178 research outputs found
Model order reduction for stochastic dynamical systems with continuous symmetries
Stochastic dynamical systems with continuous symmetries arise commonly in
nature and often give rise to coherent spatio-temporal patterns. However,
because of their random locations, these patterns are not well captured by
current order reduction techniques and a large number of modes is typically
necessary for an accurate solution. In this work, we introduce a new
methodology for efficient order reduction of such systems by combining (i) the
method of slices, a symmetry reduction tool, with (ii) any standard order
reduction technique, resulting in efficient mixed symmetry-dimensionality
reduction schemes. In particular, using the Dynamically Orthogonal (DO)
equations in the second step, we obtain a novel nonlinear Symmetry-reduced
Dynamically Orthogonal (SDO) scheme. We demonstrate the performance of the SDO
scheme on stochastic solutions of the 1D Korteweg-de Vries and 2D Navier-Stokes
equations.Comment: Minor revision
Thermal diffusion of supersonic solitons in an anharmonic chain of atoms
We study the non-equilibrium diffusion dynamics of supersonic lattice
solitons in a classical chain of atoms with nearest-neighbor interactions
coupled to a heat bath. As a specific example we choose an interaction with
cubic anharmonicity. The coupling between the system and a thermal bath with a
given temperature is made by adding noise, delta-correlated in time and space,
and damping to the set of discrete equations of motion. Working in the
continuum limit and changing to the sound velocity frame we derive a
Korteweg-de Vries equation with noise and damping. We apply a collective
coordinate approach which yields two stochastic ODEs which are solved
approximately by a perturbation analysis. This finally yields analytical
expressions for the variances of the soliton position and velocity. We perform
Langevin dynamics simulations for the original discrete system which fully
confirm the predictions of our analytical calculations, namely noise-induced
superdiffusive behavior which scales with the temperature and depends strongly
on the initial soliton velocity. A normal diffusion behavior is observed for
very low-energy solitons where the noise-induced phonons also make a
significant contribution to the soliton diffusion.Comment: Submitted to PRE. Changes made: New simulations with a different
method of soliton detection. The results and conclusions are not different
from previous version. New appendixes containing information about the system
energy and soliton profile
Numerical simulations of the stochastic KDV equation
We study the Korteweg-de Vries (KdV) equation with external noise and compare our numerical simulations to known theoretical results. By using a modification of the Zabusky-Kruskal
finite difference scheme, we are able to generate numerical solutions to the stochastic KdV.
We look at the large time behavior of the stochastic KdV and verify the diffusion of solitons. We find that the predicted large time behavior of the perturbed solution is not easily
confirmed in the simulations as the initial soliton diffuses and is lost amidst the background
noise long before the asymptotic limit is reached
Anomalous ballistic scaling in the tensionless or inviscid Kardar-Parisi-Zhang equation
The one-dimensional Kardar-Parisi-Zhang (KPZ) equation is becoming an overarching paradigm for the scaling of nonequilibrium, spatially extended, classical and quantum systems with strong correlations. Recent analytical solutions have uncovered a rich structure regarding its scaling exponents and fluctuation statistics. However, the zero surface tension or zero viscosity case eludes such analytical solutions and has remained ill-understood. Using numerical simulations, we elucidate a well-defined universality class for this case that differs from that of the viscous case, featuring intrinsically anomalous kinetic roughening (despite previous expectations for systems with local interactions and time-dependent noise) and ballistic dynamics. The latter may be relevant to recent quantum spin chain experiments which measure KPZ and ballistic relaxation under different conditions. We identify the ensuing set of scaling exponents in previous discrete interface growth models related with isotropic percolation, and show it to describe the fluctuations of additional continuum systems related with the noisy Korteweg-de Vries equation. Along this process, we additionally elucidate the universality class of the related inviscid stochastic Burgers equation.This work has been partially supported by Ministerio de Ciencia, Innovación y Universidades (Spain), Agencia Estatal de Investigación (AEI, Spain), and Fondo Europeo de Desarrollo Regional (FEDER, EU) through Grants No. PGC2018-094763-B-I00 and No. PID2019-106339GB-I00, and by Comunidad de Madrid (Spain) under the Multiannual Agreements with UC3M in the line of Excellence of University Professors (No. EPUC3M14 and No. EPUC3M23), in the context of the V Plan Regional de Investigación CientÃfica e Innovación Tecnológica (PRICIT). E.R.-F. acknowledges financial support through Contract No. 2022/018 under the EPUC3M23 line
Thermal diffusion of solitons on anharmonic chains with long-range coupling
We extend our studies of thermal diffusion of non-topological solitons to
anharmonic FPU-type chains with additional long-range couplings. The observed
superdiffusive behavior in the case of nearest neighbor interaction (NNI) turns
out to be the dominating mechanism for the soliton diffusion on chains with
long-range interactions (LRI). Using a collective variable technique in the
framework of a variational analysis for the continuum approximation of the
chain, we derive a set of stochastic integro-differential equations for the
collective variables (CV) soliton position and the inverse soliton width. This
set can be reduced to a statistically equivalent set of Langevin-type equations
for the CV, which shares the same Fokker-Planck equation. The solution of the
Langevin set and the Langevin dynamics simulations of the discrete system agree
well and demonstrate that the variance of the soliton increases stronger than
linearly with time (superdiffusion). This result for the soliton diffusion on
anharmonic chains with long-range interactions reinforces the conjecture that
superdiffusion is a generic feature of non-topological solitons.Comment: 11 figure
Wound-up phase turbulence in the Complex Ginzburg-Landau equation
We consider phase turbulent regimes with nonzero winding number in the
one-dimensional Complex Ginzburg-Landau equation. We find that phase turbulent
states with winding number larger than a critical one are only transients and
decay to states within a range of allowed winding numbers. The analogy with the
Eckhaus instability for non-turbulent waves is stressed. The transition from
phase to defect turbulence is interpreted as an ergodicity breaking transition
which occurs when the range of allowed winding numbers vanishes. We explain the
states reached at long times in terms of three basic states, namely
quasiperiodic states, frozen turbulence states, and riding turbulence states.
Justification and some insight into them is obtained from an analysis of a
phase equation for nonzero winding number: rigidly moving solutions of this
equation, which correspond to quasiperiodic and frozen turbulence states, are
understood in terms of periodic and chaotic solutions of an associated system
of ordinary differential equations. A short report of some of our results has
been published in [Montagne et al., Phys. Rev. Lett. 77, 267 (1996)].Comment: 22 pages, 15 figures included. Uses subfigure.sty (included) and
epsf.tex (not included). Related research in
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