040001 2012 A. Goñi J. Chavez Boggio, Leibniz Institut für Astrophysik
Potsdam, Germany. 040001

We present a high-speed wavelength tunable photonic crystal fiber-based source
capable of generating tunable femtosecond solitons in the infrared region.
Through measurements and numerical simulation, we show that both the
pulsewidth and the spectral width of the output pulses remain nearly constant
over the entire tuning range from $860$ to $1160$ nm. This remarkable behavior
is observed even when pump pulses are heavily chirped ($7400$ fs2), which
allows to avoid bulky compensation optics, or the use of another fiber, for
dispersion compensation usually required by the tuning device.

# High-speed tunable photonic crystal fiber-based femtosecond soliton source
without dispersion pre-compensation

Martín Caldarola  [lec] Víctor A. Bettachini E-mail<EMAIL_ADDRESS>[itba] Andrés A. Rieznik  [itba] Pablo G. König  [itba] Martín E. Masip  [lec]
Diego F. Grosz  [itba, coni] Andrea V. Bragas[lec, ifiba]

(7 July 2011; 1 February 2012)

††volume: 4

99 lec Laboratorio de Electrónica Cuántica, Departamento de Física,
Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, (C1428EHA)
Buenos Aires, Argentina. itba Instituto Tecnológico de Buenos Aires, Eduardo
Madero 399, (C1106ACD) Buenos Aires, Argentina. coni Consejo Nacional de
Investigaciones Científicas y Técnicas, Argentina. ifiba IFIBA, Consejo
Nacional de Investigaciones Científicas y Técnicas, Argentina.

##  1 Introduction

Light sources based on the propagation of solitons in optical fibers have
emerged as a compact solution to the need of a benchtop source of ultra-short
tunable pulses [1, 2, 3]. The soliton formation from femtosecond pulses
launched into an optical fiber is explained in terms of the interplay between
self-phase modulation (SPM) and group-velocity dispersion (GVD) in the
anomalous dispersion regime [4]. The wavelength tunability is a consequence of
the Raman-induced frequency shift (RIFS) produced on the pulse when traveling
through the fiber [5]. The term soliton self-frequency shift (SSFS) [6] was
coined to name this effect widely used to produce tunable femtosecond pulses
in different wavelength ranges, e.g., from $850$ to $1050$ nm [7], from $1050$
to $1690$ nm [8], and from $1566$ to $1775$ nm [1]. In most cases, photonic
crystal fibers (PCF) are used for building these sources since their GVD can
be easily tailored to produce solitons in a desired tuning range [9, 10]. For
a given choice of the PCF, full experimental characterization of the pump and
output pulses, complemented with theoretical predictions, is necessary to
understand how nonlinear effects modify the output soliton.

The wavelength tunability in a PCF-based light source is provided by the
modulation of the pump power injected into the fiber [11, 12, 13, 14]. It is
worth noting that the wavelength choice of the output pulse is done without
moving any mechanical part, which is clearly attractive for all the proposed
and imaginable applications of these soliton sources. Moreover, the wavelength
of the output pulse can be chosen as fast as one can modulate the power of the
pump pulse, as introduced in Ref. [15, 16]. By introducing an acousto-optic
modulator (AOM) in the path of the pump pulse, the output wavelength can be
changed at a speed which is ultimately limited only by the laser repetition
rate. This kind of experimental setup has been presented in some previous
reports [17, 14], with stunning applications as the one presented in Ref.
[18], where a pseudo-CW wideband source for optical coherent tomography is
introduced. However, the need to pre-compress the pump pulse to avoid the
chirp produced by the AOM contrives against the compact and mechanically
robust design of the light source. In this paper, we demonstrate that the PCF-
based source presented here is robust against chirped pump pulses. We present
a complete set of measurements showing that the temporal and spectral
characteristics of the generated solitons in the PCF remain unaltered even
when pump pulses are heavily chirped up to $\sim 7400$ fs2. Results are
presented for the whole range of tunability ($860$ nm to $1160$ nm). We also
present numerical simulations which remarkably fit the experimental data and
help to understand the soliton behavior.

This paper is organized as follows: In section II, we describe the
experimental setup. The numerical simulations are described in section III. In
section IV, we present experimental and numerical results and in section V we
further analyze the results with numerical simulations. Finally, in section
VI, we present our conclusions.

##  2 Experimental Setup

A scheme of the experimental setup is shown in Fig. 1. A Ti:Sa laser (KMLabs)
generates ultrashort transform-limited (TL) pulses of $\mathrm{\Delta t}=31$
fs (FWHM-sech2), $\mathrm{\lambda_{pump}}=830$ nm, with a spectral width
$\mathrm{\Delta\lambda}=23$ nm, and a repetition rate of $94$ MHz.

Figure 1: Experimental setup. (a) Titanium-Sapphire (Ti:Sa) laser, (b) Prism
compressor, (c) Acousto-optic modulator (AOM), (d) Coupling lens, (e) Photonic
crystal fiber (PCF), (f) Collimator objective, (g) Spatial filter, (h) Flipper
mirror, (i) Fast-scan interferometric autocorrelator, (j) Optical spectrum
analyzer (OSA).

The AOM not only allows high speed (up to MHz) and accurate control of the
soliton wavelength, as previously discussed, but also prevents feedback into
the Ti:Sa, replacing the optical isolator required in similar setups [19]. As
the AOM introduces $\sim 56$ mm of SF8 glass path, pump pulses gain a positive
chirp of about $\sim 7400$ fs2, which leads to a time spread by a factor of
$\sim 3$ in them. This can be pre-compensated, for example, by introducing an
optical fiber in the anomalous dispersion regime [20, 8] or a prism compressor
in the well-known configuration presented in [21]. In this work, the chirp was
compensated by a pair of SF18 prisms with an apex separation of $78$ cm.
Additionally, the prism compressor allowed us to up-chirp pump pulses in a
controlled fashion from TL to $\sim 1400$ fs2 by introducing an extra glass
path at the second prism of the arrangement [22]. This full or partial
compensation of the phase distortion introduced by the AOM allowed us to study
the role of different chirp figures in the temporal and spectral
characteristics of the solitons generated in the PCF.

Pump pulses were coupled into a non-polarization-maintaining microstructured
fiber commercially used for supercontinuum generation (Thorlabs,
NL-2.3-790-02). Its main parameters are listed in Table 1 and the dispersion
curve and SEM image are shown in Fig. 2.111Datasheet available in
http://www.thorlabs.com. Upon propagation down the fiber, the spectrum is
highly broadened so a spatial band-pass filter made of a prism and razor
blades, similar to the one presented in [23], allowed to filter the spectral
region of the solitonic branch (see Fig. 3) without adding any extra chirp to
the solitons.

Figure 2: Dispersion curve of the PCF, showing the zero dispersion wavelength (ZDW) at $790$ nm. The inset is the scanning electron microscope image of the PCF core. The curve and image were provided by the manufacturer. L |  $75$ cm
---|---
ZDW |  $790$ nm
$\beta_{2}$ |  $-12.4$ ps2km-1
$\beta_{3}$ |  $0.07$ ps3km-1
$\gamma(\omega)$ | $\gamma_{0}(\omega_{0})+(\omega-\omega_{0})\gamma_{1}$
$\gamma_{0}(\omega_{0})$ |  $78$ W-1km-1
$\gamma_{1}$ | $\gamma_{0}/\omega_{0}$
$\omega_{0}$ |  $2271$ THz
Table 1: PCF parameters relevant to the simulation. Further details can be
found in Ref. [14].

Once the spectral selection was achieved, a flipper mirror directed the
filtered beam for analysis either by the optical spectrum analyzer (OSA) or by
the interferometric autocorrelator. A fast-scan system [24] allows to perform
fast interferometric auto-correlations. Briefly, a platform with a hollow
retroreflector is moved sinusoidally back and forth, with a stepper motor at
11 Hz, to produce and optical delay in one of the arms of a Michelson
interferometer. The autocorrelation signal is recorded by a PMT and averaged
with an oscilloscope.

##  3 Numerical Simulations

In order to further validate experimental results, we simulated the
propagation of femtosecond pulses in the PCF by numerically solving the
generalized nonlinear Schrödinger equation (GNLSE) including dispersive, Kerr,
instantaneous and delayed Raman response, and self-steepening effects [25],
with a conservation quantity error (CQE) adaptive step-size algorithm [26].

The GNLSE reads

$\displaystyle\frac{\partial A}{\partial z}+\beta_{1}\frac{\partial
A}{\partial t}+\operatorname{i}\beta_{2}\frac{\partial^{2}A}{\partial t^{2}}$
(1) $\displaystyle-\beta_{3}\frac{\partial^{3}A}{\partial
t^{3}}+...=\operatorname{i}\gamma(\omega)\left(1+\frac{\operatorname{i}}{\omega_{0}}\frac{\partial}{\partial
t}\right)$
$\displaystyle\times\left(A(z,t)\int_{-\infty}^{\infty}R(t^{\prime})|A(z,t-t^{\prime})|^{2}dt^{\prime}\right),$
$\displaystyle R(t)=\left(1-f_{R}\right)\delta(t)+f_{R}h_{R}(t),$
$\displaystyle h_{R}(t)=\left(f_{a}+f_{c}\right)h_{a}(t)+f_{b}h_{b}(t),$
$\displaystyle
h_{a}(t)=\tau_{1}\left(\tau_{1}^{-2}+\tau_{2}^{-2}\right)\operatorname{e}^{-t/\tau_{2}}\sin{(t/\tau_{1})},$
$\displaystyle
h_{b}(t)=\left[\left(2\tau_{b}-t\right)/\tau_{b}^{2}\right]\operatorname{e}^{-t/\tau_{b}},$

where $A(z,t)$ is the complex envelope of the electric field, $\beta_{n}$ are
the expansion terms for the propagation constant around the carrier frequency
$\omega_{0}$ and $\gamma$ is the nonlinear coefficient. $f_{R}(t)$ represents
the fractional contribution of the delayed Raman effect $h_{R}$. Note that Eq.
(1) adopts a more accurate description of this effect than the one usually
used [4]. In our simulation, we adopted $\tau_{1}=12.2$ fs, $\tau_{2}=32$ fs,
$\tau_{b}=96$ fs, $f_{a}=0.75$, $f_{b}=0.21$, $f_{c}=0.04$, and $f_{R}=0.24$
[27]. The dependence of the fiber non-linear parameter $\gamma$ with the
frequency was modeled as a linear function (see Table 1).

##  4 RESULTS

###  4.1 Transform-limited pump pulses

First, we present the full characterization of the soliton source seeded by TL
pump pulses, in an extended wavelength range if compared with the results
presented in our previous paper [14].

In order to investigate the dependence of the output spectrum with the coupled
power, managed by the AOM, we skipped spectral filtering at first. Fig. 3
shows the measured spectrum at the PCF output as a function of the coupled
power. The infrared solitonic branch appears at $\sim 10$ mW and undergoes
red-shift with increasing power. The maximum wavelength attained is $1130$ nm
at $55$ mW. Spectra in Fig. 3 also shows show that some of the input energy is
converted to visible non-solitonic radiation.

Figure 3: Experimental spectra vs coupled power to the PCF with transform
limited (TL) pump pulses. The color map shows spectral intensity. The maximum
achieved soliton shift, $\lambda_{s}\simeq 1130$ nm, was reached at $55$ W.

The pulsewidth of the filtered soliton as a function of its wavelength,
$\lambda_{s}$, is shown in Fig. 4. The pulsewidth remains constant at $\sim
45$ fs, for the entire tunability range. Numerical simulations are also
plotted in the same figure, showing an excellent agreement with experimental
measurements.

Figure 4: Experimental pulsewidth of the soliton as a function of its
wavelength, pumping the PCF with TL pulses. The results for the three lower
wavelengths were already present in Ref. [14]. Full line: numerical
simulations.

###  4.2 Chirped pump pulses

The effect over the soliton produced by the chirp of pump pulses was studied
systematically by introducing a known amount of extra glass path on the second
prism of the compressor. This scheme allowed to change the GVD of pump pulses
from $0$ to $1400$ fs2. Further chirping was achieved by the complete removal
of the prism compressor, leading to a total amount of positive chirp $\sim
7400$ fs2.

Figure 5:  Soliton temporal (a) and spectral width (b) vs. chirp of input pump
pulses . Full line: numerical results. The soliton wavelength is
$\lambda_{s}=1075$ nm.

Figure 5 (a) shows the pulsewidth of solitons with wavelength
$\lambda_{s}=1075$ nm upon variation of pump pulses chirp. Even for a $\sim
7400$ fs2 chirp, the soliton output pulsewidth remained around $45$ fs.
Numerical simulations show very good agreement with these observations, as
they predict nearly constant pulsewidth regardless of the input chirp (full
line in Fig. 5). Measurements and numerical simulations in the spectral domain
also indicate that the bandwidth of the output solitons is almost unaffected
by the pump pulses chirp [see Fig. 5 (b)]. The product $\Delta t\Delta\nu$ was
found to be near $0.315$, as it is expected for transform-limited sech2
pulses.

The effect of this heavy chirping was evident in the auto-correlation traces
of pump pulses, as can be seen by comparing Fig. 6 (a) and (c). However, there
is not a clear difference between traces of the output solitons for the TL (b)
and the highly chirped ($\sim 7400$ fs2) case (d).

Figure 6:  Interferometric auto-correlation traces of TL (a) and heavily
chirped, $\sim 7400$ fs2, Ti:Sa pump pulses (c). Interferometric auto-
correlation traces of the output solitons are similar in both cases, unchirped
(b) and heavily chirped (d). Soliton wavelength is $\lambda_{s}\simeq 1075$
nm.

A color map of the spectra as function of the coupled power, for a highly
chirped pump pulse ($\sim 7400$ fs2), is shown in Fig. 7. As in the TL case,
we observe that a solitonic branch is red shifted by increasing the coupled
power. However, in this case, $80$ mW of coupled power is required to produce
a $1160$ nm soliton which represents an increment of about $\sim 45\%$ in
comparison to the TL case.

Figure 7: Spectra vs coupled power to the PCF with highly chirped ($\sim 7400$
fs2) input pulses.

A comparison of the soliton red-shift between TL and chirped pump pulse cases
is presented Fig. 8. The figure shows that more power is always required to
attain the same shift when pump pulses are heavily chirped.

Figure 8: Soliton wavelength shift for chirped (full squares) and TL pump
pulses (empty squares) vs pump pulse power. Dashed and full lines correspond
to numerical results for TL and highly chirped pump pulses, respectively.

Figure 9 (a) shows the soliton pulsewidth as a function of its wavelength,
$\lambda_{s}$, when the pump pulse is heavily chirped ($\sim 7400$ fs2). We
observe an approximately constant output pulsewidth ($\sim 45$ fs) in the
entire tuning range. Furthermore, the $\mathrm{\Delta t\Delta\nu}$ product,
shown in Fig. 9 (b), indicates that the generated pulses can be identified as
fundamental solitons (sech2-like), as in the case of TL pump pulses [14].
Numerical simulations were also performed for this case (full lines in Fig. 9)
showing an excellent agreement with experimental results.

Figure 9: (a) Soliton pulsewidth and (b) the product $\mathrm{\Delta
t\Delta\nu}$ vs wavelength in the case of highly chirped pump pulses
($\sim$7400 fs2).

##  5 DISCUSSION

###  5.1 Fiber soliton self-frequency shift effective length

In order to further analyze soliton formation, we studied the pulse evolution
along the fiber by performing numerical simulations. The spectrum evolution
along the fiber, for a given coupled power, in the TL and the chirped cases
are shown in Fig. 10 (a) and (b), respectively. These simulations show that in
the case of chirped pump pulses ($\sim 7400$ fs2), the spectrum broadening and
the soliton formation take place farther down into the fiber (see Fig. 10), as
compared to the TL case.

The delay in the formation of the soliton can be explained by an interplay of
opposite chirping effects: the positive chirp acquired by traversing the AOM
is compensated as the pulse advances into the PCF, in anomalous propagation,
leading to pulse compression. The PCF itself provides pulse compression in the
first stretch of the fiber previously to the branching of a soliton.
Therefore, the SSFS effective length, i.e., the fiber path where nonlinearity
broadens the spectrum, is longer in the TL case. If the chirp is
overcompensated and a negatively chirped pulse is fed into the fiber, these
pulses would also be compressed within the first stretch of the fiber due to
SPM [28] leading to the same behavior than in the positively chirped case,
resulting in a narrower tunability range.

Once the soliton is formed and the peak power is high enough, intrapulse Raman
scattering red-shifts the soliton as it propagates through the remaining of
the fiber. This spectral shift increases with both fiber length and soliton
peak power [4]. So the fact that the soliton is formed at different lengths
explains the red shifts observed for the same coupled power.

However, as a larger wavelength shift can be achieved with a higher input
power, this shortening in the effective length in the chirped case could be
compensated by coupling more power into the PCF [1]. Another possibility for
compensating this effect on the SSFS is using a longer PCF.

Figure 10: Simulated spectral evolution along the fiber length. Pump pulses
with identical peak power produce more soliton shifting with unchirped (a)
than with heavily chirped ($\sim$7400 fs2) (b) pump pulses.

###  5.2 Fiber power conversion efficiency

Figure 11: Simulation of the spectral evolution along the fiber showing that
the same shifting shown in Fig. 10 (a) for unchirped pump pulses is attained
with chirped ones ($\sim 7400$ fs2) with a higher pump power.

Figure 11 shows a simulation where the same shift wavelength obtained for TL
pump pulses is achieved by increasing the coupled power in the shorter
effective length fiber ($7400$ fs2 chirp). Fission of more than one soliton
branch is visible in this case, as compared to the case of TL pump pulses, for
which only a single soliton branch appears (Fig. 10). Each soliton branch
carries a fundamental soliton ($N=1$) with a peak power $P_{0}$ given by [4]

$N^{2}=1=\frac{\gamma P_{0}T_{0}^{2}}{|\beta_{2}|},$ (2)

As $\gamma$, $\beta_{2}$ and $T_{0}$ (see Figs. 4 and 5) are the same in both,
the TL and chirped cases, the peak power of the solitons is also the same.

The arising of new soliton branches partially accounts for the increased pump
power required in the chirped case (Fig. 11) to attain the same shift. Indeed,
the soliton-pump power ratio is $0.2$ in the chirped case and $0.44$ in the TL
case. This result reveals that the use of the PCF as a compressor decreases
its power conversion efficiency.

On the other hand, it is possible to achieve the same soliton shift as in the
TL case by increasing the fiber length, and keeping the same pump power. In
this case, the power conversion efficiency is even lower, $0.17$, as predicted
by simulations.

##  6 Conclusions

We have presented a high-speed tunable soliton infrared source capable of
generating $\sim 45$ fs transform-limited pulses in the range from $860$ to
$1160$ nm. Both the pulsewidth and the spectral width were shown to remain
constant over the entire tuning range, even when pump pulses were heavily
chirped up to $7400$ fs2. Insensitivity to the chirp of pump pulses points out
to the feasibility of avoiding bulky compensation optics prior to the PCF,
opening up the possibility to build reliable and compact high-speed tunable
femtosecond sources in the near infrared region. A minor drawback of this
source is that either more power needs to be coupled or a longer PCF needs to
be used in order to achieve the same tuning range obtained with transform-
limited pump pulses.

###### Acknowledgements.

This work was supported by ANPCyT PICT 2006-1594, ANPCyT PICT 2006-497 and UBA
Programación Científica 2008-2010, Proyecto N X022.

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# A Dynamical Approach to Quasi-Analytic type Problems

Ali Taghavi
Department of Mathematics, Qom university of Technology
Qom, Iran,
<EMAIL_ADDRESS>
###### Abstract

In this paper we give an alternative proof for a vanishing result about flat
functions proved in G.Stoica, ”When must a flat function be identically zero”,
The American Mathematical Monthly 125(7)648-649, 2018. With a dynamical
approach we give a generalization of this result to multidimensional
variables.

## Introduction

Let $M=(m_{k})$ be a sequence of positive real numbers. We consider the class
$C(\\{m_{n}\\})$ of all smooth functions on an interval $[a,b]$ such that for
all $x\in[a,b]$ we have $|f^{(n)}(x)|\leq b^{n}.m_{n}$ where $b$ is a constant
which depends on $f$. A class $C(\\{m_{n}\\})$ of smooth functions is called a
quasi analytic class if for every $f\in C(\\{m_{n}\\})$ if $f^{(n)}(a)=0,\quad
n=0,1,2,\ldots$ at some point $a$ then $f$ must be identically $0$ on $[a,b]$.
This concept generates several results including Denjoy-Carlman theorem. See
[1, 2]. This theory was a motivation for [5] to consider the following
question: Under what functional differential inquality imposed on a flat
function $f$ and its derivative can one conclude that $f$ identically
vanishe?. So this is an strong motivation to extend the concept of quasi
analytic functions via consideration of various functional differential
inequalities. In this direction George Stoica proved the following interesting
result in [5]:

######  Theorem 1.

Let f be a $C^{1}$ real-valued function on $[0,1]$, infinitely differentiable
at $x=0$, and such that $f^{(n)}(0)=0,\;\forall n\in\mathbb{N}\cup\\{0\\}$. If
$|xf^{\prime}(x)|\leq C|f(x)|$ for some $C>0$ and every $x\in[0,1]$, then
$f(x)=0$ for every $x\in[0,1]$

In [4] the author gives a simplified proof for the above theorem.

In this paper, apart from giving an alternative proof for this theorem, we
observe that there is a dynamical nature for the formulation of this theorem.
The method of proof of the following theeorem 2, which generalizes Theorem 1
to the multdimensional variables, represents this dynamical feature.

######  Theorem 2.

Assume that $U$ is a disc around the origin $0\in\mathbb{R}^{n}$. Let
$h:U\to\mathbb{R}^{n}$ with $h=(h_{1},h_{2},\ldots,h_{n})$ be a $C^{1}$ map
with the inner product condition $\langle h(x),x\rangle>0,\forall x\in
U\setminus\\{0\\}$. We further assume that all eigenvalues of the Jacobian
matrix $Jh(0)$ have positive real part. Assume that $f:U\to\mathbb{R}$ be a
$C^{1}$ function which is flat at $0$. If we have
$|\sum_{i=1}^{n}h_{i}\partial f/\partial x_{i}|\leq c|f(x|$ for some constant
$c$ then $f$ is identically $0$ on $U$.

######  Example.

Put $x=(x_{1},x_{2},\ldots x_{n})\in\mathbb{R}^{n}$. Let
$f:\mathbb{R}^{n}\to\mathbb{R}$ be a smooth function which is flat at $0$.
Assume that $|\sum_{i=1}^{n}x_{i}\frac{\partial f}{\partial x_{i}}|\leq c|x|$.
Then $f(x)=0,\forall x\in\mathbb{R}^{n}$. To prove this fact we apply Theorem
2 to $h(x)=x$. The Jacobian matrix $Jh=I_{n}$ whose only eigenvalue is $1$.
Moreover $\langle h(x),x\rangle>0.\quad\forall x\neq 0$

##  Backgrounds from Dynamics and Proofs of the main Theorems

In this section we first recall the Gronwal inequality and flat functions.
Then we provide necessary definitions from the theory of differential
equations and dynamical systems.

######  Gronwal Inequality.

Let $\beta$ and $u$ be real-valued continuous functions defined on an interval
$[a,b]$. If $u$ is differentiable in $(a,b)$ and satisfies the differential
inequality $u^{\prime}(t)\leq\beta(t)u(t).\;\;\forall t\in[a,b]$ then
$u(t)\leq u(a)\exp{\int_{a}^{t}\beta(s)ds}$ for all $t\in[a,b]$.

Let $f$ be a real valued function defined in a neighborhood of
$0\in\mathbb{R}$. We say that $f$ is a flat function at the origin if
$f^{(n)}(0)=0,\;\forall n\in\mathbb{N}\cup\\{0\\}$. Its multivariable version
is the following:
Assume that $U$ is an open subset of $\mathbb{R}^{n}$ containing $0$. A
$C^{1}$ function $f:U\to\mathbb{R}$ is called a flat function at the origin,
if $f(0)=0$ and all its partial derivatives at $0$ exist and vanishes. Namely
$\frac{\partial^{m}f}{\partial x_{i}^{p}\partial x_{j}^{q}}(0)=0,\;\;\forall
m\in\mathbb{N};,1\leq i,i\leq n\;\;p+q=m$. This is equivalent to say that $f$
is infinitely differentiable at the origin and

$\lim_{x\to 0}\frac{|f(x)|}{|x|^{k}}=0\quad\forall k\in\mathbb{N}$ (1)

Now we shall justify the dynamical nature of Theorem 1. First we introduce
some dynamical preliminaries which can be found in [3] :

Let $U$ be an open subset of $\mathbb{R}^{n}$ and $X:U\to\mathbb{R}^{n}$ be a
vector field with $X=(P_{1},P_{2},\ldots P_{n})$. By existence and uniqueness
theorem of the theory of ordinary differential equations we get a flow
$\phi_{t}$ associated to solution curves of

$x^{\prime}=X(x)$ (2)

namely $\phi_{t}(x)$ is the solution curve of (2) staring at point $x$, See[3,
Chapter 8]. The maximal interval of solution starting at $x$, denoted by
$I(x)$, is the maximal interval around $t=0$ where the solution $\phi_{t}(x)$
can be defined on $I(x)$. The orbit of a point $x$ is defined as

$\mathcal{O}(x)=\\{\phi_{t}(x)\mid t\in I(x)\\}$

The positive and negative semi orbits $\mathcal{O}^{\pm}$ defined as

$\\{\phi_{t}(x)\mid t\in I(x),\text{and}\quad t\geq 0(t\leq 0,\quad resp.\\}$

A singularity of $X$ is a point $a\in U$ with $X(a)=0$. A singularity $a$ of
$X$ is called a hyperbolic sink (source, resp.) if all eigenvalues of $JX(a)$,
the Jacobian matrix of $X$ at $a$, have negative(positive, resp.) real part.
The following theorem about hyperbolic sinks and source plays a crucial role
in our paper. Its proof can be found in [3]:

######  Theorem 3.

Let $a\in U$ be a sink of equation $\dot{x}=X(x)$ with flow $\phi_{t}$. Then
there exist an open set $V\subset U$ containing $a$ and constant
$\theta>0,;\lambda>0$ such that $|\phi_{t}(x)-a|\leq\theta e^{-\lambda
t}|x-a|,\quad\forall x\in V,\quad\forall t>0$

Let $X=(P_{1},P_{2},\ldots P_{n})$ be a vector field on an open set
$U\subset\mathbb{R}^{n}$. Assume that $f:U\to\mathbb{R}$ is a $C^{1}$
function. We define $X.f=\sum_{i=1}^{n}P_{i}\partial f/\partial x_{i}$. Then
$X.f$ is the derivation of $f$ along solution curves of $X$. Namely
$X.f(z)=\frac{d}{dt}(f\circ\phi_{t}(z))|_{t=0}$. The linear operator $D$ with
$D(f)=X.f$ is called ” _Derivational operator associated to X_ ”.

Assume that a vector field $X$ has a singularity at a point $a$. Assume that a
function $f$ is defined in a neighborhood of $a$ with a local minimum at point
$a$. We say that $f$ is a Liapunov function for $X$ if $X.f<0$ in a deleted
neighborhood of $a$. Existence of Liapunov functions around a singularity $a$
implies that all solution curves $\phi_{t}(x)$ starting points $x$ in a small
neighborhoods of $a$ tends to $a$ as $t$ goes to infity. This concept is
called stability. See [3, Chapter 9].

######  Remark 1.

With these notations we provided so far, the Theorem 1 can be read in the
following dynamical language:

_Let $X$ be the vector field $X(x)=x$. Assume that $f$ is a flat function at
the origin. Then if $|X.f(x)|\leq C|f(x)|$ for some constant $C$ then $f$
vanishes identically._
So it is natural to consider a dynamical nature for Theorem 1. Such a
consideration enabled us to generalize this theorem to higher dimensional
case.

Now we shall provide a new proof for the result in [5].

######  An Alternative Proof for Theorem 1.

On the contrary, assume that $f$ is not identically zero. Then there exist a
right_isolated zero for $f$. This means that there exist a point
$x_{0}\in[0,1)$ with $f(x_{0})=0$ such that

$f(x)\neq 0\qquad\forall x\in(x_{0},x_{0}+\delta]$ (3)

for some $\delta>0$. Without lose of generality we may assume that $f$ is
positive on the open interval $(x_{0},x_{0}+\delta]$, otherwise we replace $f$
by $-f$. First assume that $x_{0}\neq 0$. We define $u(t)=f(x_{0}e^{t})$ for
$t\in[0,\epsilon)$ where $\epsilon=\ln\frac{x_{0}+\delta}{x_{0}}$. Now we
differentiate $u(t)$ and apply the assumption $|xf^{\prime}(x)|\leq C|f(x)|$
of Theorem 1. Then we have

$\displaystyle u^{\prime}(t)$
$\displaystyle=x_{0}e^{t}f^{\prime}(x_{0}e^{t})\leq|x_{0}e^{t}f^{\prime}(x_{0}e^{t})|$
(4) $\displaystyle\leq C|f(x_{0}e^{t})|=Cf(x_{0}e^{t})=Cu(t)$

By Gronwal inequality we get $0\leq u(t)\leq u(0)e^{ct}=0$. This means that
$f$ vanishes in a right neighborhood of $x_{0}$. This contradicts to the
assumption (3).

Now assume that $x_{0}=0$. So $f$ does not vanish on $(0,\delta]$. For every
fixed $x\in(0,\delta)$ we define
$u(t)=f(xe^{t}),\;\;t\in[0,\ln\frac{\delta}{x}]$. We apply again the Gronwal
inequality to (4) so we obtain $f(xe^{t})\leq f(x)e^{Ct},\;\;\forall
t\in[0,\frac{\delta}{x}]$. We substitute $t=\frac{\delta}{x}$ in the latter
inequality so we have $0<f(\delta)\leq
f(x)\left(\frac{\delta}{x}\right)^{C}\Rightarrow
f(x)>\left(\frac{f(\delta)}{\delta}\right)^{C}x^{C}$. This obviously
contradicts with flatness of $f$ at the origin. This completes the proof of
Theorem 1.

Our proof of the above theorem can be generalized to higher dimensions. This
generalization is given in the proof of Theorem 2 as follows:

######  Proof of theorem 2.

The method of proof is based on usage of Theorem 3 and also on the same method
we used in the proof of Theorem 1. We apply the same method to a given typical
orbit or semi orbit $\mathcal{O}(p)$ or $\mathcal{O}^{\pm}(p)$ of the vector
field $h$ or its negative direction $Y=-h$. The vector field $h$ has a source
at the origin hence the vector field $Y=-h$ has a sink at $0$. We denote by
$\phi_{t},\psi_{t}$ the flow of $Y$ and $h$ respectively. Obviously
$\psi_{t}(x)=\phi_{-t}(x)$. For every $p\in U,\lim_{t\to+\infty}\phi_{t}(p)=0$
since

$Y.(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2})=-h.(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2})=-2\langle
h(x).x\rangle<0$

when $x=(x_{1},x_{2},\ldots,x_{n})\in U\setminus\\{0\\}$. So the function
$(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2})$ is strictly decreasing along the
solution curves of $Y$. In the other words the function
$(x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2})$ is a Lyapunov function for the vector
field $Y$. To prove Theorem 2 we assume on the contrary that there exist a
point $p\in U$ with $f(p)\neq 0$. Since $Y$ has a sink at origin which
attracts all solution curves of disc $U$, the maximal interval of solution
$I(p)$ associated to vector field $h=-Y$ is in the form

$I(x)=(-\infty,\omega(x))$ (5)

. We define a function $u(t):I(x)\to\mathbb{R}$ with $u(t)=f\circ\psi_{t}(p)$.
So $u$ is not identically zero. Furthermore the condition
$|\sum_{i=1}^{n}h_{i}\partial f/\partial x_{i}|\leq c|f(x|)$ implies that $u$
satisfies $u^{\prime}(t)\leq cu(t)$. So one of the following two situations
may occur:

  1. 1.

There exist a right_isolated root $t_{0}$ for $u$. This means that there exist
a point $t_{0}\in I(x)$ with $u(t_{0})=0$ but $u$ does not vanish on a right
neighborhood $(t_{0},t_{0}+\delta)$

  2. 2.

The function $u$ does not vanish on an interval $(-\infty,m)$ for some $m<0$

The first item 1 leads us to contradiction by Gronwal inequality. the function
$u(t)$ satisfies $u(t)\leq e^{ct}u(t_{0})=0$ which contradict to non vanishing
assumption on $(t_{0},t_{0}+\delta)$. Note that a similar argument worked in
the proof of Theorem 1. We consider the second item 2. According to Theorem 3
there exist a neighborhood $V$ around the origin and positive constants
$\theta,\lambda$ such that $|\phi_{t}(x)|\leq\theta e^{\lambda
t}|x|,\quad\forall x\in V,\;\forall t>0$. We fix a point
$q\in\mathcal{O}^{-}(p)$ which belongs $V$. For every $x=\phi_{t}(q),\quad
t>0$ we shall apply the Gronwal inequality to function $w(s)=f(\psi_{s}(x))$
defined on $[0,t]$. Again the assumptions of Theorem 2 imply that
$dw/ds=\dot{w}\leq cw$ hence $w(s)\leq e^{cs}w(0)$. So we have

$\displaystyle f(q)=w(t)\leq e^{ct}w(0)=e^{ct}f(\phi_{t}(q))\Rightarrow$
$\displaystyle f(q)\left(\frac{|\phi_{t}(q)|}{|q|}\right)^{c/\lambda}\leq
f(q){(e^{-\lambda t})}^{c/\lambda}\leq f(q)e^{-ct}\leq f(\phi_{t}(q))$

So we obtain

$k|\phi_{t}(p)|^{c/\lambda}\leq f(\phi_{t}(p))\qquad\forall t>0$

where the constant $k$ is $k=\frac{f(q)}{|q|^{c/\lambda}}$ Thanks to the
equivalent formulation of flatness property ‘(1) this contradicts to the fact
that $f$ is flat at origin since $\phi_{t}(q)$ tends to the hyperbolic sink
$0$ as $t$ goes to $-\infty$. This completes the proof of Theorem 3

## Discussions and further Reseaches

In this paper we presented a differential operator interpretation for the
assumption $|xf^{\prime}(x)|\leq c|x|$ in[5]. We considered the differential
operator $D(f)=xf^{\prime}(x)$ so we translated the main problem as follows:
Every flat function $f$ with $|D(f)(x)|\leq c|x|$ must be vanished
identically. This situation leads us to the following definition and
questions:

let $D$ be a differential operator which acts on the space of all smooth
functions on $\mathbb{R}^{n}$ or any arbitrary manifold. We say that $D$ is a
$G.S.$ operator at point $p$ if for every flat function locally defined around
$p$ which satisfies $|D(f)(x)|\leq c|x|$ then $f$ must vanish identically in a
neighborhood around $p$. We learn from [5] that the differential operator
$D(f)=xf^{\prime}(x)$ is a $G.S.$ operator at origin. Moreover we prove in
this paper that the operator $D(f)=\sum_{i=1}^{n}P_{i}(\partial f/\partial
x_{i})$ is a multidimensional $G.S.$ operator at the origin if $0$ is a
hyperbolic sink(or source) for the vector field
$X=(P_{1},P_{2},\ldots,P_{n})$. These situations suggest some proposals as
follows:

  * •

Introducing more examples of $G.S.$ differential operators and a possible
classification of all such kind of operators

  * •

Consideration of the previous proposal for the particular operator $\Delta$,
the Laplace operator on a Riemannian manifold. Determination of all Riemannian
manifold $M$ for which the Laplacian is a $G.S.$ operatore at all points. On
the opposite extreme one can think to classification of all Riemannian
manifold $M$ for which the operator $\Delta$ is a $G.S.$ operator at no point
of $M$

  * •

To a vector field $X$ with corresponding differential operator $D(f)=X.f$, we
associate the set of all points $p$ such that $D$ is a $G.S.$ operator at $p$.
So it would be interesting to study this set from a dynamical point of view.

Apart from the above geometric proposals arising from the concept introduced
in [5], in the following remark we present a different question with a
functional analysis nature:

######  Remark 2.

The assumption $|D(f)(x)|\leq c|x|$ is suggesting a concept stronger than
usual bounded ness of operator $D$ when we consider the sup norm on an
appropriate function space. So this is a motivation to ask the following
question when we reduce this strong continuity to the standard boundedness of
$D$;

Let $V=\\{f\in C^{\infty}[0,1]\mid f\quad\text{is flat at 0}\\}$. We equip $V$
with the norm $|\quad|_{\infty}$. Is there an infinite dimensional subspace
$W\subset V$ which is invariant under $D(f)=xf^{\prime}(x)$ and $D$ is a
bounded operator on $W$?

We observe that the concept of quasi analytic function was the main motivation
for consideration of the differential inequality $|xf^{\prime}(x)|\leq c|x|$
in [5]. The classical concept of quasi analytic functions involves a countable
functional inequalities. So a general and natural question is the following:

######  Question.

Can we study a typical quasi analytic class of functions via a unique
functional differential inequality? If there are some partial affirmative
answer to this question, according to existence of various dynamical
interpretations for a functional differential inequality, we ask the next
question: Is it reasonable and promising to follow some dynamical approaches
to classical quasi analytic problems?

## References

  * [1] T. Carleman, _Les fonctions quasi-analytiques_ , volume 7 of Collection de monographies sur la theorie des fonctions publiee sous la direction de M. E. Borel. Gauthier-Villars, Paris, 1926
  * [2] A. Denjoy. _Sur les fonctions quasi-analytiques de variable reelle_ , C.R. Acad. Sci. Paris, 173:1329–1331, 1921
  * [3] M.W.Hirsch,S. Smale, _Linear algebra, differential equations and dynamical systems_ , Academic Press, 1974.
  * [4] H.Lou, _Remark on “When Must a Flat Function be Identically 0?”_ , The American Mathematical Monthly, 126(10) 945-945, 2019.
  * [5] G. Stoica, _When must a flat function be identically zero_ , The American Mathematical Monthly 125(7)648-649,2018.