum_{k=1}^{\infty}c_{nk}\rho^{k},\ n\geq
1\\\ \frac{g_{n}}{g_{0}}&=\sum_{k=1}^{\infty}d_{nk}\rho^{k},\ n\geq
2,\end{split}$ (5.7)

where the series are absolutely convergent and $c_{n1}=J_{n},\
d_{n1}=J_{n-1}+J_{n}$.

###### Proof.

For brevity, our proof concerns $q_{n}$ only. Let

$\gamma_{n}=\frac{\rho}{1+\rho(2J-J_{2n})}.$

Since $\rho$ is sufficiently small by hypothesis,
$\gamma_{n}=\rho+O_{\alpha}(\rho^{2})$ by series expansion where the error
term is uniform in $n$ since $|\rho(2J-J_{2n})|<2\rho J<1$. Let $n=1$. From
(5.6a),

$\frac{q_{1}}{q_{0}}=\gamma_{1}J_{1}=\frac{\rho J_{1}}{1+\rho(2J-J_{1})},$

and hence (5.7) with $c_{n1}=J_{1}$. Suppose the claim holds for
$m=1,\dots,n-1$. Then

$\displaystyle\frac{q_{n}}{q_{0}}$
$\displaystyle=\gamma_{n}J_{n}+\gamma_{n}\sum\limits_{m=1}^{n-1}\left(J_{n-m}+J_{n+m}\right)\frac{q_{m}}{q_{0}}$
$\displaystyle=\gamma_{n}J_{n}+\gamma_{n}\sum_{k=1}^{\infty}\sum_{m=1}^{n-1}\left(J_{n-m}+J_{n+m}\right)c_{mk}\rho^{k},$
(5.8)

where the series is absolutely convergent since

$\sum_{k=1}^{\infty}\sum_{m=1}^{n-1}\left(J_{n-m}+J_{n+m}\right)|c_{mk}|\rho^{k}\leq
J\cdot\max_{1\leq m\leq
n-1}\left(\sum_{k=1}^{\infty}|c_{mk}|\rho^{k}\right)<\infty.$

The second term of (5.3) is $O(\rho^{2})$, and hence the dominant term of
$\frac{q_{n}}{q_{0}}$ is $\rho J_{n}$. ∎

By Proposition 5.3,

$\left|\frac{q_{n}/q_{0}}{\rho
J_{n}}-1\right|=J_{n}^{-1}\left|\sum_{k=2}^{\infty}c_{nk}\rho^{k-1}\right|\xrightarrow[\rho\rightarrow
0]{}0,$

and hence for any $n\in\mathbb{Z}\setminus\\{0\\}$,

$\frac{q_{n}}{q_{0}}\sim\rho J_{n},\ \rho\rightarrow 0.$ (5.9)

In fact, the convergence rate of (5.9) is uniform in $n$.

######  Proposition 5.4.

For all $\epsilon_{1},\alpha>0$, there exists
$\rho_{*}=\rho_{*}(\epsilon_{1},\alpha)>0$ such that for any
$0<\rho<\rho_{*}$,

$\begin{split}\frac{1}{1+\epsilon_{1}}&<\frac{q_{n}/q_{0}}{\rho
J_{n}}<1+\epsilon_{1},\ n\geq 1,\\\
\frac{1}{1+\epsilon_{1}}&\leq\frac{g_{n}/g_{0}}{\rho(J_{n}+J_{n-1})}\leq
1+\epsilon_{1},\ n\geq 2.\end{split}$ (5.10)

###### Proof.

The proof is for $q_{n}$ without loss of generality. Since
$n^{1+\alpha}J_{n}\xrightarrow[n\rightarrow\infty]{}A$, there exists
$N\in\mathbb{N}$ such that for any $n\geq N$, we have
$|J_{n}-\frac{A}{n^{1+\alpha}}|<\frac{A}{2n^{1+\alpha}}$. Define

$\rho_{*}=\min\left(\frac{\epsilon_{1}}{2J},\min\limits_{2\leq n\leq
2N}\left(\frac{\epsilon_{1}J_{n}}{2(1+\epsilon_{1})(n-1)J^{2}}\right),\frac{\epsilon_{1}}{3(1+2^{2+\alpha})(1+\epsilon_{1})J}\right).$

By (5.6a),

$\frac{q_{n}}{q_{0}}\geq\gamma_{n}J_{n}>\frac{\rho J_{n}}{1+2\rho
J}>\frac{\rho J_{n}}{1+\epsilon_{1}},$

and hence the lower bound of (5.10). The proof for the upper bound is by
induction. For the base case, we have

$\frac{q_{1}}{\rho J_{1}q_{0}}=\frac{1}{1+\rho(2J-J_{2})}<1+\epsilon_{1}.$

Let $C=(1+\epsilon_{1})\rho q_{0}$. For $2\leq n\leq 2N$,

$q_{n}<\rho\left(J_{n}q_{0}+C\sum_{m=1}^{n-1}(J_{n-m}+J_{n+m})J_{m}\right)\leq\rho\left(J_{n}q_{0}+2C(n-1)J^{2}\right)<CJ_{n},$

since $\rho<\rho_{*}$.

Let $n>2N$. Suppose $q_{m}<CJ_{m}$ for all $1\leq m\leq n-1$. Observe that

$\begin{split}\sum_{m=1}^{n-1}J_{n-m}q_{m}&<C\left(\sum_{1\leq
m\leq\frac{n}{2}}J_{n-m}J_{m}+\sum_{\frac{n}{2}<m\leq
n-1}J_{n-m}J_{m}\right)\\\ &<\frac{3CA}{2}\left(\sum_{1\leq
m\leq\frac{n}{2}}\frac{J_{m}}{(n-m)^{1+\alpha}}+\sum_{\frac{n}{2}<m\leq
n-1}\frac{J_{n-m}}{m^{1+\alpha}}\right)\\\ &\leq 3\cdot
2^{1+\alpha}CAn^{-(1+\alpha)}J<3\cdot 2^{2+\alpha}CJJ_{n}.\end{split}$

A similar computation yields

$\sum_{m=1}^{n-1}J_{n+m}q_{m}<3CJJ_{n},$

and altogether,

$q_{n}<\rho\left(q_{0}+3(1+2^{2+\alpha})CJ\right)J_{n}<CJ_{n}.$

This completes the induction, and the claim for $g_{n}$ follows similarly. ∎

######  Corollary 5.2.

Assuming the hypotheses of Proposition 5.4,

$\frac{q_{n}}{q_{0}}\sim\frac{A\rho}{n^{1+\alpha}},\hskip
20.0pt\frac{q_{n+1}}{q_{n}}\sim\left(\frac{n}{n+1}\right)^{1+\alpha},\hskip
20.0pt\frac{q_{2n}}{q_{n}}\sim 2^{-(1+\alpha)},$

as $\rho\rightarrow 0$ and $n\rightarrow\infty$, and similarly for
$\\{g_{n}\\}$.

As Proposition 5.1, it is shown that (5.6a), (5.6b) solve (2.3) in an
asymptotic sense.

######  Proposition 5.5.

For any $\epsilon,w,\alpha>0$, define $\\{q_{n}\\}$ by (5.6a). Then
$q_{n+1}=O(q_{n})$ as $\rho\rightarrow 0$, and

$\lim_{\rho\rightarrow
0}\sup_{n\in\mathbb{Z}}\left|\frac{-wq_{n}}{\epsilon\mathscr{L}_{\alpha}q_{n}-q_{n}^{3}}-1\right|=0,$
(5.11)

and similarly for $\\{g_{n}\\}$.

###### Proof.

The big-O bound follows from Corollary 5.2. The proof is for $q_{n}$ and
$n\geq 1$. By Proposition 5.4,

$-wq_{n}\sim-w\rho J_{n}q_{0}=-w^{\frac{3}{2}}\rho J_{n}(1+2\rho
J)^{\frac{1}{2}}.$

The interaction term is given by

$\epsilon\sum_{m\neq n}J_{|n-m|}(q_{n}-q_{m})=\epsilon
J_{n}(q_{n}-q_{0})+\epsilon\sum_{m\notin\\{0,n\\}}J_{|n-m|}(q_{n}-q_{m})=:I+II,$

and estimating the terms separately, we have

$I\sim-\epsilon J_{n}q_{0}=-w^{\frac{3}{2}}\rho J_{n}(1+2\rho
J)^{\frac{1}{2}},$

and

$II\sim\epsilon\rho q_{0}\sum_{m\notin\\{0,n\\}}J_{|n-m|}(J_{|n|}-J_{|m|})\leq
2w^{\frac{3}{2}}\rho^{2}(1+2\rho J)^{\frac{1}{2}}J^{2}.$ (5.12)

It follows that $II$ is $O(\rho^{2})$ and hence negligible. So is the
nonlinear term, which is $O(w^{\frac{3}{2}}\rho^{3})$. The uniformity in $n$
follows as Corollary 5.2. The limit (5.11) can be shown similarly for the case
$n=0$. ∎

###  5.4 Peierls-Nabarro Barrier for fDNLS.

In this part, we assume $J_{n}=n^{-(1+\alpha)}$. The PNB for fDNLS is
fundamentally different from that of the nearest-neighbor interaction. The PNB
can take both positive and negative values, and it can increase in $\epsilon$
or $w$ depending on $\alpha$; see Figures 4 and 5. This suggests that the
onsite solution need not always be the energy minimizer of the Hamiltonian. It
is of interest to further investigate the role of non-locality in the
framework of variational approach to fDNLS and the stability properties of
onsite/offsite solutions. By (2.4) and the symmetry properties of the
onsite/offsite solutions, an analogue of Proposition 5.2 is derived. The proof
follows from Corollary 5.2 and standard algebra, and thus is omitted.

######  Proposition 5.6.

Let $q_{n},g_{n}$ be given by (5.6a), (5.6b), respectively. Then,

$\begin{split}E_{A}&=\epsilon\sum_{n=1}^{\infty}\Biggl{(}\frac{|q_{n}-q_{0}|^{2}}{n^{1+\alpha}}+\frac{1}{2}\sum_{\begin{subarray}{c}m=1\\\
m\neq
n\end{subarray}}^{\infty}\left(\frac{1}{|n-m|^{1+\alpha}}+\frac{1}{|n+m|^{1+\alpha}}\right)|q_{n}-q_{m}|^{2}\Biggl{)}-\left(\frac{1}{4}q_{0}^{4}+\frac{1}{2}\sum_{n=1}^{\infty}q_{n}^{4}\right),\\\
E_{B}&=\epsilon\sum_{n=2}^{\infty}\Biggl{(}\left(\frac{1}{n^{1+\alpha}}+\frac{1}{(n-1)^{1+\alpha}}\right)|g_{n}-g_{0}|^{2}\\\
&\hskip 120.0pt+\frac{1}{2}\sum_{\begin{subarray}{c}m=2\\\ m\neq
n\end{subarray}}^{\infty}\left(\frac{1}{|n-m|^{1+\alpha}}+\frac{1}{|n+m-1|^{1+\alpha}}\right)|g_{n}-g_{m}|^{2}\Biggl{)}-\frac{1}{2}\sum_{n=1}^{\infty}g_{n}^{4}.\end{split}$
(5.13)

The analysis in Section 5.3 was simplified under the assumption
$\rho=\frac{\epsilon}{w}\ll 1$. A similar analysis is further developed in the
context of PNB using Proposition 5.6. By the conservation of particle number,
assume $w_{A}=2w_{B}$.

Figure 5: The left plot ($\epsilon=10$) shows the transient state of
PNB($\omega$) before they converge to $-\frac{w_{A}^{2}}{8}$ as
$w_{A}\rightarrow\infty$. The right plot ($w=1$) shows the PNB diverging away
from $-\frac{w_{A}^{2}}{8}$, the initial value at $\epsilon=0$, as
$\epsilon>0$ increases.

######  Corollary 5.3.

As $w_{A},w_{B}\rightarrow\infty$,

$\begin{split}E_{A}&\sim-\frac{w_{A}^{2}}{4}-(2\zeta(2+2\alpha)+\zeta(1+\alpha)^{2})\epsilon^{2},\\\
E_{B}&\sim-\frac{w_{B}^{2}}{2}-\left(2\sum_{n=2}^{\infty}\left(\frac{1}{n^{1+\alpha}}+\frac{1}{(n-1)^{1+\alpha}}\right)^{2}+\frac{1}{2}(2\zeta(1+\alpha)-1)^{2}\right)\epsilon^{2}.\end{split}$

Therefore, $E_{A}(w_{A})\sim 2E_{B}(w_{B})$ and

$\Delta E_{AB}\sim-\frac{w_{A}^{2}}{8}.$

######  Corollary 5.4.

As $\epsilon\rightarrow 0$,

$\begin{split}E_{A}&\sim-\frac{w_{A}^{2}}{4}+\left(-2\zeta(2+2\alpha)+\zeta(1+\alpha)^{2}\right)\epsilon^{2},\\\
E_{B}&\sim-\frac{w_{B}^{2}}{2}+\left(-2\sum_{n=2}^{\infty}\left(\frac{1}{n^{1+\alpha}}+\frac{1}{(n-1)^{1+\alpha}}\right)^{2}+\frac{1}{2}(2\zeta(1+\alpha)-1)^{2}\right)\epsilon^{2}.\end{split}$

Therefore, $E_{A}(w_{A})\sim 2E_{B}(w_{B})$ and

$\Delta
E_{AB}\sim-\frac{w_{A}^{2}}{8}+\left(2\sum_{n=2}^{\infty}\left(\frac{1}{n^{1+\alpha}}+\frac{1}{(n-1)^{1+\alpha}}\right)^{2}-(\zeta(1+\alpha)-1)^{2}-2\zeta(2+2\alpha)+\frac{1}{2}\right)\epsilon^{2}.$

For $w_{A}>0$ not sufficiently large, $PNB(w_{A})$ may not be well
approximated by the quadratic term; in fact, $PNB(w_{A})$ may be increasing
for $w_{A}>0$ sufficiently small. For $\epsilon>0$ not sufficiently small, the
behavior of PNB is generally not quadratic in $\epsilon$ since the higher
order terms cannot be neglected. For $\epsilon\ll 1$, note that PNB may
increase or decrease depending on the sign of the coefficient of
$\epsilon^{2}$.

######  Remark 5.3.

As $\alpha\rightarrow 0$, or more precisely, for
$\alpha\ll\min(1,\frac{\epsilon}{w})$, observe that
$\frac{1}{\alpha}<\zeta(1+\alpha)<1+\frac{1}{\alpha}$ yields

$N_{A}\sim q_{0}^{2}\sim\frac{2\epsilon}{\alpha},\ N_{B}\sim
2g_{0}^{2}\sim\frac{4\epsilon}{\alpha}.$

Since $N_{B}\sim 2N_{A}$, the conservation of particle number as a localized
wave travels along the lattice in the forms of onsite and offsite waves, if
such solutions exist at all, does not hold. This suggests that the PNB for
small $\alpha>0$ calculated from (5.13) is non-physical.

Figure 6: Position of peak intensity for traveling waves of the fDNLS for
$\epsilon=1,w=1$, $v=1$ and varying $\alpha\in\\{3.34,5.23,6.25,20\\}$.

Figure 7: Intensity plots for varying $\alpha\in\\{0.3,3.34,5,20\\}$. The top-
left plot illustrates the log intensity.

In Figures 6 and 7, the mobility/pinning of peak intensity is observed in the
nonlocal dynamics given by $J_{n}=|n|^{-(1+\alpha)}$ for $n\neq 0$ with the
initial condition as the onsite sequence defined in (5.6a). As
$\alpha\rightarrow 0$, the nonlocal coupling blows up as
$\lim\limits_{\alpha\rightarrow 0+}\zeta(1+\alpha)=\infty$. Moreover the
conservation of particle numbers between the onsite and offsite solutions
fails in the sense described in Remark 5.3, resulting in the erratic behavior
illustrated in the top-left plot of Figure 7. For $\alpha$ not too small, the
intensity drifts and eventually pins at a lattice point. More precisely as
$\alpha\to\infty$, the non-locality weakens, leading to a weaker drift
(pinning) at earlier times. See [42, Figure 1] for the drift and pinning under
the DNLS dynamics for varying mesh grid sizes $h>0$ whereas Figure 6 plots the
argmax of peak intensity for various $\alpha$.

##  6 Conclusion

The study of existence, dynamics and interactions of coherent, localized
structures in nonlinear lattices remains an active field given the wide range
of applications. Here we presented work for a model where the coupling between
units (oscillators, waveguides, resonators) of a one dimensional arrays is
global, with strength decaying algebraically with respect to the index
difference. Results include the characterization of modulational instability
and emergence of nonlinear modes for large times, obtained by numerical
simulations. By use of asymptotic techniques we show the existence of
localized modes and to assess mobility, derived formulas for the Peierls-
Nabarro barrier. In all instances we point to the behavior in terms of the
coupling strength parameter $\alpha$. This parametric dependece of the
dynamics can have potential applications. Future work will further explore
these applications, extend work to consider interaction properties
(collisions) between localized modes. Finally, there are two natural
extensions of this work; first the study of two-dimensional lattices with
similar coupling functionality and second is for the long wave limit, to
identify the continuum limit.

##  7 Acknowledgements

Both B.C. and A.M. acknowledge support from the NSF RTG award, DMS-1840260.

##  Appendix A Appendix

For our numerical simulations, a particular interaction kernel
$J_{n}=|n|^{-(1+\alpha)}$ was used. Consider $(u_{n})$ for $n=-N,\dots,N$.
Then for the Dirichlet boundary condition, we have

$\mathscr{L}_{\alpha}u_{n}=\sum_{-N\leq m\leq N,m\neq
n}\frac{u_{n}-u_{m}}{|n-m|^{1+\alpha}},$ (A.1)

where $u_{m}=0$ for all $|m|>N$.

For the periodic boundary condition, consider the modular arithmetics where
the quotient space of $\mathbb{Z}$, with $I_{N}:=\\{-N,\dots,N-1\\}$ as the
fundamental cell, is considered, as $u_{-N}=u_{N}$. Given $m\in\mathbb{Z}$,
let $m=2Nq+r$ where $q\in\mathbb{Z},\ r\in I_{N}$, and assume $u_{m}=u_{r}$.
Then for $n\in I_{N}$,

$\begin{split}\mathscr{L}_{\alpha}u_{n}&=\sum_{m\neq
n}\frac{u_{n}-u_{m}}{|n-m|^{1+\alpha}}\\\
&=2\zeta(1+\alpha)\left(1-\frac{1}{(2N)^{1+\alpha}}\right)u_{n}-\sum\limits_{r\neq
n}c_{nr}(N,\alpha)u_{r},\end{split}$ (A.2)

where

$c_{nr}:=\frac{1}{|n-r|^{1+\alpha}}+\frac{1}{(2N)^{1+\alpha}}\bigg{\\{}\zeta(1+\alpha,\frac{r-n}{2N})+\zeta(1+\alpha,-\frac{r-n}{2N})-|\frac{r-n}{2N}|^{-(1+\alpha)}(1+e^{-i(1+\alpha)\pi})\bigg{\\}},$
(A.3)

and $\zeta(s)=\sum\limits_{k=1}^{\infty}\frac{1}{k^{s}}$ is the Riemann zeta
function and $\zeta(s,a)=\sum\limits_{k=0}^{\infty}\frac{1}{(k+a)^{s}}$ is the
Hurwitz zeta function. Lastly we provide a brief derivation of (A.3). Using
$m=2Nq+r$ as above,

$\begin{split}\sum_{m\neq
n}\frac{u_{n}-u_{m}}{|n-m|^{1+\alpha}}&=2\zeta(1+\alpha)u_{n}-\sum_{m\neq
n}\frac{u_{m}}{|n-m|^{1+\alpha}}\\\ &=2\zeta(1+\alpha)u_{n}-\sum_{r\in
I_{N}\setminus\\{n\\}}\frac{u_{r}}{|n-r|^{1+\alpha}}-\sum_{r\in
I_{N}}\sum_{q\in\mathbb{Z}\setminus\\{0\\}}\frac{u_{r}}{|q(2N)+r-n|^{1+\alpha}}\end{split}$

The last sum simplifies to $2\zeta(1+\alpha)(2N)^{-(1+\alpha)}u_{n}$ when
$r=n$. When $r\neq n$,

$\begin{split}\sum_{q\in\mathbb{Z}\setminus\\{0\\}}\frac{(2N)^{1+\alpha}u_{r}}{|q(2N)+r-n|^{1+\alpha}}&=\sum_{q=1}^{\infty}\frac{1}{(q+\frac{r-n}{N})^{1+\alpha}}+\sum_{q=1}^{\infty}\frac{1}{(q-\frac{r-n}{N})^{1+\alpha}}\\\
&=\zeta(1+\alpha,\frac{r-n}{N})+\zeta(1+\alpha,-\frac{r-n}{N})-|\frac{r-n}{N}|^{-(1+\alpha)}\left(1+e^{-i(1+\alpha)\pi}\right).\end{split}$

Rearranging terms, the expression for $\mathscr{L}_{\alpha}u_{n}$ is derived
as a matrix multiplication with dense entries.

## References

  * [1] H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Physical Review Letters, vol. 81, no. 16, p. 3383, 1998.
  * [2] R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Dynamics of discrete solitons in optical waveguide arrays,” Physical Review Letters, vol. 83, no. 14, p. 2726, 1999.
  * [3] A. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, “Discrete self-trapping, soliton interactions, and beam steering in nonlinear waveguide arrays,” Physical Review E, vol. 53, no. 1, p. 1172, 1996.
  * [4] B. P. Anderson and M. A. Kasevich, “Macroscopic quantum interference from atomic tunnel arrays,” Science, vol. 282, no. 5394, pp. 1686–1689, 1998.
  * [5] G. Kalosakas, K. Rasmussen, and A. Bishop, “Delocalizing transition of bose-einstein condensates in optical lattices,” Physical review letters, vol. 89, no. 3, p. 030402, 2002.
  * [6] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. Shenoy, “Quantum coherent atomic tunneling between two trapped bose-einstein condensates,” Physical Review Letters, vol. 79, no. 25, p. 4950, 1997.
  * [7] A. Trombettoni and A. Smerzi, “Discrete solitons and breathers with dilute bose-einstein condensates,” Physical Review Letters, vol. 86, no. 11, p. 2353, 2001.
  * [8] S. F. Mingaleev, P. L. Christiansen, Y. Gaididei, M. Johansson, and K. Ø. Rasmussen, “Models for energy and charge transport and storage in biomolecules,” Journal of Biological Physics, vol. 25, pp. 41–63, Mar 1999.
  * [9] Y. B. Gaididei, S. F. Mingaleev, P. L. Christiansen, and K. O. Rasmussen, “Effects of nonlocal dispersive interactions on self-trapping excitations,” Phys. Rev. E, vol. 55, pp. 6141–6150, May 1997.
  * [10] D. S. Bassett and O. Sporns, “Network neuroscience,” Nature neuroscience, vol. 20, no. 3, pp. 353–364, 2017.
  * [11] M. Rohden, A. Sorge, M. Timme, and D. Witthaut, “Self-organized synchronization in decentralized power grids,” Physical review letters, vol. 109, no. 6, p. 064101, 2012.
  * [12] P. G. Kevrekidis, “The discrete nonlinear schrödinger equation.”
  * [13] Y. S. Kivshar and G. P. Agrawal, Optical solitons: from fibers to photonic crystals.  Academic press, 2003.
  * [14] P.-L. S. Catherine Sulem, The Nonlinear Schrödinger Equation: Self-focusing and Wave Collapse.  Applied Mathematical Sciences, Springer, 1 ed., 1999.
  * [15] Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys., vol. 83, pp. 247–305, Apr 2011.
  * [16] M. I. Molina, “The two-dimensional fractional discrete nonlinear schrödinger equation,” Physics Letters A, vol. 384, no. 33, p. 126835, 2020.
  * [17] M. Johansson, Y. B. Gaididei, P. L. Christiansen, and K. Rasmussen, “Switching between bistable states in a discrete nonlinear model with long-range dispersion,” Physical Review E, vol. 57, no. 4, p. 4739, 1998.
  * [18] K. Rasmussen, P. L. Christiansen, M. Johansson, Y. B. Gaididei, and S. Mingaleev, “Localized excitations in discrete nonlinear schrödinger systems: effects of nonlocal dispersive interactions and noise,” Physica D: Nonlinear Phenomena, vol. 113, no. 2-4, pp. 134–151, 1998.
  * [19] P. L. Christiansen, Y. B. Gaididei, F. G. Mertens, and S. F. Mingaleev, “Multi-component structure of nonlinear excitations in systems with length-scale competition,” The European Physical Journal B-Condensed Matter and Complex Systems, vol. 19, pp. 545–553, 2001.
  * [20] N. Korabel and G. M. Zaslavsky, “Transition to chaos in discrete nonlinear schrödinger equation with long-range interaction,” Physica A: Statistical Mechanics and its Applications, vol. 378, no. 2, pp. 223–237, 2007.
  * [21] S. Longhi, “Fractional schrödinger equation in optics,” Optics letters, vol. 40, no. 6, pp. 1117–1120, 2015.
  * [22] Y. S. Kivshar and D. K. Campbell, “Peierls-nabarro potential barrier for highly localized nonlinear modes,” Phys. Rev. E, vol. 48, pp. 3077–3081, Oct 1993.
  * [23] O. Bang and P. D. Miller, “Exploiting discreteness for switching in waveguide arrays,” Optics letters, vol. 21, no. 15, pp. 1105–1107, 1996.
  * [24] M. I. Weinstein, “Nonlinear schrödinger equations and sharp interpolation estimates,” Communications in Mathematical Physics, vol. 87, pp. 567–576, 1983.
  * [25] M. I. Weinstein, “The nonlinear schrödinger equation—singularity formation, stability and dispersion,” The Connection between Infinite Dimensional and Finite Dimensional Dynamical Systems, pp. 213–232, 1989.
  * [26] V. Zakharov and L. Ostrovsky, “Modulation instability: The beginning,” Physica D: Nonlinear Phenomena, vol. 238, no. 5, pp. 540–548, 2009.
  * [27] T. Dauxois and M. Peyrard, “Energy localization in nonlinear lattices,” Physical review letters, vol. 70, no. 25, p. 3935, 1993.
  * [28] I. Daumont, T. Dauxois, and M. Peyrard, “Modulational instability: first step towards energy localization in nonlinear lattices,” Nonlinearity, vol. 10, no. 3, p. 617, 1997.
  * [29] V. E. Zakharov, A. I. Dyachenko, and A. O. Prokofiev, “Freak waves as nonlinear stage of stokes wave modulation instability,” European Journal of Mechanics-B/Fluids, vol. 25, no. 5, pp. 677–692, 2006.
  * [30] V. E. Zakharov and A. Gelash, “Nonlinear stage of modulation instability,” Physical review letters, vol. 111, no. 5, p. 054101, 2013.
  * [31] A. Gelash, D. Agafontsev, V. Zakharov, G. El, S. Randoux, and P. Suret, “Bound state soliton gas dynamics underlying the spontaneous modulational instability,” Physical review letters, vol. 123, no. 23, p. 234102, 2019.
  * [32] L. Zhang, Z. He, C. Conti, Z. Wang, Y. Hu, D. Lei, Y. Li, and D. Fan, “Modulational instability in fractional nonlinear schrödinger equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 48, pp. 531–540, 2017.
  * [33] A. Aceves and A. Copeland, “Spatiotemporal dynamics in the fractional nonlinear schrödinger equation,” Frontiers, Nonlinear Photonics, vol. 3, 2022.
  * [34] V. D. Dinh, “Blow-up criteria for fractional nonlinear schrödinger equations,” Nonlinear Analysis: Real World Applications, vol. 48, pp. 117–140, 2019.
  * [35] Y. Hong and Y. Sire, “On fractional schrödinger equations in sobolev spaces,” Communications on Pure & Applied Analysis, vol. 14, no. 6, pp. 2265–2282, 2015.
  * [36] V. D. Dinh, “Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces,” International Journal of Applied Mathematics, vol. 31, pp. 483–525, Sept. 2018.
  * [37] B. Choi and A. Aceves, “Well-posedness of the mixed-fractional nonlinear schrödinger equation on r2,” Partial Differential Equations in Applied Mathematics, vol. 6, p. 100406, 2022.
  * [38] K. Kirkpatrick, E. Lenzmann, and G. Staffilani, “On the continuum limit for discrete nls with long-range lattice interactions,” Communications in mathematical physics, vol. 317, no. 3, pp. 563–591, 2013.
  * [39] Y. Hong and C. Yang, “Strong convergence for discrete nonlinear schrödinger equations in the continuum limit,” SIAM Journal on Mathematical Analysis, vol. 51, no. 2, pp. 1297–1320, 2019.
  * [40] B. Choi and A. Aceves, “Continuum limit of 2d fractional nonlinear schrödinger equation,” Journal of Evolution Equations, vol. 23, no. 2, p. 30, 2023.
  * [41] M. Jenkinson and M. I. Weinstein, “On-site and off-site bound states of the discrete nonlinear schrödinger equation and the peierls-nabarro barrier,” 2014.
  * [42] M. Jenkinson and M. I. Weinstein, “Discrete solitary waves in systems with nonlocal interactions and the peierls–nabarro barrier - communications in mathematical physics,” Feb 2017.
  * [43] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, and G. E. Karniadakis, “What is the fractional laplacian?,” 2019.
  * [44] M. I. Weinstein, “Excitation thresholds for nonlinear localized modes on lattices,” Nonlinearity, vol. 12, no. 3, p. 673, 1999.
  * [45] A. Stefanov and P. G. Kevrekidis, “Asymptotic behaviour of small solutions for the discrete nonlinear schrödinger and klein–gordon equations,” Nonlinearity, vol. 18, no. 4, p. 1841, 2005.
  * [46] E. M. Stein and T. S. Murphy, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, vol. 3.  Princeton University Press, 1993.
  * [47] M. Keel and T. Tao, “Endpoint strichartz estimates,” American Journal of Mathematics, vol. 120, no. 5, pp. 955–980, 1998.
  * [48] T. Tao, Nonlinear dispersive equations: local and global analysis.  No. 106, American Mathematical Soc., 2006.


# Noise2Noise Denoising of CRISM
Hyperspectral Data

Robert Platt1,2,3, Rossella Arcucci1,3, Cédric M. John4
1Department of Earth Science and Engineering, Imperial College London
2I-X, Imperial College London
3Data Science Institute, Imperial College London
4Digital Environment Research Institute (DERI), Queen Mary University of
London
{rp1818<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Hyperspectral data acquired by the Compact Reconnaissance Imaging Spectrometer
for Mars (CRISM) have allowed for unparalleled mapping of the surface
mineralogy of Mars. Due to sensor degradation over time, a significant portion
of the recently acquired data is considered unusable. Here a new data-driven
model architecture, Noise2Noise4Mars (N2N4M), is introduced to remove noise
from CRISM images. Our model is self-supervised and does not require zero-
noise target data, making it well suited for use in Planetary Science
applications where high quality labelled data is scarce. We demonstrate its
strong performance on synthetic-noise data and CRISM images, and its impact on
downstream classification performance, outperforming benchmark methods on most
metrics. This allows for detailed analysis for critical sites of interest on
the Martian surface, including proposed lander sites.

##  1 Introduction

The hyperspectral data collected by the Compact Reconnaissance Imaging
Spectrometer for Mars (CRISM) on-board NASA’s Mars Reconnaissance Orbiter
(MRO) mission has allowed for the highest resolution spectral mapping of the
Martian surface to date. During its operation from 2006 to 2022, CRISM
captured over 33,000 targeted observations and mapped 86% of the planetary
surface (Seelos et al., 2023). CRISM data have allowed for detailed mapping of
the surface mineralogy, and understanding of the geological evolution of the
red planet (Murchie et al., 2019).

Despite the vast quantity of imagery acquired by CRISM, much of the data
processing and image interpretation is accomplished manually. Significant
technical expertise is required to interpret the imagery, so the reliance on
manual efforts represents a large bottleneck in effective use of this data. In
addition, CRISM data quality has decreased steadily since its launch,
attributed to failing cryocooling systems (Carter et al., 2013). As a result,
much of the more recent imagery ($>$2012) contains a sufficiently high noise
fraction (see Fig. 1) to obscure absorption features, which restricts mineral
identification and analysis (Mandon et al., 2021). This limits our
understanding of key sites of interest identified more recently, for example
Oxia Planum, the proposed landing site for ESA’s ExoMars rover (Mandon et al.,
2021). The scarcity of labelled or ground truth data presents an additional
challenge, as even older CRISM data still contains noise from atmospheric and
photometric effects. Therefore, noise-free ground truth data does not exist
for Mars. This highlights the need for development of unsupervised or self-
supervised approaches (Fernandes et al., 2022; Pletl et al., 2023). To that
end, our work introduces a new, self-supervised deep learning architecture for
denoising hyperspectral signals, with no requirement for paired training data.
Additionally, our model expects target data which itself is noisy, which is
critical for application to Planetary Sciences where perfect data is rarely
available. The code used to develop this work can be found here:
https://github.com/rob-platt/n2n4m

Figure 1: Comparison of a low-noise and high-noise pixel spectra of the same
location extracted from two CRISM images (FRT0000A053 and FRS000364CA)
acquired 7 years apart. Absorption features to identify the surface material
as an Mg/Fe-Smectite are highlighted. Shown in green is the low-noise spectra
with synthetic Gaussian noise added.

###  1.1 Related Works

Two broad approaches have been developed to reduce noise and artefacts
prevalent in CRISM data. The first uses and builds upon the so-called
“Volcano-Scan” correction, the current method applied in the preprocessing
pipeline for CRISM IR images to correct for atmospheric effects (Murchie et
al., 2022; 2009; Seelos et al., 2011). The CRISM Iterative Recognition and
Removal of Unwanted Spiking (CIRRUS) and the Complement to CRISM Analysis
Toolkit (CoTCAT) use statistical methods to reduce noise and build on
“Volcano-Scan” products (Parente, 2008; Bultel et al., 2015). Similarly Carter
et al. (2012) developed a tool (OCAT) using a Fast Fourier Transform (FFT) to
generate a smoothed signal.

A second approach developed around using radiative transfer modelling,
utilizing a physical forward model (DISTORT) to simultaneously post-process
and denoise the spectra (McGuire et al., 2008). Developments included
application of Maximum Likelihood Estimation to the results (Kreisch et al.,
2017), and adding hypothesis testing for noise and data distribution (He et
al., 2019). Itoh & Parente (2020) developed a method to directly estimate the
atmospheric transmission from the image itself, to simultaneously apply
atmospheric correction and denoising. Both groups of methods have limitations
- statistical methods can over-smooth key features (Bultel et al., 2015),
whilst DISTORT-based methods have higher computational requirements and long
($\sim$2hrs per image) processing time (Itoh & Parente, 2020). Data-driven
approaches remains unexplored.

A plethora of Machine Learning (ML) approaches have been developed for
denoising of signal or image data, based on Convolutional Neural Networks
(CNNs) (Tian et al., 2020; Zhang et al., 2018), Autoencoders (Gondara, 2016),
and Diffusion models (Kawar et al., 2022). Of particular interest is the work
of (Lehtinen et al., 2018), who implemented a CNN where the target images are
not assumed to be noise free. This is critical for application to CRISM data,
where there is no noise-free ground truth.

##  2 Methods

The data used for this work is derived from the CRISM Machine Learning Toolkit
dataset (Plebani et al., 2022). This consists of 586,599 pixels extracted from
77 low-noise CRISM images, associated with 39 corresponding mineral labels.
The base Targeted Reduced Data Record (TRDR) images were processed using
MarsSI (Quantin-Nataf et al., 2018) to apply standard photometric and volcano-
scan atmospheric corrections. Each IR spectra contains 350 channels between
1.0210µm-2.6483µm and 2.8070µm-3.4769µm, with channels known to contain
artefacts excluded as in the original data (Plebani et al., 2022).
Additionally, 150,000 spectrally inert or “bland”, pixels identified by
Plebani et al. (2022) from a wider range of images were similarly processed
and added to the dataset to better reflect the true class distribution.
Synthetic noise is then added to the spectra to create high-noise training
examples. The model objective is to reconstruct the low-noise signal from the
high-noise signal as input.

###  2.1 Preprocessing

Two further preprocessing steps were applied to clean the data - imputation of
bad values (I/F $>$ 1) and of data where significant atmospheric artefact
remained between 1.91µm and 2.08µm due to issues with the volcano-scan
correction. The data was split into train, test, and validation sets by image
to ensure generalisation to unseen imagery. In addition, the Serpentine and
Muscovite mineral classes were restricted to the test set only, to provide
insight to the models ability to generalise to unseen mineral classes.

Synthetic noise was added to the low noise data to create training data (Fig.
1). The synthetic noise partially obscures some of the key absorption features
(e.g. the absorption at 1.42$\mu$m in Fig. 1) similarly to real high noise
spectra. This synthetic noise was created by sampling from a Gaussian
distribution with zero mean, following Bultel et al. (2015). An additional,
uniformly sampled random variable is added to increase the inter-channel
variability, which better reflects the observed noise distribution (Bultel et
al., 2015).

###  2.2 Model and Training Scheme

Our approach implemented a 1-Dimensional convolutional neural network (1D-CNN)
in a U-Net architecture. This was inspired by the Noise2Noise (N2N) denoising
work (Lehtinen et al., 2018), which showed that when using a U-Net
architecture, a model can be trained effectively to denoise natural images
despite the presence of noise in the target data. Our new model architecture,
Noise2Noise for Mars (N2N4M), introduces important architectural changes to
fit hyperspectral data. Instead of 2-dimensional convolution, we use
1-dimensional convolution layers to leverage features in the rich spectral
dimension unique to hyperspectral imagery, rather than spatial data.
Transposed convolution layers were used rather than nearest-neighbour
upsampling in the decoder, and kernels of size 5 were found to be the most
performant during hyperparamter tuning. N2N4M was trained for 100 epochs, and
the validation dataset was used to tune learning rate, batch size, and model
architecture. The final model architecture and exact training parameters can
be found in Appendix A. This architecture allows us to apply the model to
CRISM data, where no zero-noise ground truth data exists, which has restricted
data-driven approaches to the problem.

Figure 2: Qualitative comparsion of results on unseen spectra from two
different mineral classes. Shown are the high-noise input, low-noise target,
and high-noise data denoised by CoTCAT, and N2N4M (ours). Absorption features
for mineral identification are highlighted.

##  3 Results

Model performance was tested with three approaches; denoising ability on
synthetic noise data, downstream classification performance, and application
to real imagery. The combination of these perceptual and quantitative metrics
allow for a rigorous evaluation of the ability and limitations of the model.
The Complement to CRISM Analysis Toolkit (CoTCAT) model (Bultel et al., 2015),
and Savitzky-Golay filtering (Gorry, 1990) were used as benchmarks for
comparison.

When denoising unseen synthetic high-noise data, our model shows the lowest
reconstruction error compared to the ground truth (Table 1). A visual
inspection of the results (see Fig. 2 and Fig. 3) shows that N2N4M removes
more noise than CoTCAT whilst retaining the absorption features evident in the
ground truth which are critical for mineral identification. For downstream
classification, the Hierarchical Bayesian Model (HBM) developed by (Plebani et
al., 2022) was used, trained on low-noise data. The HBM predicts a mineral
class label for an inputted spectra, with ground truth labels included in the
original dataset by (Plebani et al., 2022). The results (Table 1) are given as
scores relative to performance on low-noise ground truth data, to highlight
the impact of different denoising methods. N2N4M denoising results in a
significant increase in most metrics over the benchmarks.

Data | Denoising Task | Downstream Classification Task
---|---|---
MSE | Relative Accuracy | Relative F1 Score | Relative Precision | Relative Recall
Ground Truth | N/A | 1.00 | 1.00 | 1.00 | 1.00
Savitzky-Golay Filter | $2.8\times 10^{-5}$ | 0.01 | 0.00 | 0.42 | 0.09
CoTCAT (Bultel et al. 2015) | $5.0\times 10^{-6}$ | 0.35 | 0.43 | 0.50 | 0.41
N2N4M (Ours) | $\mathbf{4.7\times 10^{-6}}$ | 0.52 | 0.48 | 0.50 | 0.64

Table 1: Benchmarking results for denoising, and the downstream classification
task which uses the HBM from Plebani et al. (2022). Classification metrics are
given relative to classifier performance on high quality, low-noise ground
truth data. Best performance is highlighted in bold. Figure 3: Two pairs of
test images, from top to bottom: low-noise reference image (LN), high-noise
image (HN), denoised image (DN). Images shown as the strength of absorption
features (listed at base) in each pixel, which highlight outcrops of hydrated
minerals identified in the reference image.

Image evaluation was performed on two CRISM image pairs. In each case, a low-
noise, high quality image and high-noise, low quality image had been acquired
over the same spatial location. The high-noise image in each pair was denoised
using N2N4M, before standard post-processing steps were applied to all images
using the ratioing method described by Plebani et al. (2022). Summary
parameters are a widespread method to visualise specific mineral suites in
hyperspectral images (Viviano-Beck et al., 2014) and allow for a visual
assessment of denoising quality on real data. Summary parameters for known
outcrops previously identified in the high quality imagery (Bultel et al.,
2015; Mandon et al., 2021) were calculated for all results (Fig. 3). The
denoised images show a clear increase in perceptual quality over the original,
with mineral outcrops highlighted with greater definition and clearer spatial
extents. They compare favourably with the reference images and do not exhibit
any spurious outcrops.

##  4 Discussion and Future Works

Our proposed model shows promising results on both synthetic and real noise
hyperspectral CRISM imagery. The downstream use of this data is classification
of surface mineralogy, and N2N4M denoised data shows a significant improvement
over the benchmark in enabling this task. As such automated classifiers are
used to identify outcrop shapes and extents over exact mineral species
identification (Plebani et al., 2022), the 0.23 relative increase in recall by
N2N4M over the benchmark represents a large increase in capabilities. When
real imagery is assessed, there is a clear increase in visual quality post-
denoising. Additionally, no artificial mineral outcrops are introduced which
is critical for trust in the wider use of this model outside of examples where
low noise imagery is available. Whilst N2N4M shows a clear step forward in
performance, further development is needed for truly reliable denoising and
classification. Many higher noise CRISM images display striping noise (Carter
et al., 2012), and methods to include this spatial component directly in the
model should be explored.

##  5 Conclusion

A novel, data-driven method for denoising hyperspectral imagery of Mars where
true no-noise ground truth is unavailable has been proposed. This approach is
shown to be effective at removing noise whilst retaining key absorption
features of spectra, for both synthetic noise and real example imagery. Our
method is not restricted to the particular characteristics of SWIR spectra and
could easily be applied in other domains e.g. Synthetic Aperture Radar (SAR),
or Radio signals. Our work should allow for further analysis of much of the
Martian surface mineralogy, and highlights the use-case for self-supervised ML
methods in Planetary Sciences.

## References

  * Bultel et al. (2015) Benjamin Bultel, Cathy Quantin, and Loïc Lozac’h.  Description of CoTCAT (Complement to CRISM Analysis Toolkit).  _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , 8(6):3039–3049, June 2015.  ISSN 2151-1535.  doi: 10.1109/JSTARS.2015.2405095.
  * Carter et al. (2012) J. Carter, F. Poulet, S. Murchie, and J. P. Bibring.  Automated processing of planetary hyperspectral datasets for the extraction of weak mineral signatures and applications to CRISM observations of hydrated silicates on Mars.  _Planetary and Space Science_ , 76:53–67, December 2012.  ISSN 0032-0633.  doi: 10.1016/j.pss.2012.11.007.  URL https://www.sciencedirect.com/science/article/pii/S0032063312003625.
  * Carter et al. (2013) J. Carter, F. Poulet, J.-P. Bibring, N. Mangold, and S. Murchie.  Hydrous minerals on Mars as seen by the CRISM and OMEGA imaging spectrometers: Updated global view.  _Journal of Geophysical Research: Planets_ , 118(4):831–858, April 2013.  ISSN 2169-9100.  doi: 10.1029/2012JE004145.  URL https://onlinelibrary.wiley.com/doi/abs/10.1029/2012JE004145.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1029/2012JE004145.
  * Fernandes et al. (2022) Michael Fernandes, Alexander Pletl, Nicolas Thomas, Angelo Pio Rossi, and Benedikt Elser.  Generation and Optimization of Spectral Cluster Maps to Enable Data Fusion of CaSSIS and CRISM Datasets.  _Remote Sensing_ , 14(11):2524, January 2022.  ISSN 2072-4292.  doi: 10.3390/rs14112524.  URL https://www.mdpi.com/2072-4292/14/11/2524.  Number: 11 Publisher: Multidisciplinary Digital Publishing Institute.
  * Gondara (2016) Lovedeep Gondara.  Medical Image Denoising Using Convolutional Denoising Autoencoders.  In _2016 IEEE 16th International Conference on Data Mining Workshops (ICDMW)_ , pp. 241–246, December 2016.  doi: 10.1109/ICDMW.2016.0041.  URL https://ieeexplore.ieee.org/abstract/document/7836672.  ISSN: 2375-9259.
  * Gorry (1990) Peter A. Gorry.  General least-squares smoothing and differentiation by the convolution (Savitzky-Golay) method.  _Analytical Chemistry_ , 62(6):570–573, March 1990.  ISSN 0003-2700.  doi: 10.1021/ac00205a007.  URL https://doi.org/10.1021/ac00205a007.  Publisher: American Chemical Society.
  * He et al. (2019) Linyun He, Joseph A. O’Sullivan, Daniel V. Politte, Kathryn E. Powell, and Raymond E. Arvidson.  Quantitative Reconstruction and Denoising Method HyBER for Hyperspectral Image Data and Its Application to CRISM.  _IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing_ , 12(4):1219–1230, April 2019.  ISSN 2151-1535.  doi: 10.1109/JSTARS.2019.2900644.  URL https://ieeexplore.ieee.org/document/8672185.
  * Itoh & Parente (2020) Yuki Itoh and Mario Parente.  A new method for atmospheric correction and de-noising of CRISM hyperspectral data.  _Icarus_ , 354:114024, July 2020.  ISSN 0019-1035.  doi: 10.1016/j.icarus.2020.114024.  URL https://www.sciencedirect.com/science/article/pii/S0019103520303857.
  * Kawar et al. (2022) Bahjat Kawar, Michael Elad, Stefano Ermon, and Jiaming Song.  Denoising Diffusion Restoration Models.  _Advances in Neural Information Processing Systems_ , 35:23593–23606, December 2022.  URL https://proceedings.neurips.cc/paper_files/paper/2022/hash/95504595b6169131b6ed6cd72eb05616-Abstract-Conference.html.
  * Kreisch et al. (2017) C. D. Kreisch, J. A. O’Sullivan, R. E. Arvidson, D. V. Politte, L. He, N. T. Stein, J. Finkel, E. A. Guinness, M. J. Wolff, and M. G. A. Lapôtre.  Regularization of Mars Reconnaissance Orbiter CRISM along‐track oversampled hyperspectral imaging observations of Mars.  _Icarus_ , 282:136–151, January 2017.  ISSN 0019-1035.  doi: 10.1016/j.icarus.2016.09.033.  URL https://www.sciencedirect.com/science/article/pii/S0019103516301579.
  * Lehtinen et al. (2018) Jaakko Lehtinen, Jacob Munkberg, Jon Hasselgren, Samuli Laine, Tero Karras, Miika Aittala, and Timo Aila.  Noise2Noise: Learning Image Restoration without Clean Data, October 2018.  URL http://arxiv.org/abs/1803.04189.  arXiv:1803.04189 [cs, stat].
  * Mandon et al. (2021) Lucia Mandon, Adam Parkes Bowen, Cathy Quantin-Nataf, John C. Bridges, John Carter, Lu Pan, Pierre Beck, Erwin Dehouck, Matthieu Volat, Nicolas Thomas, Gabriele Cremonese, Livio Leonardo Tornabene, and Patrick Thollot.  Morphological and Spectral Diversity of the Clay-Bearing Unit at the ExoMars Landing Site Oxia Planum.  _Astrobiology_ , 21(4):464–480, April 2021.  ISSN 1531-1074.  doi: 10.1089/ast.2020.2292.  URL https://www.liebertpub.com/doi/10.1089/ast.2020.2292.  Publisher: Mary Ann Liebert, Inc., publishers.
  * McGuire et al. (2008) Patrick C. McGuire, Michael J. Wolff, Michael D. Smith, Raymond E. Arvidson, Scott L. Murchie, R. Todd Clancy, Ted L. Roush, Selby C. Cull, Kim A. Lichtenberg, Sandra M. Wiseman, Robert O. Green, Terry Z. Marti, Ralph E. Milliken, Peter J. Cavender, David C. Humm, Frank P. Seelos, Kim D. Seelos, Howard W. Taylor, Bethany L. Ehlmann, John F. Mustard, Shannon M. Pelkey, Timothy N. Titus, Christopher D. Hash, and Erick R. Malaret.  MRO/CRISM Retrieval of Surface Lambert Albedos for Multispectral Mapping of Mars With DISORT-Based Radiative Transfer Modeling: Phase 1—Using Historical Climatology for Temperatures, Aerosol Optical Depths, and Atmospheric Pressures.  _IEEE Transactions on Geoscience and Remote Sensing_ , 46(12):4020–4040, December 2008.  ISSN 0196-2892.  doi: 10.1109/TGRS.2008.2000631.  URL http://ieeexplore.ieee.org/document/4685531/.
  * Murchie et al. (2022) Scott Murchie, Edward A. Guinness, and Susan Slavney.  CRISM Data Product Software Interface Specification Version 1.3.7.6., June 2022.
  * Murchie et al. (2009) Scott L. Murchie, Frank P. Seelos, Christopher D. Hash, David C. Humm, Erick Malaret, J. Andrew McGovern, Teck H. Choo, Kimberly D. Seelos, Debra L. Buczkowski, M. Frank Morgan, Olivier S. Barnouin-Jha, Hari Nair, Howard W. Taylor, Gerald W. Patterson, Christopher A. Harvel, John F. Mustard, Raymond E. Arvidson, Patrick McGuire, Michael D. Smith, Michael J. Wolff, Timothy N. Titus, Jean-Pierre Bibring, and Francois Poulet.  Compact Reconnaissance Imaging Spectrometer for Mars investigation and data set from the Mars Reconnaissance Orbiter’s primary science phase.  _Journal of Geophysical Research: Planets_ , 114(E2), 2009.  ISSN 2156-2202.  doi: 10.1029/2009JE003344.  URL https://onlinelibrary.wiley.com/doi/abs/10.1029/2009JE003344.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1029/2009JE003344.
  * Murchie et al. (2019) Scott L. Murchie, Jean-Pierre Bibring, Raymond E. Arvidson, Janice L. Bishop, John Carter, Bethany L. Ehlmann, Yves Langevin, John F. Mustard, Francois Poulet, Lucie Riu, Kimberly D. Seelos, and Christina E. Viviano.  Visible to Short-Wave Infrared Spectral Analyses of Mars from Orbit Using CRISM and OMEGA.  In Janice L. Bishop, James F. Bell III, and Jeffrey E. Moersch (eds.), _Remote Compositional Analysis_ , pp. 453–483. Cambridge University Press, 1 edition, November 2019.  ISBN 978-1-316-88887-2 978-1-107-18620-0.  doi: 10.1017/9781316888872.025.  URL https://www.cambridge.org/core/product/identifier/9781316888872%23CN-bp-23/type/book_part.
  * Parente (2008) Mario Parente.  A New Approach To Denoise CRISM Images.  In _39th Lunar and Planetary Science Conference_ , March 2008.  URL https://www.lpi.usra.edu/meetings/lpsc2008/pdf/2528.pdf.
  * Plebani et al. (2022) Emanuele Plebani, Bethany L. Ehlmann, Ellen K. Leask, Valerie K. Fox, and M. Murat Dundar.  A machine learning toolkit for CRISM image analysis.  _Icarus_ , 376:114849, April 2022.  ISSN 00191035.  doi: 10.1016/j.icarus.2021.114849.  URL https://linkinghub.elsevier.com/retrieve/pii/S0019103521004905.
  * Pletl et al. (2023) Alexander Pletl, Michael Fernandes, Nicolas Thomas, Angelo Pio Rossi, and Benedikt Elser.  Spectral Clustering of CRISM Datasets in Jezero Crater Using UMAP and k-Means.  _Remote Sensing_ , 15(4):939, January 2023.  ISSN 2072-4292.  doi: 10.3390/rs15040939.  URL https://www.mdpi.com/2072-4292/15/4/939.  Number: 4 Publisher: Multidisciplinary Digital Publishing Institute.
  * Quantin-Nataf et al. (2018) C. Quantin-Nataf, L. Lozac’h, P. Thollot, D. Loizeau, B. Bultel, J. Fernando, P. Allemand, F. Dubuffet, F. Poulet, A. Ody, H. Clenet, C. Leyrat, and S. Harrisson.  MarsSI: Martian surface data processing information system.  _Planetary and Space Science_ , 150:157–170, January 2018.  ISSN 0032-0633.  doi: 10.1016/j.pss.2017.09.014.  URL https://www.sciencedirect.com/science/article/pii/S0032063316304718.
  * Seelos et al. (2011) F P Seelos, S L Murchie, D C Humm, O S Barnouin, and F Mor.  Data Processing and Analysis Products Update - Calibration, Correction, and Visualisation.  In _42nd Lunar and Planetary Sciences Conference_ , 2011.
  * Seelos et al. (2023) Frank P. Seelos, Kimberly D. Seelos, Scott L. Murchie, M. Alexandra Matiella Novak, Christopher D. Hash, M. Frank Morgan, Raymond E. Arvidson, John Aiello, Jean-Pierre Bibring, Janice L. Bishop, John D. Boldt, Ariana R. Boyd, Debra L. Buczkowski, Patrick Y. Chen, R. Todd Clancy, Bethany L. Ehlmann, Katelyn Frizzell, Katie M. Hancock, John R. Hayes, Kevin J. Heffernan, David C. Humm, Yuki Itoh, Maggie Ju, Mark C. Kochte, Erick Malaret, J. Andrew McGovern, Patrick McGuire, Nishant L. Mehta, Eleanor L. Moreland, John F. Mustard, A. Hari Nair, Jorge I. Núñez, Joseph A. O’Sullivan, Liam L. Packer, Ryan T. Poffenbarger, Francois Poulet, Giuseppe Romeo, Andrew G. Santo, Michael D. Smith, David C. Stephens, Anthony D. Toigo, Christina E. Viviano, and Michael J. Wolff.  The CRISM investigation in Mars orbit: Overview, history, and delivered data products.  _Icarus_ , pp. 115612, May 2023.  ISSN 0019-1035.  doi: 10.1016/j.icarus.2023.115612.  URL https://www.sciencedirect.com/science/article/pii/S0019103523001896.
  * Tian et al. (2020) Chunwei Tian, Lunke Fei, Wenxian Zheng, Yong Xu, Wangmeng Zuo, and Chia-Wen Lin.  Deep learning on image denoising: An overview.  _Neural Networks_ , 131:251–275, November 2020.  ISSN 0893-6080.  doi: 10.1016/j.neunet.2020.07.025.  URL https://www.sciencedirect.com/science/article/pii/S0893608020302665.
  * Viviano-Beck et al. (2014) Christina E. Viviano-Beck, Frank P. Seelos, Scott L. Murchie, Eliezer G. Kahn, Kimberley D. Seelos, Howard W. Taylor, Kelly Taylor, Bethany L. Ehlmann, Sandra M. Wiseman, John F. Mustard, and M. Frank Morgan.  Revised CRISM spectral parameters and summary products based on the currently detected mineral diversity on Mars.  _Journal of Geophysical Research: Planets_ , 119(6):1403–1431, 2014.  ISSN 2169-9100.  doi: 10.1002/2014JE004627.  URL https://onlinelibrary.wiley.com/doi/abs/10.1002/2014JE004627.  _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/2014JE004627.
  * Zhang et al. (2018) Kai Zhang, Wangmeng Zuo, and Lei Zhang.  FFDNet: Toward a Fast and Flexible Solution for CNN-Based Image Denoising.  _IEEE Transactions on Image Processing_ , 27(9):4608–4622, September 2018.  ISSN 1941-0042.  doi: 10.1109/TIP.2018.2839891.  URL https://ieeexplore.ieee.org/abstract/document/8365806.  Conference Name: IEEE Transactions on Image Processing.

##  Appendix A Appendix

Figure 4: Noise2Noise 4 Mars (N2N4M) Network Architecture. All convolutional
kernels were size 5.

Hyperparamter settings for N2N4M model training: learning rate = $1\times
10^{-3}$, and batch size = 256. Model was trained for 100 epochs with early
stopping implemented after 5 epochs of no decrease in validation score. Mean
squared error (MSE) was used as the loss function, and ADAM as the optimiser.