# Gap solitons and nonlinear Bloch states in Bose-Einstein condensates with
current-dependent interactions

Jintao Xu  School of Science, Xi’an University of Posts and
Telecommunications, Xi’an, China  Qian Jia  School of Science, Xi’an
University of Posts and Telecommunications, Xi’an, China  Haibo Qiu
<EMAIL_ADDRESS>School of Science, Xi’an University of Posts and
Telecommunications, Xi’an, China  Antonio Muñoz Mateo<EMAIL_ADDRESS>Departamento de Física, Universidad de La Laguna, La Laguna, Tenerife, Spain

###### Abstract

We show how the chiral properties of Bose Einstein condensates subject to
current-density interactions and loaded in optical lattices can be observed in
the realization of nonlinear Bloch states, whose spectrum lacks the usual
periodic structure. Chirality is also manifested by spatially localized
states, or gap solitons, which are found for positive rotation rates of the
lattice at the energy gaps between the linear energy bands, whereas for
negative rotations they appear in the semi-infinite gap of the linear
spectrum. The stability of extended and localized states is checked through
the spectrum of linear excitations and nonlinear time evolution of perturbed
states, and the phenomenon of Bloch oscillations is explored. Our results are
obtained in quasi 1D ring geometries with feasible experimental parameters.

##  I Introduction

Synthetic gauge fields that depend locally on the density of matter have been
recently realized in ultracold-atom settings Clark  _et al._ (2018); Görg  _et
al._ (2019); Yao  _et al._ (2022); Frölian  _et al._ (2022). The unusual
properties of these systems were theoretically predicted by means of a non-
local unitary transformation that maps the density-dependent gauge into a
current-dependent interaction Aglietti  _et al._ (1996); Jackiw (1997). Bose-
Einstein condensates (BECs) endowed with such interparticle interactions were
shown to exhibit chiral properties in a free expansion, the onset of
persistent currents, or the center of mass oscillations Edmonds  _et al._
(2013, 2015); additionally, in the absence of external potential, it was
demonstrated that chiral bright solitons can exist only if they move along one
(but not the opposite) direction Aglietti  _et al._ (1996); Dingwall and
Öhberg (2019); Bhat  _et al._ (2021); Frölian  _et al._ (2022), and that
collisions between them differ significantly from those between regular
solitons Dingwall  _et al._ (2018); Jia  _et al._ (2022).

Many aspects of this chiral theory remain unexplored, and very recent
experimental realizations Frölian  _et al._ (2022) open new prospects for
testing the theoretical predictions. A significant subject that was previously
restricted to solid state systems, the matter dynamics in periodic potentials,
became also accessible to the field of ultracold gases with the realization of
optical lattices Morsch and Oberthaler (2006); Schäfer  _et al._ (2020). As
far as we know, this subject has still not been addressed within the chiral
theory.

In this work, we focus on BECs that are loaded in optical lattices and subject
to interactions that depend locally on the current density. The lattice is
assumed to be imprinted on a quasi-1D ring, as generated by a tight transverse
confinement of the atoms, and to be able to rotate. In particular, within the
framework of a generalized Gross-Pitaevskii equation, we study the properties
of nonlinear Bloch waves and gap solitons, and demonstrate their unusual
properties. While the energy dispersion of the former states loses the usual
periodic structure, and new non-regular Bloch states emerge, the situation of
gap solitons within the energy gaps changes drastically with the direction of
the rotation rate. We analyze the spectrum of linear excitations and show that
the stability of both extended and localized states is also conditioned by the
chiral properties. Finally, we perform numerical simulations of the equation
of motion to explore the stability of stationary states against small
perturbations, and also the existence of Bloch oscillations in the presence of
current-density interactions.

##  II Model

We assume that the condensate wave function $\psi(x,t)$ follows a generalized
Gross-Pitaevskii equation in a ring of radius $R$:

$i\hbar\frac{\partial\psi}{\partial
t}=\left[\frac{(-i\hbar\partial_{x}-m\Omega R)^{2}}{2m}+U_{\rm
latt}+\hbar\kappa J\right]\psi,$ (1)

where $U_{\rm latt}(x)=U_{0}\,\sin^{2}(\pi x/d)$ is the lattice potential,
with amplitude $U_{0}$ and lattice spacing $d$, which can rotate with angular
velocity $\Omega$. The strength of the current-dependent mean field is
measured by the dimentionless parameter $\kappa$, and
$J(x,t)=\hbar(\psi^{*}\partial_{x}\psi-\psi\partial_{x}\psi^{*})/(i2m)$ is the
current density in the lab frame. The total number of particles $N=\int
dx|\psi|^{2}$, and the total energy (which does not include an explicit
dependency on the current density)

$E=\int dx\,\psi^{*}\left[\frac{(-i\hbar\partial_{x}-m\Omega
R)^{2}}{2m}+U_{\rm latt}\right]\psi,$ (2)

are conserved quantities Aglietti  _et al._ (1996). We particularize our
analysis on an $M-$site lattice over the ring, so that $2\pi R=M\,d$. As
energy reference, we will make use of the lattice recoil energy
$E_{L}=\hbar^{2}(\pi/d)^{2}/(2m)$ Morsch and Oberthaler (2006).

The stationary states take the form $\psi(x,t)=\psi(x)\,\exp(-i\mu t/\hbar)$,
with $\mu$ as the energy eigenvalue. In the search of stationary states, it is
worth noticing that, from the continuity equation
$\partial_{t}|\psi|^{2}+\partial_{x}(J-|\psi|^{2}\,\Omega\,R)=0$, the current
density fulfills $J-|\psi|^{2}\,\Omega\,R=J_{0}$, where $J_{0}$ is a constant,
thus it transforms the equation of motion (1) into the regular time-
independent Gross-Pitaevskii equation

$\displaystyle(\mu-\hbar\kappa\,J_{0})\psi=\left[\frac{(-i\hbar\partial_{x}-m\Omega
R)^{2}}{2m}+U_{\rm latt}+g_{\Omega}\,|\psi|^{2}\right]\psi,$ (3)

where the effective constant-interaction strength is
$g_{\Omega}=\hbar\kappa\Omega R$. Furthermore, if the lattice does not rotate,
the initial nonlinear equation (1) is transformed into the time-independent
Schrödinger equation

$\displaystyle(\mu-\hbar\kappa\,J_{0})\psi$
$\displaystyle=\left[-\frac{\hbar^{2}}{2m}\partial_{x}^{2}+U_{\rm
latt}\right]\psi,$ (4)

with the current constraint
$J_{0}=|\psi(x)|^{2}\,{\hbar}\partial_{x}\theta(x)/m$, where
$\theta=\arg\,\psi(x)$ is the phase.

The linear excitations $\delta\psi_{j}=[u_{j},\,v_{j}]^{T}$ of stationary
states $\psi(x,t)\rightarrow\exp(-i\mu
t/\hbar)\,\\{\psi(x)+\sum_{j}[u_{j}(x)\,\exp(-i\omega_{j}t)+v_{j}(x)^{*}\,\exp(i\omega_{j}^{*}t)]\\}$,
with $j$ being a mode index, can be obtained through the Bogoliubov’s
equations $B\delta\psi_{j}=\hbar\omega_{j}\,\delta\psi_{j}$. The Bogoliubov
matrix is given by

$\displaystyle B=\begin{pmatrix}H_{\mbox{\tiny GP}}+i\kappa B_{uu}-\mu&i\kappa
B_{uv}\\\ i\kappa B^{*}_{uv}&-H_{\mbox{\tiny GP}}^{*}+i\kappa
B^{*}_{uu}+\mu\end{pmatrix},$ (5)

where $H_{\mbox{\tiny GP}}=(-i\hbar\partial_{x}-m\Omega R)^{2}/2m+U_{\rm
latt}+\hbar\kappa J$ is the Hamiltonian operator in Eq. (1),
$B_{uu}=\hbar^{2}(\psi{\partial_{x}\psi^{*}}-|\psi|^{2}{\partial_{x}})/2m$,
and $B_{uv}=-\hbar^{2}(\psi\partial_{x}\psi-\psi^{2}{\partial_{x}})/2m$.
Linear excitations with complex frequencies, $\Im(\omega_{j})\neq 0$, lead to
the exponential growth (in the linear regime) of small perturbations on the
stationary state that can produce its decay during time evolution.

##  III Extendend and localized eigenstates

In a non-rotating linear system ($\Omega=0$ and $\kappa=0$), the dispersion
relations of quantum states in a ring lattice consist of energy bands
separated by energy gaps Kittel and McEuen (2018). The corresponding spectrum
of eigenstates can be described in terms of Bloch waves
$\psi_{n,k}(x,t)=\exp[i(kx-\epsilon_{n,k}t/\hbar)]\,u_{n,k}(x)$, with
eigenenergies $\epsilon_{n,k}$, where $n=1,\,2\,,\dots$ identifies the band
number, and $k=q\,k_{0}$, with $q=0,\,\pm 1,\,\pm 2,\dots$ and $k_{0}=1/R$, is
the wavenumber associated with the quasimomentum $\hbar k$. The functions
$u_{n,k}(x)$ share spatial period with the lattice $u_{n,k}(x+d)=u_{n,k}(x)$,
so that the probability density profile is homogeneous over the lattice sites.
If the lattice is finite, and contains $M$ sites, there are just $M$ values of
quasimomentum Kittel and McEuen (2018).

In a system with varying contact interactions, the linear Bloch waves have
been shown to find continuation as nonlinear Bloch waves when the interactions
are switched on Morsch and Oberthaler (2006); Muñoz Mateo  _et al._ (2019). We
will show that this continuation also exists in the presence of current-
density interactions. In addition, differently to the case of contact
interactions, there exist new extended states with a non-homogeneous density
profile over the lattice sites.

###  III.1 Nonlinear Bloch waves

First, we focus on the dispersion of the system at $\Omega=0$ for varying
quasimomentum. Insight can be obtained from the comparison with a system
subject to contact interactions (hence following the usual GP equation); in
this case, the linear energy bands are shifted to higher energies when the
interaction is repulsive, whereas the opposite happens for attractive
interaction. Therefore, in the presence of current-density interaction, where
the effective interaction changes from repulsive to attractive according to
the sign of the particle current, the resulting dispersion curves are expected
to be asymmetric with respect to the value of quasimomentum, with energies
higher than the linear bands for states with positive currents, and lower than
the linear bands for states with negative currents.

As can be seen in Fig. LABEL:fig:bands(a), this is indeed the scenario shown
by our numerical results for a ring lattice with $M=10$ sites and shallow
depth $s=2$. The number of particles has been fixed for the nonlinear states
considered to give $\kappa N/(2\pi R)=0.47/d$; correspondingly, the order of
magnitude of the nonlinearity in Eq. (3) is $g_{\Omega}N/(2\pi
R)=4.7(R/d)\hbar\Omega$. For comparison, contact interaction cases are
represented, and have been parameterized by the non-dimensional quantity
$g=m\,d\,g_{\rm 1D}/\hbar^{2}$, where $g_{\rm 1D}$ is the one-dimensional
contact interaction strength, so that $|g|=\kappa$. The lowest energy band of
the linear system (dotted line) lays in between the lowest chemical potential
bands of nonlinear systems with positive (open symbols joined by dashed lines
to guide the eye) and negative (open symbols joined by dot-dashed lines)
contact interactions, whereas the energy eigenvalues of the system with
current-density interaction (filled symbols) vary as predicted. The states
with minimum (zero) and maximum ($k=5\,k_{0}$) values of quasimomentum have no
particle currents, thus they follow a linear (Schrödinger) equation of motion,
and match the energy of the linear bands.

Four instances of Bloch states, with quasimomentum wavenumber
$k/k_{0}=1,\,-1,\,5,$ represented by their density and phase profiles, are
shown in Fig. LABEL:fig:bands(b). While states with opposite currents but
equal absolute value of quasimomentum $|k|$ do not show appreciable
differences in the density profile, states with different $|k|$ do, reflecting
the increasing interaction associated to higher $|k|$.

###  III.2 Non-regular Bloch states

An interesting novelty of the systems with current-density interactions,
contrary to the case of contact interactions, is the existence of extended
stationary states that do not conform to the usual picture of Bloch states,
since they present a non-homogeneous density profile over the lattice sites.
They do not conform either to the features of alternative states hosting dark
solitons in the lattice Machholm  _et al._ (2004). We will refer to them as
non-regular Bloch states, since the quasimomentum, associated with the phase
winding number $q=k/k_{0}$, is still a well defined quantity. As a general
aspect, the higher the variation between density peaks of lattice sites that
they present, the lower the constant current density becomes. Our numerical
results for characteristic quantities suggest a continuum of non-regular Bloch
states, for we were able to find close states with very small differences, of
the order of 1‰ in energies. The linear Eq. (4) sheds light on this
phenomenon, since it admits solutions as linear superpositions of states with
the same eigenvalue $\mu-\hbar\kappa\,J_{0}$ and current density $J_{0}$; such
superpositions are possible because the linear Bloch states are doubly
degenerate.

Figure LABEL:fig:bands(c) shows the density (top) and phase (bottom) profiles
of three non-regular Bloch states with winding number $q=k/k_{0}=1$. They
range from almost homogeneous $\psi_{1a}$, to intermediate $\psi_{1b}$, and up
to large variation $\psi_{1c}$, in the density peaks. The density modulation
over the whole lattice has the form $n_{k}\,[1+\beta\,\sin(k_{0}\,x)]$, where
$n_{k}$ is the density of the homogenous Bloch state, and $\beta$ varies from
almost zero for $\psi_{1a}$, to $\beta=0.6$ for $\psi_{1c}$. Despite the large
differences in the density profile, their energy eigenvalues differ in less
than 1$\%$, and their energies in less that 1‰. The phase profiles show also
small differences and follow a monotonic increase in the range $[0,\,2\pi]$.

Figure 2:  Trajectories of nonlinear Bloch waves (solid and dashed lines) and
gap solitons (thin lines with symbols) in a ring lattice moving with angular
rotation $\Omega$. All states contain the same number of particles, so that
the varying rotation translates into a varying interaction. Nonlinear Bloch
waves are characterized by the wavenumber $k$ that indexes their
quasimomentum, whereas gap solitons trajectories differ for positive (open
circles) and negative (open triangles) currents. Left: Energy eigenstates
measured in units of the lattice recoil energy $E_{L}$; the underlying energy
bands of the linear problem (shaded regions) are represented for comparison.
Right: Average current density in units of $J_{0}=N\Omega_{0}$.

###  III.3 Lattice rotation and gap solitons

Gap solitons are localized states in systems loaded in optical lattices; they
usually occupy a few sites, a small region of the whole lattice. Although
their existence can be easily understood in systems with attractive
interactions, similarly as in translation-invariant settings, their emergence
in the presence of repulsive interactions is a priori not that evident Louis
_et al._ (2003), and can be explained through the sign change induced by the
lattice in the effective mass of the particles (see for instance Ref. Kramer
_et al._ (2003)). In this work, the current-density interaction provides both
possibilities for the emergence of gap solitons, which become distinct for
positive and negative current densities. Since the interaction (or
nonlinearity) is necessary for the solitons to exist, the ring lattice has to
rotate in order for the gap solitons to emerge.

Although the finite lattice considered here, having $M$ sites, allows for only
$M$ Bloch waves, the introduction of rotation gives access to the continuous
spectrum of the infinite lattice Muñoz Mateo  _et al._ (2019). In systems with
Galilean symmetry (as happens for contact interactions), for varying rotation
rate $\Omega\in(-M/2,M/2]\times\Omega_{0}$, where $\Omega_{0}=\hbar/(mR^{2})$,
the energy of each Bloch wave $\epsilon_{n,k}(\Omega)$ in the finite lattice
[as obtained from Eq.(2)] reproduces the energy band profile against
quasimomentum in the first Brillouin zone $k\in(-\pi/d,\pi/d]$ of the infinite
lattice with $\Omega=0$. The dipersion graph, $\epsilon_{n,k}$ versus
$\Omega$, is also useful in understanding the emergence of gap solitons; the
energy degeneracies found in this graph for the linear system, which
correspond to crossings of Bloch-wave trajectories, provide the origin of gap
solitons when the interparticle interactions are switched on. Thus, gap
solitons are the nonlinear continuation of linear states made of Bloch-wave
superpositions Muñoz Mateo  _et al._ (2019).

Figure 2 shows our numerical results for stationary states in a moving ring
lattice with current density interactions and same parameters as in Fig.
LABEL:fig:bands. The eigenenergy (left) and current density (right) of both
nonlinear Bloch states and gap solitons are represented for fixed number of
particles. Gap solitons spread when approaching the energy bands (light-gray
shaded regions in the graph), and then become dynamically unstable Louis  _et
al._ (2003). Eventually, as happens in the present case at the bottom of the
second energy band, they extend to the whole system (or stop existing in an
infinite lattice) when entering a band Muñoz Mateo  _et al._ (2019). Overall,
the chirality of the system manifests as asymmetric trajectories for positive
and negative rotation rates of the lattice. In addition, as we demonstrate
next, apparent differences arise in the states belonging to these families,
which show distinct density profiles and stability properties.

Figure LABEL:fig:gs(a) shows the density and phase profiles of two typical gap
solitons with the same number of particles and opposite lattice rotation. For
positive rotation rate (top panel) the soliton is situated between the first
and second energy bands, corresponding to the family represented by lines with
open circles in Fig. 2. For negative rotation (bottom panel) the soliton
belongs to the family lying in the semi-infinite gap, indicated by lines with
open triangles in Fig. 2. The latter soliton, having negative current and then
effective attractive interparticle interaction, is comparatively more compact
than the former, and occupies just one lattice site. On the contrary, as can
be seen in Fig. LABEL:fig:gs(b), the density profiles of two nonlinear Bloch
waves with equal quasimomentum $k=5\,k_{0}$ but opposite lattice rotations,
hence opposite current densities, are almost indistinguishable
(notwithstanding, the differences become clearer for increasing number of
particles).

###  III.4 Linear stability analysis

We have studied the linear stability of the stationary states reported in Fig.
2 by numerically solving the corresponding Bogoliubov’s equations (5). Before
analyzing our results, it is insightful to remind the scenario of equivalent
states with contact interparticle interactions; we particularize it for
otherwise equal parameters as in Fig. 2. In such a case, while there is no
difference regarding the sign of the quasimomentum, the dynamical stability
depends strongly on the character of the contact interactions, either
repulsive or attractive. For the former case, it is known that Bloch states
close to the edge of the Brillouin zone become unstable; how close depends in
turn on the strength of the interactions and the lattice depth Wu and Niu
(2003). The opposite happens for attractive contact interactions, where Bloch
states close to zero quasimomentum become unstable; in this latter situation,
one can understand the source of instability as associated with smoother,
slowly varying density profiles, on which modulation instability can operate
due to the existence of lower energy states with localized density profiles.
Regarding fundamental (one main peak) gap solitons, althouth instabilities can
be found when their chemical potential approaches a linear energy band, they
are usually stable states.

Our results in current-interacting systems show, in general, a trend similar
to the scenario of contact interactions. The main difference resides in the
asymmetry between positive and negative quasimomenta, which can be mapped into
systems with effective positive and negative contact interactions,
respectively. Another particular difference is observed in the stability of
gap solitons with positive rotation rates for the lattice depth considered in
Fig. 2, $s=2$, for which we have not found stable cases, despite the fact that
the energy of the unstable modes (having complex frequencies) can be very
small in comparison with the corresponding energy eigenvalue $\mu$. Still, we
did find stability for these solitons at higher values of the lattice depth.

Several examples of these general features on linear stability are presented
in Fig. LABEL:fig:modes; both the real and imaginary part of the excitation
modes are shown for each state considered. The panels (a) show the excitation
energies of two nonlinear Bloch waves with the same absolute value of
quasimomentum $|k|=4k_{0}$ (close to the edge of the Brillouin zone) in a
lattice at rest $\Omega=0$. While the state with negative quasimomentum is
dynamically stable, the positive-quasimomentum state is not. The scenario is
analogous to systems with contact interparticle interactions, hence our
results are the opposite (stable for positive rotation and unstable for
negative rotation) for states with quasimomentum $|k|=k_{0}$ (not shown).

The panels in Fig. LABEL:fig:modes(b) represent the energy excitations of the
two gap solitons shown in Fig. LABEL:fig:gs. As anticipated, the soliton with
negative rotation is dynamically stable, while the soliton with positive
rotation, $\Omega=\Omega_{0}$, presents complex frequencies that can cause the
dynamical decay of this stationary state in a real-time evolution (see next
section). However, for the same parameters but in a deeper lattice, $s=3$, we
have found that the corresponding soliton becomes stable.

##  IV Dynamics

Despite the linear analysis performed in the previous section, the stability
of stationary states can only be ensured through the nonlinear time evolution
of the system. To this end, we have solved the time-dependent Eq. (1) for
initial stationary states on which perturbative noise has been added. Although
our results for the subsequent time evolution are consistent with the
predictions of the linear stability analysis, we have also found some
interesting cases whose dynamics show features of structural stability (small
variations of the stationary state profile that do not break qualitatively its
structure) despite the presence of unstable linear modes.

Figure LABEL:fig:evolution(a) depicts three snapshots for selected times
($t=0,\,T/2,\,T$) of a time evolution, with total time $T=25\;md^{2}/\hbar$,
of the linearly-unstable soliton of Fig. LABEL:fig:gs(a) with positive
rotation rate. As can be seen, it confirms the linear prediction of
instability shown in Fig. LABEL:fig:modes(b), since it displays the tunneling
of particles into nearby lattice sites as time passes. The linear analysis
predicts also the instability of the non-linear Bloch states with $k=5k_{0}$
and opposite rotation rates shown in Fig. LABEL:fig:gs(b), with unstable
linear modes of higher energy in the case of positive rotation. However the
time evolution, Fig. LABEL:fig:evolution(b), shows a distinct nonlinear
dynamics for them. While the positive rotation Bloch state evolves into a non-
symmetric structure that is highly variable in time, the negative rotation one
exhibits a small breathing dynamics that do not alter the geomteric shape of
the initial state. We have checked that different types of small perturbations
lead to the same conclusions, and not different dynamics is shown for
evolution times much longer (up to ten times) than the case shown in Fig.
LABEL:fig:modes(b).

One of the most interesting features of the dynamics in optical lattices is
the emergence of Bloch oscillations. This phenomenon has been shown to appear
in BECs with contact interparticle interactions when the lattice velocity is
slowly ramped up Choi and Niu (1999); Morsch  _et al._ (2001). It manifests
the periodic nature of the system through the ground state transit from
positive to negative quasimomentum states over the first energy band. As a
consequence, the system’s average velocity oscillates with respect to the
lattice velocity. But differently to linear systems, the nonlinearity
introduced by the interactions in BECs can lead to the breakdown of Bloch
oscillations, which is associated with the instability of Bloch states close
to the edge of the Brillouin zone Wu and Niu (2003). In addition, the energy
band structure changes with increasing interactions, so that, precisely at the
edge of the Brillouin zone, it develops a cusp, first, and a swallow tail
configuration, later, that prevents the adiabatic transit between different
quasimomentum statesWu and Niu (2003).

In what follows, we make an exploration of Bloch oscillations in systems with
current-dependent interactions. On examination of the dispersion of nonlinear
Bloch states for varying rotation, Fig. 2, and despite the lack of symmetry
with respect to the rotation direction, one can expect the transit from
positive to negative velocities to take place if the states passed across for
varying rotation are dynamically stable, or if, these states being unstable,
their instability modes grow at slower rate than the transit speed.

The expected period of Bloch oscillations is

$T_{B}=\frac{2\pi\hbar}{m\alpha Rd},$ (6)

where $\alpha R$ is a constant acceleration in the ring, since this is the
time taken to cross the first Brillouin zone from quasi-momentum $k=-\pi/d$ to
$k=\pi/d$. The influence of the nonlinearity on the oscillations in systems
with contact interactions can be captured, at least for smooth density
profiles, by effectively modifying the lattice depth as
$U_{0}^{\rm(eff)}=U_{0}/(1+4{g_{\rm 1D}\bar{n}}/{E_{L}})$ Choi and Niu (1999),
where $\bar{n}$ is the average number density. This effective potential
provides us with a way to account for the current-density interactions on
Bloch oscillations by means of an equivalent effective lattice depth

$U_{0}^{\rm(eff)}=U_{0}\,\left(1+4\frac{\hbar\kappa\bar{J}}{E_{L}}\right)^{-1},$
(7)

where $\bar{J}$ is the average current density.

Guided by Eqs. (6) and (7), we have chosen the initial ground state in Fig.
LABEL:fig:bands, with $k=0$, and have imparted a constant angular acceleration
$\alpha$ to the ring lattice. The results are shown in Fig. LABEL:fig:oscill
for the absolute value of the angular acceleration
$|\alpha|/\Omega_{0}^{2}=2$. Figure LABEL:fig:oscill(a) represents the
relative rotation of the evolved state with respect to the lattice for
negative and positive signs of the acceleration. The observed period of the
Bloch oscillations is consistent with Eq. (6), with the zero crossings (or
points of zero relative velocity) reached at times $t_{j}$ that are integer
multiples of $T_{B}/2$, i.e $t_{j}=jT_{B}/2$, and $j=0,1,2,\dots$ For $j$ odd,
the system transits through the edge of the Brillouin zone and the wave
function presents M nodes (associated with the formation of standing waves by
Bragg reflections Kittel and McEuen (2018)). The latter fact can be
interpreted as the presence of M dark solitons caused by the M cores of
corresponding vortices in their transition from the outer part to the inner
part of the ring Muñoz Mateo  _et al._ (2019). This view is better understood
by monitoring the expectation value of the linear velocity measured in ring
units $\hbar/mR$, as depicted in the bottom panel of Fig. LABEL:fig:oscill(a).
As can be seen, the linear velocity is approximately constant around integer
values of $t/T_{B}$, that is around $t_{j}=l\,T_{B}$, for $j=2l$ even, and
$l=0,1,2,\dots$, when the system transits through the center of the Brillouin
zone and the velocity takes the value $\sim lM\hbar/mR$, as due to $l\times M$
vortices that have entered the inner part of the ring.

Apparent differences can be observed in the smoothness (and also in the
duration, see below) of the Bloch oscillations that depend on the sign of the
acceleration. Qualitatively, they can be explained by the effective lattice
depth of Eq. (7). Since positive (respectively negative) rotations translate
into positive (resp. negative) current densities, they decrease (resp.
increase) the effective lattice depth and make the lattice progressively less
(resp. more) relevant. Contrary to the case of contact interactions, this
causes the monotonic variation of the amplitude of Bloch oscillations (the
maximum relative speed) after each zero crossing; it keeps increasing (resp.
decreasing) for positive (resp. negative) acceleration. For high negative
values of the current density the superfluid features of the system tend to
disappear, the system state becomes strongly localized, and the current
density approaches the lattice rotation, which is reflected in the bottom
panel of Fig. LABEL:fig:oscill(a) by a progressively straightened curve,
without the marked plateaus of the curve with positive acceleration. On the
other hand, the latter adiabatic curve becomes accompanied at higher currents
by higher frequency oscillations, related to Landau-Zener tunneling Choi and
Niu (1999), between configurations with different density modulation [see Fig.
LABEL:fig:oscill(b)].

We finish by commenting briefly on the effect of different accelerations. As
shown in Fig. LABEL:fig:oscill, for $|\alpha|/\Omega_{0}^{2}=2$ the decay of
Bloch oscillations does not appear before $2T_{B}$. However, at the lower
acceleration of $\alpha=-0.5\Omega_{0}^{2}$ (not shown) the oscillations
hardly complete one whole period $T_{B}$ before breaking down, producing the
decay of the homogeneous profile over the lattice sites into more localized
density peaks. This decay resembles the action of modulation instabilities
Zakharov and Ostrovsky (2009). Therefore, our results point to the fact that
Bloch oscillations that proceed with (still adiabatic, but) higher
acceleration last longer, which we attribute to the fact that in this case the
unstable modes have a shorter time to grow.

##  V Conclusions

We have reported on gap solitons and nonlinear Bloch states in a rotating ring
lattice within a theory with current-density interactions. Our results show
that the presence of chirality is manifest in all the states considered,
including their spectrum of linear excitations and the display of Bloch
oscillations for constant angular acceleration. A novelty is the existence of
stationary and dynamically stable non-regular Bloch states characterized by a
modulated density profile.

The recent experimental achievement of this theory in Bose-Einstein
condensates of ultracold atoms Frölian  _et al._ (2022) opens the way for the
experimental realization of the states and phenomena that we have described.
Although currently optical lattice potentials in a ring are experimentally
available Sakamoto  _et al._ (2013); Yamane  _et al._ (2016), our results are
not restricted to this geometry, and can also be realized in one dimensional
linear lattices, as has been routinely done in the presence of contact
interatomic interactions Morsch and Oberthaler (2006); Schäfer  _et al._
(2020).

Future prospects of our work include the study of fundamental and higher-order
soliton states in different energy gaps, and the extension to realistic 2D or
3D systems that reach the quasi-1D regime.

###### Acknowledgements.

JX, QJ, and HQ acknowledge support by the National Natural Science Foundation
of China (Grant No. 11402199), the Natural Science Foundation of Shaanxi
Province(Grant No. 2022JM-004, No. 2018JM1050), and the Education Department
Foundation of Shaanxi Province(Grant No. 14JK1676).

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On a stationary random knot
Andrey A.Dorogovtsev

Institute of mathematics

National Academy of Sciences of Ukraine

<EMAIL_ADDRESS>
Abstract

It this article the construction of a stationary random knot is proposed. The
corresponding smooth random curve has no self-intersections in deterministic
moments of time and changes its topological type at random moments.

Mathematical subject classification: 60B99, 60G60, 60H10.

Key words: random curve, random knot, random field, equation with interaction.

## Introduction

In this article the construction of a stationary process of closed smooth
random curves in ${\mathbb{R}}^{3}$ is proposed. The main advantage of this
construction consists of two features. Firstly, for fixed deterministic moment
of time obtained random curve has no self-intersections. Thus it is indeed a
random knot. Secondly, it changes topological type with time. The interest to
such an object arose in the mathematical modeling of linear polymers. There
exists a well-developed theory of polymer models based on the modeling random
walk or Brownian motion in Euclid space (see for example [1-3]). One of the
essential problems here is the existence of the self-intersection points which
are not present in the real linear polymer. To overcame this difficulty some
counting and penalization of self-intersections is provided [4 – 11]. Another
important feature of models based on the Brownian motion is the
nondifferentiability of trajectories. This causes problems with geometric
properties of the obtained random curve. It leads to the use of the Haussdorff
measure or consideration of self-intersection renormalized local times as the
Minkowski functionals in tube formula [11] Such complicated approach works
well for Brownian motion or for Gaussian process of similar structure [9 – 11]
But if we want to consider a moving curve it is not natural to assume that a
curve that changes its shape with time can still be described as Brownian
motion.

In this paper we construct the stationary smooth closed random curve using a
stochastic differential equation with interaction. Such kind of equations were
introduced by the author [13 – 17] in order to describe the motion of large
systems of interacting particles together with their mass distribution. It
occuers that if one takes as initial mass distribution the visitation measure
of a certain curve then equation can be interpreted as a rule of motion of
this curve in the space. This idea has one disadvantage. Namely, all obtained
curves are homotopic because the flow generated by the equation with
reasonably good coefficients consists of homeomorphisms. Hence the moving
curve will keep its topological type. To allow the changing of the type (which
also means that in some moments curve must have intersections) we consider not
the solution to equation with interaction but its image under the random
mapping from ${\mathbb{R}}^{3}$ to ${\mathbb{R}}^{3}.$ This random mapping is
choosen to be centered Gaussian with smooth rapidly decreasing covariance.
Obtained random curve has interesting properties. On the one hand it has no
self-intersections, but on the other hand it changes its topological type.
This, in particular, means that there are self-intersections at random moments
of time. It must be noted that equations with interaction were used for
description of the motion of random curves or surfaces in [18], but in that
article non-smooth surfaces where considered. Also, they remain to be
homotopic to initial surface and the self-intersection local times were
investigated as their geometrical characteristics. In contrast to the
mentioned work, the stationary random knot, which is built here, is smooth and
changes its topological type.To present this construction, the paper is
devided onto three sections. In the first section we consider the image of the
smooth deterministic closed curve under the random mapping. This random
mapping itself is a rotation and translation invariant centered Gaussian
random field. As an example of deterministic curve we consider the circle on
the plane. It is proved in the first section, that obtained random curve has
no self-intersections and can be of arbitrary topological type with positive
probability. Of cause, all statements from this section remain valid for an
arbitrary initial closed curve, not just a circle. In the second section
equations with interaction are applyed to the description of the motion of the
curve. Here the image of the moving measure under the random mapping is
introduced and its mean characteristics are included into coefficients of
equation. Last section contains the construction of the stationary random knot
as an application of the statements from the previous two sections.

##  1 Curves under stationary random mappings

In this section we consider the image of the smooth curve on the plane under
stationary random mapping acting from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{3}$
and discuss its topological properties. The construction will be as follows.
Let $\vec{\xi}=(\xi_{1},\xi_{2},\xi_{3}):{\mathbb{R}}^{2}\to{\mathbb{R}}^{3}$
be a centered Gaussian random field with independent and identically
distributed coordinates with the covariance

$E\xi_{1}(\vec{u})\xi_{1}(\vec{v})=\Gamma(\vec{u}-\vec{v})=e^{-\|\vec{u}-\vec{v}\|^{2}}.$
(1.1)

Since the covariance (1.1) is infinitely differentiable, then due to
Gaussianity, $\vec{\xi}$ has an infinitely differentiable on
${\mathbb{R}}^{2}$ modification. Define a random curve $\gamma$ in
${\mathbb{R}}^{3}$ using parametrization $\vec{\gamma}(t)=\vec{\xi}(\cos
t,\sin t),\ t\in[0;2\pi].$ So, $\gamma$ is an image of the unit circle $S$ on
the plane under the mapping $\vec{\xi}.$ Here we prove the following
statements about the random curve $\gamma.$

######  Theorem 1.1.

With probability one $\gamma$ has no points of self-intersection, namely

$P\\{\exists\,t,s\in[0;2\pi),t\neq s:\ \
\vec{\gamma}(t)=\vec{\gamma}(s)\\}=0.$

######  Theorem 1.2.

Let $\vec{f}\in C^{2}([0;2\pi])$ be a parametrization of a smooth closed curve
in ${\mathbb{R}}^{3}$ without self-intersections. Then the set of
$\omega\in\Omega$ such that $\vec{\xi}(\gamma_{0})$ is homotopic to $\vec{f}$
is a random event of positive probability.

We prove first some auxiliary lemmas. Denote

$\zeta_{1}(t)=\xi_{1}(\cos t,\sin t),\ t\in[0;2\pi].$

######  Lemma 1.1.

If $\mu_{\gamma}$ is the distribution of the Gaussian random function
$\zeta_{1}(t),t\in[0;2\pi]$ in the space $C_{p}([0;2\pi])$ of continuous
periodic functions, then

$\rm{supp}\,\mu_{\gamma}=C_{p}([0;2\pi]).$

###### Proof.

Since $\mu_{\gamma}$ is Gaussian measure, then $\rm{supp}\,\mu_{\gamma}$ is a
closed linear subspace in $C_{p}([0;2\pi]).$ Suppose, that
$\rm{supp}\,\mu_{\gamma}$ is strictly less then $C_{p}([0;2\pi]).$ Then by the
Hahn-Banach theorem there exists non-trivial continuous linear functional
$\varkappa\in C_{p}([0;2\pi])^{*}$ such that its kernel contains
$\rm{supp}\,\mu_{\gamma}.$ The functional $\varkappa$ can be interpreted as a
signed measure on $S.$ Then for the random field $\xi_{1}$

$\int_{S}\xi_{1}(\vec{v})\varkappa(d\vec{v})=0.$

Hence

$\forall\ \vec{u}\in{\mathbb{R}}^{2}:\ \
\int_{S}e^{-\|\vec{u}-\vec{v}\|^{2}}\varkappa(d\vec{v})=E\vec{\xi}_{1}(\vec{u})\int_{S}\xi_{1}(\vec{v})\varkappa(d\vec{v})=0.$
(1.2)

Then

$\forall\ \vec{\lambda}\in{\mathbb{R}}^{2}:\ \
\hat{\Gamma}_{\frac{1}{4}}(\vec{\lambda})\hat{\varkappa}(\vec{\lambda})=0,$
(1.3)

where $\hat{\Gamma}_{\frac{1}{4}},\hat{\varkappa}$ are the Fourier transforms
of ${\Gamma}_{\frac{1}{4}}$ and $\varkappa.$ It follows from (1.3) that
$\varkappa$ can not be nontrivial and our assumption about
$\rm{supp}\,\mu_{\gamma}$ was wrong. Lemma is proved. ∎

Define on ${\mathbb{R}}^{3}$ the following family of functions approximating
$\delta$-function at $\vec{0}$

$h_{\varepsilon}(\vec{x})=\frac{1}{\frac{4}{3}\pi\varepsilon^{3}}{1\\!\\!\,{\rm
I}}_{B(\vec{0},\varepsilon)}(\vec{x}),\ \vec{x}\in{\mathbb{R}}^{3},\
\varepsilon>0.$

Here

$B(\vec{0},\varepsilon)=\\{\vec{x}:\|\vec{x}\|<\varepsilon\\}.$

######  Lemma 1.2.

There exists a finite limit

$\lim_{\varepsilon\to
0}\int^{b}_{a}\int^{d}_{c}Eh_{\varepsilon}(\vec{\gamma}(t_{1})-\vec{\gamma}(t_{2}))dt_{1}dt_{2}=$
$=\int^{b}_{a}\int^{d}_{c}\frac{dt_{1}dt_{2}}{(2\pi(2-2\Gamma(\vec{u}(t_{2})-\vec{u}(t_{1}))))^{3/2}},$

where $0\leq a<b<c<d<2\pi,\ \vec{u}(t)=(\cos t,\ \sin t).$

Proof of the lemma easily follows from the Lebesgue dominated convergence
theorem.

As a consequence of Lemma 1.2 one can obtain a proof of Theorem 1.1.

###### Proof of Theorem 1.1.

It is enough to check that the pieces of $\gamma,$ $\vec{\gamma}([a;b])$ and
$\vec{\gamma}([c;d]),$ have no intersections with probability one for $0\leq
a<b<c<d<2\pi.$ First note, that the set $A$ of $\omega\in\Omega$ such that
$\exists\ t_{1}\in[a;b],t_{2}\in[c;d]:$

$\vec{\gamma}(t_{1})(\omega)=\vec{\gamma}(t_{2})(\omega)$ (1.4)

is measurable. Indeed, the restrictions of $\vec{\gamma}(\cdot)$ on $[a;b]$
and $[c;d]$ are continuous functions. Consequently, the opposite property to
(1.4) looks like
$\exists\ n\geq 1:\forall\ r_{1}\in{\mathbb{Q}}\cap[a;b],\ \forall\
r_{2}\in{\mathbb{Q}}\cap[c;d]:$

$\|\vec{\gamma}(r_{1})(\omega)-\vec{\gamma}(r_{2})(\omega)\|>\frac{1}{n}.$
(1.5)

Suppose now, that $P(A)>0.$ Take $\omega\in A$ and consider

$\int^{b}_{a}\int^{d}_{c}h_{\varepsilon}(\vec{\gamma}(t_{1})(\omega)-\vec{\gamma}(t_{2})(\omega))dt_{1}dt_{2}.$
(1.6)

If there exists a pair $(t^{0}_{1},t^{0}_{2})\in[a;b]\times[c;d]$ such that

$\vec{\gamma}(t^{0}_{1})(\omega)=\vec{\gamma}(t^{0}_{2})(\omega),$

then, due to differentiability of $\vec{\gamma}(\cdot)(\omega)$ at these
points, there exists positive constant (depending on $\omega$) $C(\omega)>0$
such that integral in (1.6) satisfies the inequality
$\exists\ \varepsilon_{0}:\forall\ \varepsilon<\varepsilon_{0}:$

$\int^{b}_{a}\int^{d}_{c}h_{\varepsilon}(\vec{\gamma}(t_{1})(\omega)-\vec{\gamma}(t_{2})(\omega))dt_{1}dt_{2}\geq
C(\omega)\frac{1}{\varepsilon}.$ (1.7)

Together with the Fatou lemma (1.7) and the hypothesis $P(A)>0$ contradict to
the statement of Lemma 1.2. Consequently, our assumption that $P(A)>0$ was
wrong. The theorem is proved. ∎

It was proved, that the random curve $\vec{\xi}(\gamma_{0})$ with probability
one has no self-intersection. So $\vec{\xi}(\gamma_{0})$ is a random knot. Now
let us prove Theorem 1.2.

###### Proof.

We start with a purely deterministic construction. For $n\geq 1$ consider the
uniform partition $0=t_{0}<\ldots<t_{n}=2\pi$ (i.e. $t_{k}=\frac{k2\pi}{n}$).
Define $\vec{f}_{n}$ as a polygonal interpolation of $\vec{f}$ with its values
at $t_{0},\ldots,t_{n}.$ Then there exists $n_{0}$ such, that for $n\geq
n_{0}$ $\vec{f}_{n}$ is homotopic to $\vec{f}$ (i.e. the polygonal line and
the curve described by $\vec{f}_{n}$ and $\vec{f}$ define the knot of the same
topological type). It was proved in [19]. Consequently, to check the homotopy
between $\vec{\xi}(\gamma_{0})$ and $\vec{f},$ it is enough to check the
homotopy of corresponding polygonal lines. To check that two polygonal lines
are homotopic it is enough to consider flat diagrams and try to get one from
another by the finite sequence of Reidemeister moves [21]. Eeach moving is
applied to a certain crossroad. So, one have to consider finite sequences of
Reidemeister moves and crossroads. Hence the occurrence of homotopy between
random and fixed polygonal lines is a random event.

Now we will prove that this event has a positive probability. Here we need
auxilary notations and statement. Denote by $C^{2}_{p}([0;2\pi])$ the set of
all functions which are two times continuously differentiable and
$2\pi$-periodic. Let $\eta(t)=\xi(\vec{\gamma}(t)),t\in[0;2\pi]$ be a Gaussian
random process defined with the help of the centered Gaussian random field
$\xi,$ which is equidistributed with $\xi_{i},i=1,2,3.$ Let $\mu$ be the
distribution of $\eta^{\prime\prime}$ in the space of all functions

$M=\\{g:g=f^{\prime\prime},\ f\in C^{2}_{p}([0;2\pi])\\}$

with uniform distance.

######  Lemma 1.3.

$\rm{supp}\,\mu=M.$

###### Proof.

The proof is similar to the proof of Lemma 1.3, but we include it for the
convenience. Since $\mu$ is centered Gaussian measure in $M,$ then it is
enough to check, that there is no such linear continuous functional
$\varkappa$ on $M,$ that

$\exists\ g\in M:\langle g,\varkappa\rangle\neq 0,$ $\forall\
h\in\rm{supp}\,\mu:\langle h,\varkappa\rangle=0.$

To check this, suppose that such $\varkappa$ exsists and get contradiction.
Indeed, since $M$ can be viewed as a subspace of $C([0;2\pi]),$ then
$\varkappa$ can be identified with a finite signed measure on $[0;2\pi].$
Correspondingly for random process $\eta$ one can write

$\int^{2\pi}_{0}\eta(t)^{\prime\prime}\varkappa(dt)=0.$

Then

$\forall\ \vec{u}\in{\mathbb{R}}^{2}:\
\int^{2\pi}_{0}(E\xi(\vec{u})\eta(t))^{\prime\prime}\varkappa(dt)=0.$

Hence

$\forall\ \vec{u}\in{\mathbb{R}}^{2}:\
\int^{2\pi}_{0}\Gamma(\vec{u}-\vec{\gamma}(t))^{\prime\prime}\varkappa(dt)=0,$

where as before

$\Gamma(\vec{u}-\vec{v})=e^{-\|\vec{u}-\vec{v}\|^{2}}$

is a covariance function of the random field $\xi.$ Let us define the
generalized function $\widetilde{\varkappa}$ on ${\mathbb{R}}^{2}$ as follows
$\forall\varphi\in S({\mathbb{R}}^{2}):$

$\langle\varphi,\widetilde{\varkappa}\rangle:=\int^{2\pi}_{0}\varphi(\vec{\gamma}(t))^{\prime\prime}\varkappa(dt).$

Then for the convolution $\Gamma*\widetilde{\varkappa}$ we have

$\Gamma*\widetilde{\varkappa}\equiv 0.$

Consequently, the Fourier transform

$\widehat{\Gamma*\widetilde{\varkappa}}=\widehat{\Gamma}\cdot\widehat{\widetilde{\varkappa}}\equiv
0.$

Since

$\forall\ \vec{\lambda}\in{\mathbb{R}}^{2}:\
\widehat{\Gamma}(\vec{\lambda})\neq 0,$

then

$\widehat{\widetilde{\varkappa}}\equiv 0.$

It means, that the generalized function $\widetilde{\varkappa}$ equals zero,
Consequently,
$\forall\varphi\in S({\mathbb{R}}^{2}):$

$\int^{2\pi}_{0}\varphi(\vec{\gamma}(t))^{\prime\prime}\varkappa(dt)=0.$

It remains to note, that the set

$\\{\varphi(\vec{\gamma}(\cdot))^{\prime\prime}:\ \varphi\in
S({\mathbb{R}}^{2})\\}$

is dence in $M$ with respect to uniform distance. Finally our measure
$\varkappa$ is such, that
$\forall\ g\in M:$

$\int^{2\pi}_{0}g(t)\varkappa(dt)=0.$

This means, that related functional $\varkappa$ on $M$ is zero. Obtained
contradiction proves the statement of the lemma.

∎

Now let us back to the main statement. Consider a curve $F_{0},$ which is
parametrized by the infinitely differentiable periodic function $\vec{f}:\
[0;2\pi]\to{\mathbb{R}}^{3}$ with the following property of its projection
$\vec{f}_{*}$ on the first two coordinates. For all $\delta\in(0;\pi)$ there
exists $\varepsilon_{1}>0$ such that
$\forall\ t_{1},t_{2}\in[0;2\pi],\
\min(|t_{1}-t_{2}|,2\pi-|t_{1},t_{2}|)>\delta:$

$\|\vec{f}_{*}(t_{1})-\vec{f}_{*}(t_{2})\|>\varepsilon_{1}.$

Consider another function $\vec{g}:[0;2\pi]\to{\mathbb{R}}^{3}$ which is
periodic, infinitely differentiable and such that for some positive
$\varepsilon$

$\max_{[0;2\pi]}\|\vec{f}^{\prime\prime}(t)-\vec{g}^{\prime\prime}(t)\|<\varepsilon.$

Due to periodicity the following relationships hold

$f(t)=c_{1}+\frac{t}{2\pi}\int^{2\pi}_{0}sf^{\prime\prime}(s)ds+\int^{t}_{0}(t-s)f^{\prime\prime}(s)ds,$
$g(t)=c_{2}+\frac{t}{2\pi}\int^{2\pi}_{0}sg^{\prime\prime}(s)ds+\int^{t}_{0}(t-s)g^{\prime\prime}(s)ds.$

Suppose that a planar curve parametrized by $\vec{f}_{*}$ has no self-
intersections. Prove that the same holds for function $\vec{g}_{*}.$ Consider
new function $\vec{\widetilde{f}}$ defined as

$\vec{\widetilde{f}}(t)=\vec{c}_{2}+\frac{t}{2\pi}\int^{2\pi}_{0}s{\vec{f}}^{\prime\prime}(s)ds+\int^{t}_{0}s{\vec{f}}^{\prime\prime}(s)ds.$

The curves parametrized by the functions $\vec{f}$ and $\vec{\widetilde{f}}$
are connected by parallel translation on $\vec{c}_{2}-\vec{c}_{1}$ and,
consequently have the same type. So, it is enough to compare
$\vec{\widetilde{f}}$ and $\vec{g}.$ Due to the choice of $\vec{g}$ one has

$\max_{[0;2\pi]}\|\vec{\widetilde{f}}(t)-\vec{g}(t)\|\leq
2\pi^{2}\varepsilon,$
$\max_{[0;2\pi]}\|\vec{\widetilde{f}}^{\prime}(t)-\vec{g}^{\prime}(t)\|\leq
2\pi^{2}\varepsilon.$

Consider projections of $\vec{\widetilde{f}}$ and $\vec{g}$ onto the first two
coordinates. Let us prove that projection of $\vec{g}$ parametrizes curve
without self-intersections. Suppose that for different
$t_{1},t_{2}\in[0;2\pi],t_{1}<t_{2}$

$\vec{g}_{*}(t_{1})=\vec{g}_{*}(t_{2}).$

Then one can find $s_{1},s_{2}\in[t_{1};t_{2}]$ such, that the angle between
$\vec{g}_{*}^{\prime}(s_{1})$ and $\vec{g}_{*}^{\prime}(s_{2})$ equals $\pi.$
Also note, that due to the condition on $\vec{f}_{*},$ if
$2\pi\varepsilon^{2}<\varepsilon_{1},$ then

$t_{2}-t_{1}<\delta.$

Hence

$\|\vec{g}_{*}^{\prime}(s_{2})-\vec{g}_{*}^{\prime}(s_{1})\|\leq\delta\max_{[0;2\pi]}\|\vec{g}_{*}^{\prime\prime}(s)\|.$

But

$\max_{[0;2\pi]}\|\vec{g}_{*}^{\prime}(s)-\vec{\widetilde{f}}^{\prime}(s)\|\leq
2\pi^{2}\varepsilon.$

Consequently,

$2R-4\pi^{2}\varepsilon\leq\delta\max_{[0;2\pi]}\|\vec{g}_{*}^{\prime\prime}(s)\|\leq$
$\leq\delta(\max_{[0;2\pi]}\|\vec{f}_{*}^{\prime\prime}(s)\|+\varepsilon).$

Finally

$2R\leq\delta(\max_{[0;2\pi]}\|\vec{f}_{*}^{\prime\prime}(s)\|+\varepsilon)+4\pi^{2}\varepsilon.$

Now it remains to choose $\delta$ in a such way, that

$\delta(\max_{[0;2\pi]}\|\vec{f}_{*}^{\prime\prime}(s)\|<\frac{1}{2}R,$

then take such $\varepsilon>0,$ which satisfies inequalities

$2\pi\varepsilon^{2}<\frac{1}{2}\varepsilon_{1},\ \
\varepsilon\delta+4\pi\varepsilon^{2}<\frac{1}{2}R.$

With such a choice $\varepsilon,\delta$ we can not find such pair $t_{1}\neq
t_{2},$ that $\vec{g}_{*}(t_{1})=\vec{g}_{*}(t_{2}).$ Now it follows from
these consideration and previous lemma, that with positive probability the
random curve $\gamma$ is a trivial knot. To prove, that with positive
probability $\gamma$ can be a nontrivial knot let us consider the function
$\vec{f}_{*}:[0;2\pi]\to{\mathbb{R}}^{2}$ with the same as above properties.
Bat, in contrast to previous case, suppose, that the curve, which is
parametrized by $\vec{f}_{*}$ has a finite number of self-intersection points.
Denote by $t_{1}<s_{1},\ldots,t_{n}<s_{n}$ corresponding values of parameters.
Take positive $\delta$ such that $\delta$-neighbourhoods of
$t_{1},\ldots,t_{n},\ldots,s_{n}$ have no intersections. Consider a function
$f_{3}\in C^{2}_{p}([0;2\pi])$ such that $|f_{3}|>c>0$ on all
$\delta$-neighbourhoods of $t_{1},\ldots,t_{n},s_{n}.$ Then, it can be checked
similarly to the previous considerations, that if the function $\vec{g}\in
C^{2}_{p}([0;2\pi],{\mathbb{R}}^{3})$ approximates $\vec{f},$ then the knots
parametrized by $\vec{f}$ and $\vec{g}$ have the same to pological type. This
completes the proof of the statement, that the random curve $\gamma$ with
positive probability has an arbitrary topological type.

∎

##  2 Moving random curve in ${\mathbb{R}}^{3}$

In this section we propose a model of a moving random curve in
${\mathbb{R}}^{3}$ based on equation with interaction [14 – 18] and random map
$\vec{\xi}$ from the previous section. An equation with interaction is an
equation of the following kind

$d\vec{x}(\vec{u},t)=\vec{a}(\vec{x}(\vec{u},t),\mu_{t})dt+\int_{{\mathbb{R}}^{d}}b(\vec{x}(\vec{u},t),\mu_{t},\vec{}p)\vec{W}(d\vec{p},dt),$
(2.1) $\vec{x}(\vec{u},0)=\vec{u},\ \vec{u}\in{\mathbb{R}}^{d},\
\mu_{t}=\mu_{0}\circ\vec{x}(\cdot,t)^{-1}.$

Here $\mu_{0}$ is the initial mass distribution of the system. $\vec{W}$ is an
${\mathbb{R}}$-valued Wiener sheet on ${\mathbb{R}}^{d}\times[0;+\infty).$
$\vec{W}$ has independent coordinates, each of which is a centered Gaussian
random measure on ${\mathbb{R}}^{d}\times[0;+\infty)$ with independent values
on disjoint sets and structural measure, which is equal to Lebesgue measure.
$\vec{W}$ plays the role of an outer random media, which perturbs the motion
of particles. The particles start from every point of the space. Infinitesimal
increments of the particle trajectory depend not only on the position of the
particle, but also on the mass distribution of all particles. Such form of the
equation allows to consider infinite systems of interacting particles in
random media. Note, that the initial mass distribution is not necessary
discrete. It can be continuous and even have density with respect to the
Lebesgue measure. For us will be important the case when $\mu_{0}$ is a
visitation measure of a certain smooth closed curve $\gamma_{0}.$ Namely,
suppose, that $\vec{f}\in C^{2}_{p}([0;1],{\mathbb{R}}^{3})$ and $\gamma_{0}$
is parametrised by $\vec{f}.$ Then define $\mu_{0}$ as follows

$\mu_{0}(\Delta)=\int^{1}_{0}{1\\!\\!\,{\rm I}}_{\Delta}(\vec{f}(t))dt.$

Note, that $\mu_{0}$ is a probability measure on ${\mathbb{R}}^{3}$ depending
on the parametrization of $\gamma_{0}.$ If we start from such $\mu_{0},$ then
at the moment $t>0$ $\mu_{t}$ will be a visitation measure of the curve
$\gamma_{t}=\vec{x}(\gamma_{0},t),$ which is parametrized by the function
$\vec{x}(\vec{f}(\cdot),t).$ In a such way equation (2.1) describes the
evolution of the curve. However such model has an essential disadvantage. If
the coefficients of the equation have two bounded continuous derivatives with
respect to a spatial variable, then the solution $\vec{x}$ with probability
one is jointly continuous with respect $\vec{u}$ and $t$ and diffeomorphic as
a map from ${\mathbb{R}}^{3}$ to ${\mathbb{R}}^{3}$ for a fixed $t.$ This
means that $\gamma_{0}$ is homotopic to $\gamma_{t}.$ Or, that the curve
$\gamma_{t}$ has the same topological type as $\gamma_{0}.$ In order to get
around such difficulty, we consider in this article not the evolution of
$\gamma_{t}$ itself, but its image in ${\mathbb{R}}^{3}$ under the mapping
$\vec{\xi}.$ To present the corresponding equation, we need to specify the
coefficients, which depend on the visitation measure of $\gamma_{t}.$ Consider
the function

$\vec{h}:\ {\mathbb{R}}^{3}\times{\mathbb{R}}^{3}\to{\mathbb{R}}^{3},$

which is bounded and satisfies Lipschitz condition with respect to both
variablises. As a space of probability measures we will use the set
${\mathfrak{M}}_{2}$ of all probability measures on the Borel $\sigma$-field
of ${\mathbb{R}}^{3}$ equipped with Wasserstain distance of order 2. This
distance is defined as follows [20]. For $\mu,\nu\in{\mathfrak{M}}$ consider
$C(\mu,\nu)$ a set of all probability distributions on
${\mathbb{R}}^{3}\times{\mathbb{R}}^{3}$ which have $\mu$ and $\nu$ as its
marginals. Then the Wasserstain distance of order 2 between $\mu$ and $\nu$ is

$\gamma_{2}(\mu,\nu)=\inf_{c(\mu,\nu)}\Big{(}\iint_{{\mathbb{R}}^{3}}\|\vec{u}-\vec{v}\|^{2}\varkappa(d\vec{u},d\vec{v})\Big{)}^{\frac{1}{2}}.$

It is known [20], that $({\mathfrak{M}}_{2},\gamma_{2})$ is a complete
separable metric space. Now consider a random mapping
$\vec{\xi}:{\mathbb{R}}^{3}\to{\mathbb{R}}^{3}$ which is organized at the same
way as in the previous section (so, we use the same letter for notation).
Namely, the coordinates of $\vec{\xi}$ are independent centered Gaussian
random fields with the covariance

$E\xi_{1}(\vec{u})\xi_{1}(\vec{v})=e^{-\|\vec{u}-\vec{v}\|^{2}}.$

For $\mu\in{\mathfrak{M}}_{2}$ denote by $\mu_{\xi}$ its image under the
mapping $\vec{\xi}$

$\mu_{\xi}=\mu\circ\vec{\xi}^{-1}.$

Since $\vec{\xi}$ is infinitely differentiable random field, then $\mu_{\xi}$
is a random measure on ${\mathbb{R}}^{3}.$ Now consider a function
$\vec{a}:{\mathbb{R}}^{3}\times{\mathfrak{M}}\to{\mathbb{R}}^{3},$ which is
defined as follows

$\vec{a}(\vec{u},\mu)=E\int_{{\mathbb{R}}^{3}}h(\vec{u},\vec{v})\mu_{\xi}(d\vec{v}).$

######  Lemma 2.1.

If $h$ satisfies the Lipschitz condition, then the function $\vec{a}$
satisfies the Lipschitz condition with respect to both variables.

###### Proof.

For $\mu_{1},\mu_{2}\in{\mathfrak{M}}_{2}$ take $\varkappa\in
C(\mu_{1},\mu_{2}).$ Then for $u_{1},u_{2}\in{\mathbb{R}}^{3}$

$\|\vec{a}(\vec{u}_{1},\vec{\mu}_{1})-\vec{a}(\vec{u}_{2},{\mu}_{2})\|\leq$
$\leq
E\|\int_{{\mathbb{R}}^{3}}h(\vec{u}_{1},\vec{v}_{1})\mu_{1,\xi}(d\vec{v}_{1})-\int_{{\mathbb{R}}^{3}}h(\vec{u}_{2},\vec{v}_{2})\mu_{2,\xi}(\vec{v}_{2})\|=$
$=E\|\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}(h(\vec{u}_{1},\vec{v}_{1})-h(\vec{u}_{2},\vec{v}_{2}))\varkappa_{\xi}(d\vec{v}_{1},d\vec{v}_{2})\|\leq$
$\leq
E\|\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}L(\|\vec{u}_{1}-\vec{u}_{2}\|+\|\vec{v}_{1}-\vec{v}_{2}\|)\varkappa_{\xi}(d\vec{v}_{1},d\vec{v}_{2})=$
$=L\|\vec{u}_{1}-\vec{u}_{2}\|+L\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}E\|\vec{\xi}(\vec{v}_{1})-\vec{\xi}(\vec{v}_{2})\|\varkappa(d\vec{v}_{1},d\vec{v}_{2})\leq$
$\leq
L\|\vec{u}_{1}-\vec{u}_{2}\|+L\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}\Big{(}E\|\vec{\xi}(\vec{v}_{1})-\vec{\xi}(\vec{v}_{2})\|^{2}\Big{)}^{\frac{1}{2}}\varkappa(d\vec{v}_{1},d\vec{v}_{2})=$
$=L\|\vec{u}_{1}-\vec{u}_{2}\|+L\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}\Big{(}6(1-e^{-\|\vec{v}_{1}-\vec{v}_{2}\|^{2}})\Big{)}^{\frac{1}{2}}\varkappa(d\vec{v}_{1},d\vec{v}_{2})\leq$
$\leq
L\|\vec{u}_{1}-\vec{u}_{2}\|+\sqrt{6}L\Big{(}\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}\Big{(}1-e^{-\|\vec{v}_{1}-\vec{v}_{2}\|^{2}}\Big{)}\varkappa(d\vec{v}_{1},d\vec{v}_{2})\Big{)}^{\frac{1}{2}}\leq$
$\leq
L\|\vec{u}_{1}-\vec{u}_{2}\|+\sqrt{6}L\Big{(}\int_{{\mathbb{R}}^{3}}\int_{{\mathbb{R}}^{3}}\|\vec{v}_{1}-\vec{v}_{2}\|^{2}\varkappa(d\vec{v}_{1},d\vec{v}_{2})\Big{)}^{\frac{1}{2}}.$

Now taking $\inf$ with respect to $\varkappa\in C(\mu_{1},\mu_{2})$ one can
get the inequality

$\|\vec{a}(\vec{u}_{1},\mu_{1})-\vec{a}(\vec{u}_{2},\mu_{2})\|\leq
L\|\vec{u}_{1}-\vec{u}_{2}\|+\sqrt{6}L\gamma_{2}(\mu_{1},\mu_{2}).$

The lemma is proved. ∎

Consider the following partial case of equation with interaction with the
coefficient $\vec{a}$ from the previous lemma.

$d\vec{x}(\vec{u},t)=\vec{a}(\vec{x}(\vec{u},t),\mu_{t})dt+\int_{{\mathbb{R}}^{3}}b(\vec{x}(\vec{u},t),\vec{p})\vec{W}(d\vec{p},dt),$
(2.2) $\vec{x}(\vec{u},0)=\vec{u},\ \vec{u}\in{\mathbb{R}}^{3},\
\mu_{t}=\mu_{0}\circ\vec{x}(\cdot,t)^{-1}.$

Suppose, that $\vec{\xi}$ and $\vec{W}$ are independent and $\vec{h}$ has two
continuous bounded derivatives on ${\mathbb{R}}^{3}\times{\mathbb{R}}^{3}.$
Suppose, also, that $b$ has two continuous bounded derivatives as a mapping
from ${\mathbb{R}}^{3}$ to $L_{2}({\mathbb{R}}^{3}).$ Under such assumptions
the solution to the equation with interaction exists and is unique. Namely,
the following statement is a partial case of the general theorem from [17].

######  Theorem 2.1.

Suppose that all formulated above assumptions are satisfied. Then, for the
initial mass distribution $\mu_{0}\in{\mathfrak{M}}_{2}$ there exists a unique
solution $x$ to Cauchy problem (2.2) such that

1) with probability one $\vec{x}$ is continuous with respect to $\vec{u}$ and
$t,$ for a fixed $t\geq 0\ \vec{x}(\cdot,t)$ is a diffeomorphism
${\mathbb{R}}^{3}$ on itself,

2) for all $t\geq 0\ \mu_{t}$ is a random element in ${\mathfrak{M}}_{2}.$

Now suppose, that $\mu_{0}$ is a visitation measure of a closed smooth curve
$\gamma_{0}$ in ${\mathbb{R}}^{3}$ which is parametrized by the function
$\vec{f}\in C^{2}_{p}([0;2\pi],{\mathbb{R}}^{3}).$ Then, due to Theorem 2.1
equation (2.2) will define the evolution $\\{\gamma_{t},t\geq 0\\}$ of the
curve $\gamma_{0}.$ As mentioned above all $\gamma_{t},t\geq 0$ are homotopic
and, therefore, have the same topological type. But we will consider curves

$\Gamma_{t}=\vec{\xi}(\gamma_{t}),\ t\geq 0.$

As we have already proved they can change its type with time. $\Gamma_{t}$ can
be considered as a moving random knot in ${\mathbb{R}}^{3}.$ In the next
section we will consider an example of such stationary motion.

##  3 Ornstein–Uhlenbeck random curves process

In this section we will consider a partial case of equation (2.2), which has a
stationary solution. Namely, suppose that the motion of $\gamma$ is described
by the following equation with interaction

$\begin{split}&d\vec{x}(\vec{u},t)=A(\vec{x}(\vec{u},t)-\int_{{\mathbb{R}}^{3}}\vec{x}(\vec{v},t)\mu_{0}(d\vec{v}))dt+d\vec{w}(t),\\\
&\vec{x}(\vec{u},0)=\vec{u},\ \vec{u}\in{\mathbb{R}}^{3}.\end{split}$ (3.1)

Here $\mu_{0}$ is a visitation measure of $\gamma_{0}.$ As before we suppose
that $\gamma_{0}$ is parametrized by the function $\vec{f}_{0}\in
C^{2}_{p}([0;2\pi],{\mathbb{R}}^{3}).$ Then $\mu_{0}$ has the form

$\forall\ \Delta\in{\mathcal{B}}({\mathbb{R}}^{3}):\
\mu_{0}(\Delta)=\int^{2\pi}_{0}{1\\!\\!\,{\rm I}}_{\Delta}(\vec{f}_{0}(s))ds.$

In (3.1) $\vec{w}$ is a standard Wiener process in ${\mathbb{R}}^{3},$ $A$ is
a generator of unitary semi-group in ${\mathbb{R}}^{3}.$ It means that the
solution to Cauchy problem for matrix-valued equation

$\begin{cases}dU_{t}=AU_{t}dt,\\\ U_{0}=I\end{cases}$

consists of orthogonal operators, i.e.

$\forall\ t\geq 0\ \ U^{*}_{t}=U^{-1}_{t}.$

Cauchy problem (3.1) is partial case of (2.2) and, consequently, has a unique
solution. Let us clarify the precise form of solution to (3.1). Define for
every $t\geq 0$ the center of mass for $\mu_{t}$ as

$\vec{m}_{t}=\int_{{\mathbb{R}}^{3}}\vec{v}\mu_{t}(d\vec{v})=\int_{{\mathbb{R}}^{3}}\vec{x}(\vec{u},t)\mu_{0}(d\vec{u}).$

Then, integrating (3.1) against $\mu_{0}$ one can get the equation for
$\vec{m}_{t}$

$d\vec{m}_{t}=d\vec{w}(t).$

Now, for any $\vec{u}\in{\mathbb{R}}^{3}$

$d(\vec{x}(\vec{u},t)-\vec{m}(t))=A(\vec{x}(\vec{u},t)-\vec{m}_{t})dt.$

Consequently,

$\vec{x}(\vec{u},t)-\vec{m}(t)=U_{t}(\vec{u}-\vec{m}_{0}).$

Hence, the motion of the curve $\gamma$ can be described as follows. Its
center of mass is floating over ${\mathbb{R}}^{3}$ as a Brownian particle and
simultaneously $\gamma$ is rotating around $\vec{m}_{t}$ accordingly to
orthogonal transformation $U_{t}.$ This character of motion causes the
stationarity of the random knot $\\{\Gamma_{t};t\geq 0\\}.$ Let us recall,
that

$\Gamma_{t}=\vec{\xi}(\gamma_{t}).$

Consequently, $\Gamma_{t}$ is parametrized by the random function
$\vec{\xi}(\vec{x}(\vec{f}_{0}(s),t)),\ s\in[0;2\pi].$ Let us shortly denote
it by $\vec{\eta}_{t}.$

######  Theorem 3.1.

$\\{\eta_{t};t\geq 0\\}$ is a stationary random process in
$C^{2}_{p}([0;2\pi]).$

###### Proof.

For $n,m\geq 1$ consider sets of numbers
$\\{s_{1},\ldots,s_{m}\\}\subset[0;2\pi],\ 0\leq t_{1}<\ldots<t_{n}.$ Consider
the distribution of the random vector
$\vec{\eta}_{t_{1}}(s_{1}),\ldots,\vec{\eta}_{t_{1}}(s_{m}),\ldots,\vec{\eta}_{t_{n}}(s_{1}),\ldots,\vec{\eta}_{t_{n}}(s_{m}).$
Conditionally on $\vec{x},$ the distribution of this vector is Gaussian with
zero mean and a covariance matrix whose elements are of the form

$e^{\|\vec{x}(\vec{f}_{0}(s_{k_{1}}),t_{j_{1}})-\vec{x}(\vec{f}_{0}(s_{k_{2}}),t_{j_{2}})\|^{2}}.$

Let us rewrite

$\vec{x}(\vec{f}_{0}(s_{k_{1}}),t_{j_{1}})-\vec{x}(\vec{f}_{0}(s_{k_{2}}),t_{j_{2}})=$
$=U_{t_{j_{1}}}(\vec{f}_{0}(s_{k_{1}})-\vec{m}_{0})-U_{t_{j_{2}}}(\vec{f}_{0}(s_{k_{2}})-\vec{m}_{0})+$
$+\vec{w}(t_{j_{1}})-\vec{w}(t_{j_{2}})=U_{t_{1}}(U_{t_{j_{1}}-t_{1}}(\vec{f}_{0}(s_{k_{1}})-m_{0})-$
$-U_{t_{j_{2}}-t_{1}}(\vec{f}_{0}(s_{k_{2}})-m_{0})+U^{-1}_{t_{1}}(\vec{w}(t_{j_{1}})-\vec{w}(t_{j_{2}}))$

Note, that due to the choice of covariance function the random field
$\vec{\xi}$ has the following property
$\forall\ \vec{u}_{1},\ldots,\vec{u}_{n}\in{\mathbb{R}}^{3}$

$(\vec{\xi}(\vec{u}_{1}),\ldots,\vec{\xi}(\vec{u}_{n}))\overset{d}{=}(\vec{\xi}(U^{-1}_{t_{1}}\vec{u}_{1}),\ldots,\vec{\xi}(U^{-1}_{t_{1}}\vec{u}_{n})).$

Also note, that $U^{-1}_{t_{1}}\vec{w}(t),t\geq 0$ is a standard Wiener
process. Consequently, the distribution of
$(\vec{\eta}_{t_{1}}(s_{1}),\ldots,\vec{\eta}_{t_{n}}(s_{m}))$ depends only on
the differences between $t_{1},\ldots,t_{n}.$ This means stationarity of the
considered process. The theorem is proved. ∎

Let us check that the random knot $\Gamma_{t}$ changes its topological types.
Consider two deterministic knots $G_{1},G_{2},$which are not of the same type.
It is enough to prove, that there exists such $t>0$ that the probability

$P\\{\Gamma_{0}\ \mbox{has type}\ G_{1},\ \Gamma_{t}\ \mbox{has type}\ G_{2}\
\\}$ (3.2)

is positive. We will check this by using a rapid decrease of covariance of the
random field $\vec{\xi}$ and the independence of $\vec{\xi}$ and $\vec{w}.$

######  Theorem 3.2.

There exists such $t_{0},$ for which probability (3.2) is positive.

###### Proof.

First, note that for an arbitrary $t\geq 0$ and $\vec{u}\in{\mathbb{R}}^{3}$

$P\\{\vec{\xi}(U_{t}\gamma_{0}+\vec{u})\ \mbox{has type}\ G_{2}\
\\}=P\\{\vec{\xi}(\gamma_{0})\ \mbox{has type}\ G_{2}\ \\}.$

This follows from the form of covariance of $\vec{\xi}.$ Since the covariance
of $\vec{\xi}$ decrease to 0 on infinity, there exists such $R>0,$ that
$\forall\ t>0\ \forall\ \vec{u}\in{\mathbb{R}}^{3},\|\vec{u}\|\geq R$

$|P\\{\vec{\xi}(\gamma_{0})\ \mbox{has type}\
G_{1},\vec{\xi}(U_{t}\gamma_{0}+\vec{u})\ \mbox{has type}\ G_{2}\ \\}-$
$P\\{\vec{\xi}(\gamma_{0})\ \mbox{has type}\ G_{1}\\}\cdot
P\\{\vec{\xi}(U_{t}\gamma_{0}+\vec{u})\ \mbox{has type}\ G_{2}\ \\}|<$
$<\frac{1}{2}P\\{\vec{\xi}(\gamma_{0})\ \mbox{has type}\ G_{1}\ \\}\cdot
P\\{\vec{\xi}(U_{t}\gamma_{0}+\vec{u})\ \mbox{has type}\ G_{2}\ \\}.$

Now the statement of the theorem follows from the fact, that in
${\mathbb{R}}^{3}$

$\|\vec{w}(t)\|\to+\infty,\ t\to\infty\ \mbox{a.s.}$

and the independence of $\vec{w}$ and $\vec{\xi}.$ The theorem is proved. ∎

Remark. Since it was proved in Section 1 that for fixed $t\geq 0$ $\Gamma_{t}$
has no self-intersections with probability one, then the constructed
stationary random knot $\\{\Gamma_{t};t\geq 0\\}$ changes its type at random
moments of time.

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