Chapter 18 - Stability of Travelling Waves

[1]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[2]  N. G. Parke,et al.  Ordinary Differential Equations. , 1958 .

[3]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[4]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[5]  John Evans.  Nerve Axon Equations: III Stability of the Nerve Impulse , 1972 .

[6]  Stability , 1973 .

[7]  A. Aziz,et al.  Numerical solutions of boundary value problems for ordinary differential equations : [proceedings of a symposium held at the University of Maryland, Baltimore County Campus, June 10-12, 1974] , 1975 .

[8]  David H. Sattinger,et al.  On the stability of waves of nonlinear parabolic systems , 1976 .

[9]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[10]  K. Kirchgässner.  Wave-solutions of reversible systems and applications , 1982 .

[11]  Patrick S. Hagan,et al.  Spiral Waves in Reaction-Diffusion Equations , 1982 .

[12]  André Vanderbauwhede,et al.  Local bifurcation and symmetry , 1982 .

[13]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[14]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[15]  J. Prüss.  On the Spectrum of C 0 -Semigroups , 1984 .

[16]  Christopher Jones,et al.  Stability of the travelling wave solution of the FitzHugh-Nagumo system , 1984 .

[17]  Kenneth J. Palmer,et al.  Exponential dichotomies and transversal homoclinic points , 1984 .

[18]  Jan Prüss,et al.  On the spectrum of ₀-semigroups , 1984 .

[19]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[20]  J. Maddocks.  Errata: Restricted Quadratic Forms and Their Application to Bifurcation and Stability in Constrained Variational Principles , 1985 .

[21]  E. Yanagida.  Stability of fast travelling pulse solutions of the FitzHugh—Nagumo equations , 1985 .

[22]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[23]  R. Gardner.  Existence of multidimensional travelling wave solutions of an initial-boundary value problem , 1986 .

[24]  Yasumasa Nishiura,et al.  Stability of singularly perturbed solutions to systems of reaction-diffusion equations , 1987 .

[25]  Eiji Yanagida,et al.  Branching of double pulse solutions from single pulse solutions in nerve axon equations , 1987 .

[26]  J. Keener,et al.  Singular perturbation theory of traveling waves in excitable media (a review) , 1988 .

[27]  M. Grillakis,et al.  Linearized instability for nonlinear Schr?odinger and Klein-Gordon equations , 1988 .

[28]  Kenneth J. Palmer,et al.  Exponential dichotomies and Fredholm operators , 1988 .

[29]  Eiji Yanagida,et al.  Stability of double-pulse solutions in nerve axon equations , 1989 .

[30]  Jack K. Hale,et al.  Slow-motion manifolds, dormant instability, and singular perturbations , 1989 .

[31]  M. Grillakis,et al.  Analysis of the linearization around a critical point of an infinite dimensional Hamiltonian system , 1990 .

[32]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

[33]  J. Carr,et al.  Metastable patterns in solutions of ut = ϵ2uxx − f(u) , 1989 .

[34]  Peter W. Bates,et al.  Invariant Manifolds for Semilinear Partial Differential Equations , 1989 .

[35]  P. Glendinning.  Subsidiary bifurcations near bifocal homoclinic orbits , 1989, Mathematical Proceedings of the Cambridge Philosophical Society.

[36]  Xiao-Biao Lin,et al.  Using Melnikov's method to solve Silnikov's problems , 1990, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[37]  R. Gardner,et al.  TRAVELING WAVES OF A PERTURBED DIFFUSION EQUATION ARISING IN A PHASE FIELD MODEL , 1990 .

[38]  A. Spence,et al.  Continuation and Bifurcations: Numerical Techniques and Applications , 1990 .

[39]  S. Chow,et al.  Homoclinic bifurcation at resonant eigenvalues , 1990 .

[40]  Wolf-Jürgen Beyn.  Global Bifurcations and their Numerical Computation , 1990 .

[41]  Wolf-Jürgen Beyn,et al.  The Numerical Computation of Connecting Orbits in Dynamical Systems , 1990 .

[42]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[43]  Masayasu Mimura,et al.  Singular limit analysis of stability of traveling wave solutions in bistable reaction-diffusion systems , 1990 .

[44]  Ehud Meron,et al.  Patterns of propagating pulses , 1990 .

[45]  J. Alexander,et al.  A topological invariant arising in the stability analysis of travelling waves. , 1990 .

[46]  Christopher K. R. T. Jones,et al.  Stability of travelling wave solutions of diffusive predator-prey systems , 1991 .

[47]  The bifurcations of countable connections from a twisted heteroclinic loop , 1991 .

[48]  Mark J. Friedman,et al.  Numerical computation and continuation of invariant manifolds connecting fixed points , 1991 .

[49]  B. Deng.  The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations , 1991 .

[50]  E. Meron.  Pattern formation in excitable media , 1992 .

[51]  J. Eckmann,et al.  Diffusive repair for the Ginzburg-Landau equation , 1992 .

[52]  P. C. Hohenberg,et al.  Fronts, pulses, sources and sinks in generalized complex Ginzberg-Landau equations , 1992 .

[53]  R. Pego,et al.  On asymptotic stability of solitary waves , 1992 .

[54]  J. Eckmann,et al.  Diffuse repair for the Ginzburg-Landau equation , 1992 .

[55]  Antti Kupiainen,et al.  Renormalization Group and the Ginzburg-Landau equation , 1992 .

[56]  Bernold Fiedler,et al.  Homoclinic period blow-up in reversible and conservative systems , 1992 .

[57]  Klaus Kirchgässner,et al.  On the nonlinear dynamics of travelling fronts , 1992 .

[58]  Michael I. Weinstein,et al.  Eigenvalues, and instabilities of solitary waves , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[59]  Antti Kupiainen,et al.  Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations , 1993, chao-dyn/9306008.

[60]  J. Maddocks,et al.  On the stability of KdV multi‐solitons , 1993 .

[61]  J. Alexander,et al.  Existence and stability of asymptotically oscillatory triple pulses , 1993 .

[62]  Hiroshi Kokubu,et al.  Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields , 1993 .

[63]  R. Gardner,et al.  ON THE STRUCTURE OF THE SPECTRA OF PERIODIC TRAVELING WAVES , 1993 .

[64]  Yu. A. Kuznetsov,et al.  NUMERICAL DETECTION AND CONTINUATION OF CODIMENSION-TWO HOMOCLINIC BIFURCATIONS , 1994 .

[65]  Todd Kapitula,et al.  STABILITY OF TRAVELING WAVES FOR NONCONVEX SCALAR VISCOUS CONSERVATION-LAWS , 1993 .

[66]  On Šilnikov's homoclinic-saddle-focus theorem , 1993 .

[67]  T. Gallay,et al.  Local stability of critical fronts in nonlinear parabolic partial differential equations , 1994 .

[68]  Vitaly Volpert,et al.  Traveling Wave Solutions of Parabolic Systems , 1994 .

[69]  Martin Krupa,et al.  The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit , 1994, Ergodic Theory and Dynamical Systems.

[70]  J. Eckmann,et al.  The non-linear stability of front solutions for parabolic partial differential equations , 1994 .

[71]  Alan R. Champneys,et al.  Subsidiary homoclinic orbits to a saddle focus for reversible systems , 1994 .

[72]  Lloyd N. Trefethen,et al.  Pseudospectra of the Convection-Diffusion Operator , 1993, SIAM J. Appl. Math..

[73]  Yasumasa Nishiura,et al.  Coexistence of Infinitely Many Stable Solutions to Reaction Diffusion Systems in the Singular Limit , 1994 .

[74]  $$L^2$$ of periodic equilibria of reaction diffusion systems , 1994 .

[75]  J. Bricmont,et al.  Stability of moving fronts in the Ginzburg-Landau equation , 1993, chao-dyn/9306009.

[76]  J. Alexander,et al.  Existence and stability of asymptotically oscillatory double pulses. , 1994 .

[77]  On the Stability of Traveling Waves in Weighted L∞ Spaces , 1994 .

[78]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[79]  June-Gi Kim.  HOMOCLINIC ORBITS FOR HAMILTONIAN SYSTEMS , 1995 .

[80]  Thomas J. Bridges,et al.  A proof of the Benjamin-Feir instability , 1995 .

[81]  Christopher Jones,et al.  Geometric singular perturbation theory , 1995 .

[82]  THE PRINCIPLE OF LINEARIZED INSTABILITY FOR SPACE-PERIODIC EQUILIBRIA OF NAVIER-STOKES ON AN INFINITE PLATE , 1995 .

[83]  J. Maddocks,et al.  Constrained Variational Principles and Stability in Hamiltonian Systems , 1995 .

[84]  Robert D. Russell,et al.  Numerical solution of boundary value problems for ordinary differential equations , 1995, Classics in applied mathematics.

[85]  A. Bose,et al.  Stability of the in-phase travelling wave solutions in a pair of coupled nerve fibers , 1995 .

[86]  新居 俊作.  An extension of the stability index for travelling wave solutions and its application for bifurcations , 1995 .

[87]  G. Schneider.  Diffusive stability of spatial periodic solutions of the Swift-Hohenberg equation , 1996 .

[88]  A. Mielke,et al.  Dynamics of Nonlinear Waves in Dissipative Systems Reduction, Bifurcation and Stability , 1996 .

[89]  Hsueh-Chia Chang,et al.  Local stability theory of solitary pulses in an active medium , 1996 .

[90]  J. Roquejoffre,et al.  Global stability of traveling fronts and convergence towards stacked families of waves in monotone parabolic systems , 1996 .

[91]  J. Eckmann,et al.  Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation , 1996, patt-sol/9612002.

[92]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[93]  H. Engel,et al.  Chemical Waves and Patterns , 1996 .

[94]  Y. Nishiura,et al.  Stability of traveling waves and a relation between the Evans function and the SLEP equation. , 1996 .

[95]  J. Nathan Kutz,et al.  Stability of Pulses in Nonlinear Optical Fibers Using Phase-Sensitive Amplifiers , 1996, SIAM J. Appl. Math..

[96]  Björn Sandstede,et al.  A numerical toolbox for homoclinic bifurcation analysis , 1996 .

[97]  Th. Gallay,et al.  Stability of travelling waves for a damped hyperbolic equation , 1996 .

[98]  Thomas J. Bridges,et al.  Absolute and convective instabilities of spatially periodic flows , 1996, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[99]  Alan R. Champneys,et al.  Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system , 1996 .

[100]  B. Sandstede,et al.  Fast and Slow Waves in the FitzHugh–Nagumo Equation , 1997 .

[101]  Todd Kapitula,et al.  Stability criterion for bright solitary waves of the perturbed cubic-quintic Schro¨dinger equation , 1997, patt-sol/9701011.

[102]  Shunsaku Nii.  Stability of travelling multiple-front (multiple-back) wave solutions of the Fitzhugh-Nagumo equations , 1997 .

[103]  Björn Sandstede,et al.  Stability of pulses on optical fibers with phase-sensitive amplifiers , 1997 .

[104]  Michael I. Weinstein,et al.  Convective Linear Stability of Solitary Waves for Boussinesq Equations , 1997 .

[105]  Alexander Mielke,et al.  Instability and Stability of Rolls in the Swift–Hohenberg Equation , 1997 .

[106]  Lloyd N. Trefethen,et al.  Pseudospectra of Linear Operators , 1997, SIAM Rev..

[107]  Björn Sandstede,et al.  Convergence estimates for the numerical approximation of homoclinic solutions , 1997 .

[108]  Shunsaku Nii.  An extension of the stability index for traveling-wave solutions and its application to bifurcations , 1997 .

[109]  Adrian Ankiewicz,et al.  Solitons : nonlinear pulses and beams , 1997 .

[110]  Martin Golubitsky,et al.  Meandering of the Spiral Tip: An Alternative Approach , 1997 .

[111]  Björn Sandstede,et al.  Dynamics of Spiral Waves on Unbounded Domains Using Center-Manifold Reductions , 1997 .

[112]  R. Gardner,et al.  Spectral analysis of long wavelength periodic waves and applications. , 1997 .

[113]  Björn Sandstede,et al.  Exponential Dichotomies for Solitary-Wave Solutions of Semilinear Elliptic Equations on Infinite Cylinders , 1997 .

[114]  Björn Sandstede,et al.  Existence and stability of N -pulses on optical fibers with phase-sensitive amplifiers , 1997 .

[115]  Bjorn Sandstede.  Instability of localized buckling modes in a one-dimensional strut model , 1997, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[116]  Alan R. Champneys,et al.  Localization and solitary waves in solid mechanics , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[117]  Markus Bär,et al.  Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model , 1997 .

[118]  Robert Gardner,et al.  Stability analysis of singular patterns in the 1-D Gray-Scott model I: a matched asymptotics approach , 1998 .

[119]  J. A. Kuznecov.  Elements of applied bifurcation theory , 1998 .

[120]  Jörg Härterich.  Cascades of reversible homoclinic orbits to a saddle-focus equilibrium , 1998 .

[121]  A. Mielke,et al.  Diffusive Mixing of Stable States in the Ginzburg–Landau Equation , 1998, patt-sol/9801003.

[122]  Björn Sandstede,et al.  Stability of multiple-pulse solutions , 1998 .

[123]  Y. A. Li,et al.  Structural stability of non-ground state traveling waves of coupled nonlinear Schro , 1998 .

[124]  Hsueh-Chia Chang,et al.  Generation and Suppression of Radiation by Solitary Pulses , 1998, SIAM J. Appl. Math..

[125]  Bernold Fiedler,et al.  Normal Forms, Resonances, and Meandering Tip Motions near Relative Equilibria of Euclidean Group Actions , 1998 .

[126]  T. Gallay,et al.  Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system , 1997, patt-sol/9705008.

[127]  Todd Kapitula,et al.  Instability mechanism for bright solitary-wave solutions to the cubic–quintic Ginzburg–Landau equation , 1998 .

[128]  G. Schneider.  Nonlinear Stability of Taylor Vortices in Infinite Cylinders , 1998 .

[129]  Björn Sandstede,et al.  Stability of N -fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation , 1998 .

[130]  Arnd Scheel,et al.  Bifurcation to spiral waves in reaction-diffusion systems , 1998 .

[131]  A. Bruno.  Normal forms , 1998 .

[132]  Kevin Zumbrun,et al.  The gap lemma and geometric criteria for instability of viscous shock profiles , 1998 .

[133]  Todd Kapitula,et al.  Stability of bright solitary-wave solutions to perturbed nonlinear Schro , 1998 .

[134]  B. Fiedler,et al.  Large patterns of elliptic systems in infinite cylinders , 1998 .

[135]  Klaus Kirchgässner,et al.  Stability of Fronts for a KPP-System, II: The Critical Case , 1998 .

[136]  D. Peregrine.  A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95. , 1998, Journal of Fluid Mechanics.

[137]  Gunilla Kreiss,et al.  Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals , 1999 .

[138]  Todd Kapitula,et al.  The Evans function and generalized Melnikov integrals , 1999 .

[139]  James Demmel,et al.  LAPACK Users' Guide, Third Edition , 1999, Software, Environments and Tools.

[140]  Frédéric Dias,et al.  NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES , 1999 .

[141]  H. Uecker.  Diffusive stability of rolls in the two–dimensional real and complex swift–hohenberg equation , 1999 .

[142]  Shunsaku Nii,et al.  A Topological Proof of Stability of N-Front Solutions of the FitzHugh–Nagumo Equations , 1999 .

[143]  T. Bridges,et al.  The Symplectic Evans Matrix,¶and the Instability of Solitary Waves and Fronts , 2001 .

[144]  Björn Sandstede,et al.  Bifurcations and Dynamics of Spiral Waves , 1999 .

[145]  W. Saarloos,et al.  Front propagation into unstable states : universal algebraic convergence towards uniformly translating pulled fronts , 2000, cond-mat/0003181.

[146]  S. Levandosky.  A stability analysis of fifth-order water wave models , 1999 .

[147]  J. Rubin,et al.  Stability, bifurcations and edge oscillations in standing pulse solutions to an inhomogeneous reaction-diffusion system , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[148]  J. Alexander,et al.  LINEAR INSTABILITY OF SOLITARY WAVES OF A BOUSSINESQ-TYPE EQUATION: A COMPUTER ASSISTED COMPUTATION , 1999 .

[149]  Thomas J. Bridges,et al.  Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry , 1999, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[150]  A. Scheel,et al.  Essential instability of pulses and bifurcations to modulated travelling waves , 1999, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[151]  E.C.M. Crooks.  STABILITY OF TRAVELLING-WAVE SOLUTIONS FOR REACTION-DIFFUSION-CONVECTION SYSTEMS , 2000 .

[152]  Keith Promislow,et al.  The Mechanism of the Polarizational Mode Instability in Birefringent Fiber Optics , 2000, SIAM J. Math. Anal..

[153]  Björn Sandstede,et al.  Numerical Computation of Solitary Waves in Infinite Cylindrical Domains , 2000, SIAM J. Numer. Anal..

[154]  Shunsaku Nii,et al.  The accumulation of eigenvalues in a stability problem , 2000 .

[155]  Alice C. Yew,et al.  Stability Analysis of Multipulses in Nonlinearly-Coupled Schrodinger Equations , 2000 .

[156]  Björn Sandstede,et al.  Gluing unstable fronts and backs together can produce stable pulses , 2000 .

[157]  Thierry Gallay,et al.  Scaling Variables and Stability of Hyperbolic Fronts , 1998, SIAM J. Math. Anal..

[158]  Thomas J. Bridges.  Universal geometric condition for the transverse instability of solitary waves , 2000, Physical review letters.

[159]  Markus Bär,et al.  Stable bound states of pulses in an excitable medium , 2000 .

[160]  Instability of multiple pulses in coupled nonlinear Schrodinger equations , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[161]  Björn Sandstede,et al.  Absolute and Convective Instabilities of Waves on Unbounded and Large Bounded Domains , 2022 .

[162]  Tsutomu Ikeda,et al.  Bifurcation Phenomena from Standing Pulse Solutions in Some Reaction-Diffusion Systems , 2000 .

[163]  Björn Sandstede,et al.  Spectral stability of modulated travelling waves bifurcating near essential instabilities , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[164]  G. Schneider.  Existence and stability of modulating pulse-solutions in a phenomenological model of nonlinear optics , 2000 .

[165]  Todd Kapitula,et al.  Existence and stability of standing hole solutions to complex Ginzburg-Landau equations , 1999, patt-sol/9902002.

[166]  Jack Xin,et al.  Front Propagation in Heterogeneous Media , 2000, SIAM Rev..

[167]  Scheel,et al.  Absolute versus convective instability of spiral waves , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[168]  Hannes Uecker,et al.  Stable Modulating Multipulse Solutions for Dissipative Systems with Resonant Spatially Periodic Forcing , 2001, J. Nonlinear Sci..

[169]  Björn Sandstede,et al.  On the Structure of Spectra of Modulated Travelling Waves , 2001 .

[170]  Robert Gardner,et al.  Large stable pulse solutions in reaction-diffusion equations , 2001 .

[171]  Björn Sandstede,et al.  On the Stability of Periodic Travelling Waves with Large Spatial Period , 2001 .

[172]  Björn Sandstede,et al.  Essential instabilities of fronts: bifurcation, and bifurcation failure , 2001 .

[173]  Xiao-Biao Lin,et al.  Construction and asymptotic stability of structurally stable internal layer solutions , 2001 .

[174]  A. Yew,et al.  Multipulses of Nonlinearly-coupled Schrr Odinger Equations , 1999 .

[175]  W. Beyn,et al.  Chapter 4 – Numerical Continuation, and Computation of Normal Forms , 2002 .

[176]  Shin-Ichiro Ei,et al.  The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems , 2002 .

[177]  Kevin Zumbrun,et al.  Errata to: 'Pointwise semigroup methods and stability of viscous shock waves' , 2002 .

[178]  Robert Gardner,et al.  A stability index analysis of 1-D patterns of the Gray-Scott model , 2002 .

[179]  Alexander Mielke,et al.  The Ginzburg-Landau Equation in Its Role as a Modulation Equation , 2002 .

[180]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .