Nonlinear Stability of Sources
|
|
- Lizbeth Miller
- 6 years ago
- Views:
Transcription
1 Nonlinear Stability of Sources Björn Sandstede Arnd Scheel Margaret Beck Toan Nguyen Kevin Zumbrun Spiral waves [Li, Ouyang, Petrov, Swinney] [Nettesheim, von Oertzen, Rotermund, Ertl] Dynamics of core / spiral tip Modulations of wave trains in far field [Li, Ouyang, Petrov, Swinney]
2 One-dimensional defects space time Chloride-iodide-malonic acid reaction (CIMA) [Perraud, De Wit, Dulos, De Kepper, Dewel, Borckmans] Standing time-periodic structures space One-dimensional defects Surface waves [Pastur et al.] space Light-sensitive BZ-reaction [Yoneyama, Fujii, Maeda]
3 One-dimensional defects wave train defect wave train time asymptotically periodic in space time-periodic in co-moving frame space Overview: wave trains & group velocities sources existence & bifurcations spectral & nonlinear stability Dynamics of wave trains c wave train k wavenumber ω=ω(k) temporal frequency local wavenumber slowly varying modulations of wavenumber Spectrum of wave trains λ(iγ) Im λ Re λ λ(iγ) = -ic g γ - dγ + Ο( γ 3 )
4 Dynamics of wave trains c wave train k wavenumber ω=ω(k) temporal frequency local wavenumber slowly varying modulations of wavenumber c g q(x,t) q t = c g v x group velocity: direction of transport Spectrum of wave trains λ(iγ) Im λ Re λ λ(iγ) = -ic g γ - dγ + Ο( γ 3 ) Dynamics of wave trains q(x,t) local wavenumber c g slowly varying modulations of wavenumber near k 0 on scale X=ε(x-c gt) and T=ε t/ for 0<ε<<1 Viscous = (k 0 ) q X Formal derivation: [Howard & Kopell], [Kuramoto] Validity over natural time scale 1/ε : [Doelman, S., Scheel, Schneider] Stability of wave trains: [S., Scheel, Schneider, Uecker], [Johnson, Zumbrun] Anticipated dynamics: Zero-mean perturbations converge to zero Lax shocks and rarefaction waves
5 Sources Sources: outgoing transport group velocities point away from core c g transport c g Existence: how do sources arise? Spectral and linear stability: linearized equation is time-periodic Nonlinear stability: previous methods do not apply Essential Hopf instabilities of pulses standing pulse + Im λ Re λ Hopf instability of rest state wave trains Theorem [S., Scheel] k flip-flop target source μ target flip-flop
6 Spatial dynamics time u x = v v x = D 1 [u t c d v f(u)] space u H 1 (S 1 ) L (S 1 ) v wave train = periodic orbit defect = heteroclinic orbit wave train = periodic orbit Spectra of sources Reaction-diffusion system: Standing sources are time-periodic: Floquet spectrum determines spectral stability time space Spectrum of wave trains Evans-function analysis (eigenfunctions u x and u t) L space t>>1 c g<0 exponential weight c g>0 t>>1 exponentially weighted L space defect core x
7 Expected dynamics Burgers equation with advection in far field exponential adjustment of position and phase x=-c g t t position/phase adjustment x=c g t defect core Gaussians error terms x Nonlinear stability Theorem [Beck, Nguyen, S., Zumbrun]: Assume u * (x,t) is a spectrally stable source and let u(x,0)=u * (x,0)+v 0 (x) where v 0 (x)exp(x /M) <ε is sufficiently small. Then there are constants p, φ <ε such that u(x,t)-u * (x-p,t-φ ) < εc exp(-ηt) for (x,t) in Ω 1 and u(x,t)-u * (x,t) < εc exp(-ηt) for (x,t) in Ω. x=-c g t t x=c g t Ω 1 Ω Ω defect core x
8 Nonlinear stability proofs Define appropriate offset from source: Derive equation for offset: Variation-of-constants formula: Fixed-point argument in appropriate function space: No decay in L spaces Decay in weighted L spaces, but nonlinearity not well defined Caveats Long-time dynamics for small localized initial data Heat equation Reaction-diffusion equation Burgers equation Heat equation Reaction diffusion Burgers equation Reason: Gaussian * Gaussian Gaussian Differentiated Gaussian * Gaussian Gaussian
9 Nonlinear stability proofs Define appropriate offset from source: Derive equation for offset: Variation-of-constants formula: Fixed-point argument in appropriate function space: No decay in L spaces Decay in weighted L spaces, but nonlinearity not well defined Define offsets Let u(x,t) be a solution of u t = Du xx + cu x + f(u) near a given defect u * (x,t) Define p(x,t) and φ(x,t) so that p(x,t) and φ(x,t): space-time shift v(x,t): profile changes Substitution gives the following system for the functions p, φ, and v:
10 Solve linearized system: Green s function Linearization about defect: Solve via Green s function: Expansion of Green s function: x=-c gt projection x=c gt t error function plateau y Gaussians error terms x a j(x,t;y,s): G R(x,t;y,s): scalar functions composed of error function plateaus in (x,t-s) times a localized projection function in y plus Gaussian errors sum of moving differentiated Gaussians Variation-of-constants formula Equation for offsets: Variation-of-constants formula: Expansion of Green s function: Collect terms:
11 Nonlinear iterations Equation for offsets: differentiated Gaussian * Gaussian Gaussian Expansion of Green s function: x=-c gt w j(y) x=c gt t Gaussians err(x,t-s) y a j(x,t;y,s) = err(x,t-s) w j(y) + Gaussians G R(x,t;y,s) differentiated Gaussians x Nonlinear iterations Equation for offsets: differentiated Gaussian * Gaussian Gaussian Expansion of Green s function: a j (x,t;y,s) = err(x,t-s) w j (y) + Gaussians G R(x,t;y,s) differentiated Gaussians Nonlinear iteration using templates: v and (p,φ) x sum of moving Gaussians Completes nonlinear stability result
12 Expansion of Green s function Green s function G(x,t;y,s) satisfies Laplace transform: Resolvent kernel (x,t;y,s,λ) is π-periodic in (t,s) and satisfies Makes connection with spatial-dynamics formulation of defects: u x = v u v x = D 1 H 1 (S 1 ) L (S 1 ) [u t c d v f(u)] v Linearize about defect and expand in λ gives bounds needed for (x,t;y,s,λ) Summary: Summary + Outlook Proved that spectrally stable sources are nonlinearly stable in an appropriate sense Outlook: Obtain expansions of dynamics in wedge-shape interface regions: Stability analysis of contact defects (c g =0): contact defect Interaction of sink-source pairs:
Nonlinear stability of time-periodic viscous shocks. Margaret Beck Brown University
Nonlinear stability of time-periodic viscous shocks Margaret Beck Brown University Motivation Time-periodic patterns in reaction-diffusion systems: t x Experiment: chemical reaction chlorite-iodite-malonic-acid
More informationStability of nonlinear waves: pointwise estimates
Stability of nonlinear waves: pointwise estimates Margaret Beck July 24, 25 Abstract This is an expository article containing a brief overview of key issues related to the stability of nonlinear waves,
More informationNonlinear stability of semidiscrete shocks for two-sided schemes
Nonlinear stability of semidiscrete shocks for two-sided schemes Margaret Beck Boston University Joint work with Hermen Jan Hupkes, Björn Sandstede, and Kevin Zumbrun Setting: semi-discrete conservation
More informationSelf-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model
Letter Forma, 15, 281 289, 2000 Self-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model Yasumasa NISHIURA 1 * and Daishin UEYAMA 2 1 Laboratory of Nonlinear Studies and Computations,
More informationCover Page. The handle holds various files of this Leiden University dissertation.
Cover Page The handle http://hdl.handle.net/1887/45233 holds various files of this Leiden University dissertation. Author: Rijk, B. de Title: Periodic pulse solutions to slowly nonlinear reaction-diffusion
More informationNonlinear convective stability of travelling fronts near Turing and Hopf instabilities
Nonlinear convective stability of travelling fronts near Turing and Hopf instabilities Margaret Beck Joint work with Anna Ghazaryan, University of Kansas and Björn Sandstede, Brown University September
More informationSlow Modulation & Large-Time Dynamics Near Periodic Waves
Slow Modulation & Large-Time Dynamics Near Periodic Waves Miguel Rodrigues IRMAR Université Rennes 1 France SIAG-APDE Prize Lecture Jointly with Mathew Johnson (Kansas), Pascal Noble (INSA Toulouse), Kevin
More informationSpectral stability of periodic waves in dispersive models
in dispersive models Collaborators: Th. Gallay, E. Lombardi T. Kapitula, A. Scheel in dispersive models One-dimensional nonlinear waves Standing and travelling waves u(x ct) with c = 0 or c 0 periodic
More informationAbsolute and convective instabilities of waves on unbounded and large bounded domains
Physica D 145 (2000) 233 277 Absolute and convective instabilities of waves on unbounded and large bounded domains Björn Sandstede a, Arnd Scheel b, a Department of Mathematics, Ohio State University,
More informationIntroduction LECTURE 1
LECTURE 1 Introduction The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in
More informationThe Dynamics of Reaction-Diffusion Patterns
The Dynamics of Reaction-Diffusion Patterns Arjen Doelman (Leiden) (Rob Gardner, Tasso Kaper, Yasumasa Nishiura, Keith Promislow, Bjorn Sandstede) STRUCTURE OF THE TALK - Motivation - Topics that won t
More informationA review of stability and dynamical behaviors of differential equations:
A review of stability and dynamical behaviors of differential equations: scalar ODE: u t = f(u), system of ODEs: u t = f(u, v), v t = g(u, v), reaction-diffusion equation: u t = D u + f(u), x Ω, with boundary
More informationStability Analysis of Stationary Solutions for the Cahn Hilliard Equation
Stability Analysis of Stationary Solutions for the Cahn Hilliard Equation Peter Howard, Texas A&M University University of Louisville, Oct. 19, 2007 References d = 1: Commun. Math. Phys. 269 (2007) 765
More informationA Survey of Computational High Frequency Wave Propagation I
A Survey of Computational High Frequency Wave Propagation I Bjorn Engquist University of Texas at Austin CSCAMM Workshop on High Frequency Wave Propagation, University of Maryland, September 19-22, 2005
More informationRELATIVE MORSE INDICES, FREDHOLM INDICES, AND GROUP VELOCITIES. Björn Sandstede. Arnd Scheel. (Communicated by the associate editor name)
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX RELATIVE MORSE INDICES, FREDHOLM INDICES, AND GROUP VELOCITIES Björn Sandstede Department of
More informationNonlinear stability of source defects in the complex Ginzburg-Landau equation
Nonlinear stability of source defects in the complex Ginzburg-Landau equation Margaret Beck Toan T. Nguyen Björn Sandstede Kevin Zumbrun February 12, 214 Abstract In an appropriate moving coordinate frame,
More informationScaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations
Journal of Statistical Physics, Vol. 122, No. 2, January 2006 ( C 2006 ) DOI: 10.1007/s10955-005-8006-x Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations Jan
More informationThe stability of travelling fronts for general scalar viscous balance law
J. Math. Anal. Appl. 35 25) 698 711 www.elsevier.com/locate/jmaa The stability of travelling fronts for general scalar viscous balance law Yaping Wu, Xiuxia Xing Department of Mathematics, Capital Normal
More informationCoherent structures near the boundary between excitable and oscillatory media
Coherent structures near the boundary between excitable and oscillatory media Jeremy Bellay University of Minnesota Department of Computer Science 200 Union St. S.E. Minneapolis, MN 55455, USA Arnd Scheel
More informationScaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations
Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Jan Wehr and Jack Xin Abstract We study waves in convex scalar conservation laws under noisy initial perturbations.
More informationLocalized structures as spatial hosts for unstable modes
April 2007 EPL, 78 (2007 14002 doi: 10.1209/0295-5075/78/14002 www.epljournal.org A. Lampert 1 and E. Meron 1,2 1 Department of Physics, Ben-Gurion University - Beer-Sheva 84105, Israel 2 Department of
More informationChapter Three Theoretical Description Of Stochastic Resonance 24
Table of Contents List of Abbreviations and Symbols 5 Chapter One Introduction 8 1.1 The Phenomenon of the Stochastic Resonance 8 1.2 The Purpose of the Study 10 Chapter Two The Experimental Set-up 12
More informationTHE problem of phase noise and its influence on oscillators
IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 54, NO. 5, MAY 2007 435 Phase Diffusion Coefficient for Oscillators Perturbed by Colored Noise Fergal O Doherty and James P. Gleeson Abstract
More informationin Bounded Domains Ariane Trescases CMLA, ENS Cachan
CMLA, ENS Cachan Joint work with Yan GUO, Chanwoo KIM and Daniela TONON International Conference on Nonlinear Analysis: Boundary Phenomena for Evolutionnary PDE Academia Sinica December 21, 214 Outline
More informationLecture 10: Whitham Modulation Theory
Lecture 10: Whitham Modulation Theory Lecturer: Roger Grimshaw. Write-up: Andong He June 19, 2009 1 Introduction The Whitham modulation theory provides an asymptotic method for studying slowly varying
More informationNotes: Outline. Shock formation. Notes: Notes: Shocks in traffic flow
Outline Scalar nonlinear conservation laws Traffic flow Shocks and rarefaction waves Burgers equation Rankine-Hugoniot conditions Importance of conservation form Weak solutions Reading: Chapter, 2 R.J.
More informationCoherent structures generated by inhomogeneities in oscillatory media
Coherent structures generated by inhomogeneities in oscillatory media Richard Kollár University of Michigan Department of Mathematics 53 Church Street Ann Arbor, MI 4819, USA Arnd Scheel University of
More informationNonlinear stability at the Eckhaus boundary
Nonlinear stability at the Eckhaus boundary Julien Guillod, Guido Schneider 2, Peter Wittwer, Dominik Zimmermann 2 LJLL, Sorbonne Université, 4 Place Jussieu, F-755 Paris, France 2 IADM, Universität Stuttgart,
More informationComputing absolute and essential spectra using continuation
Computing absolute and essential spectra using continuation Jens D.M. Rademacher a,1 Björn Sandstede b, Arnd Scheel c a Centre for Mathematics and Computer Science (CWI), Kruislaan 413, 1098 SJ Amsterdam,
More informationBIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION
BIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION JUNGHO PARK AND PHILIP STRZELECKI Abstract. We consider the 1-dimensional complex Ginzburg Landau equation(cgle) which
More informationStationary radial spots in a planar threecomponent reaction-diffusion system
Stationary radial spots in a planar threecomponent reaction-diffusion system Peter van Heijster http://www.dam.brown.edu/people/heijster SIAM Conference on Nonlinear Waves and Coherent Structures MS: Recent
More informationMS: Nonlinear Wave Propagation in Singular Perturbed Systems
MS: Nonlinear Wave Propagation in Singular Perturbed Systems P. van Heijster: Existence & stability of 2D localized structures in a 3-component model. Y. Nishiura: Rotational motion of traveling spots
More informationSeparation of variables in two dimensions. Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 )
Separation of variables in two dimensions Overview of method: Consider linear, homogeneous equation for u(v 1, v 2 ) Separation of variables in two dimensions Overview of method: Consider linear, homogeneous
More informationA bifurcation approach to non-planar traveling waves in reaction-diffusion systems
gamm header will be provided by the publisher A bifurcation approach to non-planar traveling waves in reaction-diffusion systems Mariana Haragus 1 and Arnd Scheel 2 1 Université de Franche-Comté, Département
More informationComparative Analysis of Packet and Trigger Waves Originating from a Finite Wavelength Instability
11394 J. Phys. Chem. A 2002, 106, 11394-11399 Comparative Analysis of Packet and Trigger Waves Originating from a Finite Wavelength Instability Vladimir K. Vanag*, and Irving R. Epstein Department of Chemistry
More informationA Theory of Spatiotemporal Chaos: What s it mean, and how close are we?
A Theory of Spatiotemporal Chaos: What s it mean, and how close are we? Michael Dennin UC Irvine Department of Physics and Astronomy Funded by: NSF DMR9975497 Sloan Foundation Research Corporation Outline
More informationInstabilities of Wave Trains and Turing Patterns in Large Domains
Instabilities of Wave Trains and Turing Patterns in Large Domains Jens D.M. Rademacher Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39 10117 Berlin, Germany Arnd Scheel University
More informationTitleScattering and separators in dissip. Author(s) Nishiura, Yasumasa; Teramoto, Takas
TitleScattering and separators in dissip Author(s) Nishiura, Yasumasa; Teramoto, Takas Citation Physical review. E, Statistical, no physics, 67(5): 056210 Issue Date 2003 DOI Doc URLhttp://hdl.handle.net/2115/35226
More informationForced patterns near a Turing-Hopf bifurcation
Macalester College From the SelectedWorks of Chad M. Topaz 21 Forced patterns near a Turing-Hopf bifurcation Chad M. Topaz, Macalester College Anne Catlla, Wofford College Available at: https://works.bepress.com/chad_topaz/16/
More informationLooking Through the Vortex Glass
Looking Through the Vortex Glass Lorenz and the Complex Ginzburg-Landau Equation Igor Aronson It started in 1990 Project started in Lorenz Kramer s VW van on the way back from German Alps after unsuccessful
More information1 Introduction Travelling-wave solutions of parabolic equations on the real line arise in a variety of applications. An important issue is their stabi
Essential instability of pulses, and bifurcations to modulated travelling waves Bjorn Sandstede Department of Mathematics The Ohio State University 231 West 18th Avenue Columbus, OH 4321, USA Arnd Scheel
More informationPHYSICAL REVIEW E VOLUME 62, NUMBER 6. Target waves in the complex Ginzburg-Landau equation
PHYSICAL REVIEW E VOLUME 62, NUMBER 6 DECEMBER 2000 Target waves in the complex Ginzburg-Landau equation Matthew Hendrey, Keeyeol Nam, Parvez Guzdar,* and Edward Ott University of Maryland, Institute for
More information0.3.4 Burgers Equation and Nonlinear Wave
16 CONTENTS Solution to step (discontinuity) initial condition u(x, 0) = ul if X < 0 u r if X > 0, (80) u(x, t) = u L + (u L u R ) ( 1 1 π X 4νt e Y 2 dy ) (81) 0.3.4 Burgers Equation and Nonlinear Wave
More informationLast time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler:
Lecture 7 18.086 Last time: Diffusion - Numerical scheme (FD) Heat equation is dissipative, so why not try Forward Euler: U j,n+1 t U j,n = U j+1,n 2U j,n + U j 1,n x 2 Expected accuracy: O(Δt) in time,
More informationFREDHOLM ALTERNATIVE. y Y. We want to know when the
FREDHOLM ALTERNATIVE KELLY MCQUIGHAN (I) Statement and Theory of Fredholm Alternative Let X, Y be Banach spaces, A L(X, Y), equation Ax = y is solvable for x X. y Y. We want to know when the (a) Intuition
More informationof the Schnakenberg model
Pulse motion in the semi-strong limit of the Schnakenberg model Newton Institute 2005 Jens Rademacher, Weierstraß Institut Berlin joint work with Michael Ward (UBC) Angelfish 2, 6, 12 months old [Kondo,
More informationTRAVELLING WAVES. Morteza Fotouhi Sharif Univ. of Technology
TRAVELLING WAVES Morteza Fotohi Sharif Univ. of Technology Mini Math NeroScience Mini Math NeroScience Agst 28 REACTION DIFFUSION EQUATIONS U = DU + f ( U ) t xx x t > U n D d 1 = d j > d n 2 Travelling
More informationNon-linear Scalar Equations
Non-linear Scalar Equations Professor Dr. E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro August 24, 2014 1 / 44 Overview Here
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationDestabilization mechanisms of periodic pulse patterns near a homoclinic limit
Destabilization mechanisms of periodic pulse patterns near a homoclinic limit Arjen Doelman, Jens Rademacher, Björn de Rijk, Frits Veerman Abstract It has been observed in the Gierer-Meinhardt equations
More informationThe Evans function and the stability of travelling waves
The Evans function and the stability of travelling waves Jitse Niesen (University of Leeds) Collaborators: Veerle Ledoux (Ghent) Simon Malham (Heriot Watt) Vera Thümmler (Bielefeld) PANDA meeting, University
More informationControl of spiral instabilities in reaction diffusion systems*
Pure Appl. Chem., Vol. 77, No. 8, pp. 1395 1408, 2005. DOI: 10.1351/pac200577081395 2005 IUPAC Control of spiral instabilities in reaction diffusion systems* Hui-Min Liao 1, Min-Xi Jiang 1, Xiao-Nan Wang
More informationLecture 2 Supplementary Notes: Derivation of the Phase Equation
Lecture 2 Supplementary Notes: Derivation of the Phase Equation Michael Cross, 25 Derivation from Amplitude Equation Near threshold the phase reduces to the phase of the complex amplitude, and the phase
More informationLecture Notes on Numerical Schemes for Flow and Transport Problems
Lecture Notes on Numerical Schemes for Flow and Transport Problems by Sri Redeki Pudaprasetya sr pudap@math.itb.ac.id Department of Mathematics Faculty of Mathematics and Natural Sciences Bandung Institute
More informationHI CAMBRIDGE n S P UNIVERSITY PRESS
Infinite-Dimensional Dynamical Systems An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors JAMES C. ROBINSON University of Warwick HI CAMBRIDGE n S P UNIVERSITY PRESS Preface
More informationShadow system for adsorbate-induced phase transition model
RIMS Kôkyûroku Bessatsu B5 (9), 4 Shadow system for adsorbate-induced phase transition model Dedicated to Professor Toshitaka Nagai on the occasion of his sixtieth birthday By Kousuke Kuto and Tohru Tsujikawa
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationLecture Notes on Numerical Schemes for Flow and Transport Problems
Lecture Notes on Numerical Schemes for Flow and Transport Problems by Sri Redeki Pudaprasetya sr pudap@math.itb.ac.id Department of Mathematics Faculty of Mathematics and Natural Sciences Bandung Institute
More informationNon-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3
Non-linear Wave Propagation and Non-Equilibrium Thermodynamics - Part 3 Tommaso Ruggeri Department of Mathematics and Research Center of Applied Mathematics University of Bologna January 21, 2017 ommaso
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More informationSpatial decay of rotating waves in parabolic systems
Spatial decay of rotating waves in parabolic systems Nonlinear Waves, CRC 701, Bielefeld, June 19, 2013 Denny Otten Department of Mathematics Bielefeld University Germany June 19, 2013 CRC 701 Denny Otten
More informationIMA Preprint Series # 2054
COMPUTING ABSOLUTE AND ESSENTIAL SPECTRA USING CONTINUATION By Jens D.M. Rademacher Björn Sandstede and Arnd Scheel IMA Preprint Series # 054 ( June 005 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
More informationMyopic Models of Population Dynamics on Infinite Networks
Myopic Models of Population Dynamics on Infinite Networks Robert Carlson Department of Mathematics University of Colorado at Colorado Springs rcarlson@uccs.edu June 30, 2014 Outline Reaction-diffusion
More informationA stochastic particle system for the Burgers equation.
A stochastic particle system for the Burgers equation. Alexei Novikov Department of Mathematics Penn State University with Gautam Iyer (Carnegie Mellon) supported by NSF Burgers equation t u t + u x u
More informationHandbook of Stochastic Methods
Springer Series in Synergetics 13 Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences von Crispin W Gardiner Neuausgabe Handbook of Stochastic Methods Gardiner schnell und portofrei
More informationCoarsening fronts. Arnd Scheel University of Minnesota School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA.
Coarsening fronts Arnd Scheel University of Minnesota School of Mathematics 206 Church St. S.E. Minneapolis, MN 55455, USA Abstract We characterize the spatial spreading of the coarsening process in the
More informationAPPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems
APPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fourth Edition Richard Haberman Department of Mathematics Southern Methodist University PEARSON Prentice Hall PEARSON
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Partial differential equations arise in a number of physical problems, such as fluid flow, heat transfer, solid mechanics and biological processes. These
More informationExistence of (Generalized) Breathers in Periodic Media
Existence of (Generalized) Breathers in Periodic Media Guido Schneider Lehrstuhl für Analysis und Modellierung www.iadm.uni-stuttgart.de/lstanamod/schneider/ Collaborators:. Martina Chirilus-Bruckner,
More informationHOPF DANCES NEAR THE TIPS OF BUSSE BALLOONS. Arjen Doelman. (Communicated by the associate editor name)
Manuscript submitted to AIMS Journals Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX HOPF DANCES NEAR THE TIPS OF BUSSE BALLOONS Arjen Doelman Mathematisch Instituut, Universiteit
More informationCharacteristics for IBVP. Notes: Notes: Periodic boundary conditions. Boundary conditions. Notes: In x t plane for the case u > 0: Solution:
AMath 574 January 3, 20 Today: Boundary conditions Multi-dimensional Wednesday and Friday: More multi-dimensional Reading: Chapters 8, 9, 20 R.J. LeVeque, University of Washington AMath 574, January 3,
More informationDrift velocity of rotating spiral waves in the weak deformation approximation
JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 8 22 AUGUST 2003 Drift velocity of rotating spiral waves in the weak deformation approximation Hong Zhang a) Bambi Hu and Department of Physics, University
More information1D spirals: is multi stability essential?
1D spirals: is multi staility essential? A. Bhattacharyay Dipartimento di Fisika G. Galilei Universitá di Padova Via Marzolo 8, 35131 Padova Italy arxiv:nlin/0502024v2 [nlin.ps] 23 Sep 2005 Feruary 8,
More informationPattern formation in Nikolaevskiy s equation
Stephen Cox School of Mathematical Sciences, University of Nottingham Differential Equations and Applications Seminar 2007 with Paul Matthews, Nottingham Outline What is Nikolaevskiy s equation? Outline
More informationMetastability of solitary roll wave solutions of the St. Venant equations with viscosity
Metastability of solitary roll wave solutions of the St. Venant equations with viscosity Blake Barker Mathew A. Johnson L.Miguel Rodrigues Kevin Zumbrun March 9, Keywords: solitary waves; St. Venant equations;
More informationTravelling waves. Chapter 8. 1 Introduction
Chapter 8 Travelling waves 1 Introduction One of the cornerstones in the study of both linear and nonlinear PDEs is the wave propagation. A wave is a recognizable signal which is transferred from one part
More informationThe Model and Preliminaries Problems and Solutions
Chapter The Model and Preliminaries Problems and Solutions Facts that are recalled in the problems The wave equation u = c u + G(x,t), { u(x,) = u (x), u (x,) = v (x), u(x,t) = f (x,t) x Γ, u(x,t) = x
More informationThe Nonlinear Schrodinger Equation
Catherine Sulem Pierre-Louis Sulem The Nonlinear Schrodinger Equation Self-Focusing and Wave Collapse Springer Preface v I Basic Framework 1 1 The Physical Context 3 1.1 Weakly Nonlinear Dispersive Waves
More informationDestabilization mechanisms of periodic pulse patterns near a homoclinic limit
Destabilization mechanisms of periodic pulse patterns near a homoclinic limit Arjen Doelman, Jens Rademacher, Björn de Rijk, Frits Veerman Abstract It has been observed in the Gierer-Meinhardt equations
More informationDecay profiles of a linear artificial viscosity system
Decay profiles of a linear artificial viscosity system Gene Wayne, Ryan Goh and Roland Welter Boston University rwelter@bu.edu July 2, 2018 This research was supported by funding from the NSF. Roland Welter
More informationPrimary, secondary instabilities and control of the rotating-disk boundary layer
Primary, secondary instabilities and control of the rotating-disk boundary layer Benoît PIER Laboratoire de mécanique des fluides et d acoustique CNRS Université de Lyon École centrale de Lyon, France
More informationStability and instability of nonlinear waves:
Stability and instability of nonlinear waves: Introduction 1. Nonlinear Waves 2. Stability problems 3. Stability of pulses and fronts 4. Stability of periodic waves 1 Nonlinear waves particular solutions
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationASYMPTOTICS TOWARD THE PLANAR RAREFACTION WAVE FOR VISCOUS CONSERVATION LAW IN TWO SPACE DIMENSIONS
TANSACTIONS OF THE AMEICAN MATHEMATICAL SOCIETY Volume 35, Number 3, Pages 13 115 S -9947(999-4 Article electronically publishe on September, 1999 ASYMPTOTICS TOWAD THE PLANA AEFACTION WAVE FO VISCOUS
More informationNumerical Methods for Hyperbolic Conservation Laws Lecture 4
Numerical Methods for Hyperbolic Conservation Laws Lecture 4 Wen Shen Department of Mathematics, Penn State University Email: wxs7@psu.edu Oxford, Spring, 018 Lecture Notes online: http://personal.psu.edu/wxs7/notesnumcons/
More informationFDM for wave equations
FDM for wave equations Consider the second order wave equation Some properties Existence & Uniqueness Wave speed finite!!! Dependence region Analytical solution in 1D Finite difference discretization Finite
More informationTurbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg-Landau equation
Turbulence control by developing a spiral wave with a periodic signal injection in the complex Ginzburg-Landau equation Hong Zhang, 1, * Bambi Hu, 1,2 Gang Hu, 1,3 Qi Ouyang, 4 and J. Kurths 5 1 Department
More informationsystem CWI, Amsterdam May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit
CWI, Amsterdam heijster@cwi.nl May 21, 2008 Dynamic Analysis Seminar Vrije Universiteit Joint work: A. Doelman (CWI/UvA), T.J. Kaper (BU), K. Promislow (MSU) Outline 1 2 3 4 Outline 1 2 3 4 Paradigm U
More information2.3. Quantitative Properties of Finite Difference Schemes. Reading: Tannehill et al. Sections and
3 Quantitative Properties of Finite Difference Schemes 31 Consistency, Convergence and Stability of FD schemes Reading: Tannehill et al Sections 333 and 334 Three important properties of FD schemes: Consistency
More informationChaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 6 JUNE 2000 Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves David Cai a) and David W. McLaughlin Courant Institute
More informationNo-hair and uniqueness results for analogue black holes
No-hair and uniqueness results for analogue black holes LPT Orsay, France April 25, 2016 [FM, Renaud Parentani, and Robin Zegers, PRD93 065039] Outline Introduction 1 Introduction 2 3 Introduction Hawking
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationBorel Summability in PDE initial value problems
Borel Summability in PDE initial value problems Saleh Tanveer (Ohio State University) Collaborator Ovidiu Costin & Guo Luo Research supported in part by Institute for Math Sciences (IC), EPSRC & NSF. Main
More informationarxiv: v1 [physics.flu-dyn] 14 Jun 2014
Observation of the Inverse Energy Cascade in the modified Korteweg de Vries Equation D. Dutykh and E. Tobisch LAMA, UMR 5127 CNRS, Université de Savoie, Campus Scientifique, 73376 Le Bourget-du-Lac Cedex,
More informationApplied Asymptotic Analysis
Applied Asymptotic Analysis Peter D. Miller Graduate Studies in Mathematics Volume 75 American Mathematical Society Providence, Rhode Island Preface xiii Part 1. Fundamentals Chapter 0. Themes of Asymptotic
More informationStability of Travelling Wave Solutions for Coupled Surface and Grain Boundary Motion
Stability of Travelling Wave Solutions for Coupled Surface and Grain Boundary Motion Margaret Beck Zhenguo Pan Brian Wetton May 14, 2010 Abstract We investigate the spectral stability of the travelling
More informationThe Application of Extreme Learning Machine based on Gaussian Kernel in Image Classification
he Application of Extreme Learning Machine based on Gaussian Kernel in Image Classification Weijie LI, Yi LIN Postgraduate student in College of Survey and Geo-Informatics, tongji university Email: 1633289@tongji.edu.cn
More informationFinite Difference Method
Capter 8 Finite Difference Metod 81 2nd order linear pde in two variables General 2nd order linear pde in two variables is given in te following form: L[u] = Au xx +2Bu xy +Cu yy +Du x +Eu y +Fu = G According
More informationChapter 1. Introduction to Nonlinear Space Plasma Physics
Chapter 1. Introduction to Nonlinear Space Plasma Physics The goal of this course, Nonlinear Space Plasma Physics, is to explore the formation, evolution, propagation, and characteristics of the large
More informationOscillatory Turing Patterns in a Simple Reaction-Diffusion System
Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007, pp. 234 238 Oscillatory Turing Patterns in a Simple Reaction-Diffusion System Ruey-Tarng Liu and Sy-Sang Liaw Department of Physics,
More information