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2015 | OriginalPaper | Buchkapitel

17. Spatial Dynamics

verfasst von : Christian Kuehn

Erschienen in: Multiple Time Scale Dynamics

Verlag: Springer International Publishing

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Abstract

In this chapter, the main topic is traveling waves for time-dependent spatially extended systems in one space dimension. Note that we have already extensively discussed various techniques to prove the existence of waves for partial differential equations (PDEs); see, e.g., Chapter 6 Hence, we focus here on further topics beyond the existence of waves in PDEs.

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Literatur
[ABBG02]
Zurück zum Zitat P. Ashwin, M.V. Bartuccelli, T.J. Bridges, and S.A. Gourley. Travelling fronts for the KPP equation with spatio-temporal delay. Zeitschr. Angewand. Math. Phys., 53(1):103–122, 2002.MATHMathSciNet P. Ashwin, M.V. Bartuccelli, T.J. Bridges, and S.A. Gourley. Travelling fronts for the KPP equation with spatio-temporal delay. Zeitschr. Angewand. Math. Phys., 53(1):103–122, 2002.MATHMathSciNet
[ACY03]
Zurück zum Zitat S. Ai, S.-N. Chow, and Y. Yi. Travelling wave solutions in a tissue interaction model for skin pattern formation. J. Dyn. Diff. Eq., 15(2):517–534, 2003.MATHMathSciNet S. Ai, S.-N. Chow, and Y. Yi. Travelling wave solutions in a tissue interaction model for skin pattern formation. J. Dyn. Diff. Eq., 15(2):517–534, 2003.MATHMathSciNet
[ADD12]
Zurück zum Zitat E.O. Alzahrani, F.A. Davidson, and N. Dodds. Reversing invasion in bistable systems. J. Math. Biol., 65:1101–1124, 2012.MATHMathSciNet E.O. Alzahrani, F.A. Davidson, and N. Dodds. Reversing invasion in bistable systems. J. Math. Biol., 65:1101–1124, 2012.MATHMathSciNet
[AG13]
Zurück zum Zitat G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear. G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.
[AGJ90]
Zurück zum Zitat J.C. Alexander, R.A. Gardner, and C.K.R.T. Jones. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410:167–212, 1990.MATHMathSciNet J.C. Alexander, R.A. Gardner, and C.K.R.T. Jones. A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math., 410:167–212, 1990.MATHMathSciNet
[AGK+11]
Zurück zum Zitat Z. Artstein, C.W. Gear, I.G. Kevrekidis, M. Slemrod, and E.S. Titi. Analysis and computation of a discrete KdV-Burgers type equation with fast dispersion and slow diffusion. SIAM J. Numer. Anal., 49(5):2124–2143, 2011.MATHMathSciNet Z. Artstein, C.W. Gear, I.G. Kevrekidis, M. Slemrod, and E.S. Titi. Analysis and computation of a discrete KdV-Burgers type equation with fast dispersion and slow diffusion. SIAM J. Numer. Anal., 49(5):2124–2143, 2011.MATHMathSciNet
[AHM08]
Zurück zum Zitat M. Alfaro, D. Hilhorst, and H. Matano. The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system. J. Differen. Equat., 245(2):505–565, 2008.MATHMathSciNet M. Alfaro, D. Hilhorst, and H. Matano. The singular limit of the Allen–Cahn equation and the FitzHugh–Nagumo system. J. Differen. Equat., 245(2):505–565, 2008.MATHMathSciNet
[Ai07]
Zurück zum Zitat S. Ai. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differential Equat., 232(1):104–133, 2007.MATHMathSciNet S. Ai. Traveling wave fronts for generalized Fisher equations with spatio-temporal delays. J. Differential Equat., 232(1):104–133, 2007.MATHMathSciNet
[Ai10]
Zurück zum Zitat S. Ai. Traveling waves for a model of a fungal disease over a vineyard. SIAM J. Math. Anal., 42(2): 833–856, 2010.MATHMathSciNet S. Ai. Traveling waves for a model of a fungal disease over a vineyard. SIAM J. Math. Anal., 42(2): 833–856, 2010.MATHMathSciNet
[AK13b]
Zurück zum Zitat F. Achleitner and C. Kuehn. On bounded positive stationary solutions for a nonlocal Fisher-KPP equation. arXiv:1307.3480, pages 1–20, 2013. F. Achleitner and C. Kuehn. On bounded positive stationary solutions for a nonlocal Fisher-KPP equation. arXiv:1307.3480, pages 1–20, 2013.
[AMPP87]
Zurück zum Zitat S.B. Angenent, J. Mallet-Paret, and L.A. Peletier. Stable transition layers in a semilinear boundary value problem. J. Differential Equat., 67(2):212–242, 1987.MATHMathSciNet S.B. Angenent, J. Mallet-Paret, and L.A. Peletier. Stable transition layers in a semilinear boundary value problem. J. Differential Equat., 67(2):212–242, 1987.MATHMathSciNet
[AS12]
Zurück zum Zitat F. Achleitner and P. Szmolyan. Saddle-node bifurcation of viscous profiles. Physica D, 241(20): 1703–1717, 2012.MATHMathSciNet F. Achleitner and P. Szmolyan. Saddle-node bifurcation of viscous profiles. Physica D, 241(20): 1703–1717, 2012.MATHMathSciNet
[Bar83]
Zurück zum Zitat J.W. Barker. Interactions of fast and slow waves in hyperbolic systems with two time scales. Math. Methods Appl. Sci., 5(3):292–307, 1983.MATHMathSciNet J.W. Barker. Interactions of fast and slow waves in hyperbolic systems with two time scales. Math. Methods Appl. Sci., 5(3):292–307, 1983.MATHMathSciNet
[Bar84]
Zurück zum Zitat J.W. Barker. Interactions of fast and slow waves in problems with two time scales. SIAM J. Math. Anal., 15(3):500–513, 1984.MATHMathSciNet J.W. Barker. Interactions of fast and slow waves in problems with two time scales. SIAM J. Math. Anal., 15(3):500–513, 1984.MATHMathSciNet
[BD97]
Zurück zum Zitat E. Brunet and B. Derrida. Shift in the velocity front due to a cutoff. Phys. Rev. E, 56(3):2597–2604, 1997.MathSciNet E. Brunet and B. Derrida. Shift in the velocity front due to a cutoff. Phys. Rev. E, 56(3):2597–2604, 1997.MathSciNet
[BDK06]
Zurück zum Zitat M. Beck, A. Doelman, and T.J. Kaper. A geometric construction of traveling waves in a bioremediation model. J. Nonlinear Sci., 16(4):329–349, 2006.MATHMathSciNet M. Beck, A. Doelman, and T.J. Kaper. A geometric construction of traveling waves in a bioremediation model. J. Nonlinear Sci., 16(4):329–349, 2006.MATHMathSciNet
[BE95]
Zurück zum Zitat V. Booth and T. Erneux. Understanding propagation failure as a slow capture near a limit point. SIAM J. Appl. Math., 55(5):1372–1389, 1995.MATHMathSciNet V. Booth and T. Erneux. Understanding propagation failure as a slow capture near a limit point. SIAM J. Appl. Math., 55(5):1372–1389, 1995.MATHMathSciNet
[BJSW08]
Zurück zum Zitat M. Beck, C.K.R.T. Jones, D. Schaeffer, and M. Wechselberger. Electrical waves in a one-dimensional model of cardiac tissue. SIAM J. Appl. Dyn. Syst., 7(4):1558–1581, 2008.MATHMathSciNet M. Beck, C.K.R.T. Jones, D. Schaeffer, and M. Wechselberger. Electrical waves in a one-dimensional model of cardiac tissue. SIAM J. Appl. Dyn. Syst., 7(4):1558–1581, 2008.MATHMathSciNet
[BNS85]
Zurück zum Zitat H. Berestycki, B. Nicolaenko, and B. Scheurer. Traveling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal., 16(6):1207–1242, 1985.MATHMathSciNet H. Berestycki, B. Nicolaenko, and B. Scheurer. Traveling wave solutions to combustion models and their singular limits. SIAM J. Math. Anal., 16(6):1207–1242, 1985.MATHMathSciNet
[Bos00]
Zurück zum Zitat A. Bose. A geometric approach to singularly perturbed nonlocal reaction–diffusion equations. SIAM J. Math. Anal., 31(2):431–454, 2000.MATH A. Bose. A geometric approach to singularly perturbed nonlocal reaction–diffusion equations. SIAM J. Math. Anal., 31(2):431–454, 2000.MATH
[Bre12b]
Zurück zum Zitat P.C. Bressloff. Spatiotemporal dynamics of continuum neural fields. J. Phys. A: Math. Theor., 45:(033001), 2012. P.C. Bressloff. Spatiotemporal dynamics of continuum neural fields. J. Phys. A: Math. Theor., 45:(033001), 2012.
[BST08]
Zurück zum Zitat W.-J. Beyn, S. Selle, and V. Thümmler. Freezing multipulses and multifronts. SIAM J. Appl. Dyn. Syst., 7(2):577–608, 2008.MATHMathSciNet W.-J. Beyn, S. Selle, and V. Thümmler. Freezing multipulses and multifronts. SIAM J. Appl. Dyn. Syst., 7(2):577–608, 2008.MATHMathSciNet
[BW11]
Zurück zum Zitat M. Beck and C.E. Wayne. Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity. SIAM Rev., 53(1):129–153, 2011.MATHMathSciNet M. Beck and C.E. Wayne. Using global invariant manifolds to understand metastability in the Burgers equation with small viscosity. SIAM Rev., 53(1):129–153, 2011.MATHMathSciNet
[CB02]
Zurück zum Zitat A. Carpio and L.L. Bonilla. Pulse propagation in discrete systems of coupled excitable cells. SIAM J. Appl. Math., 63(2):619–635, 2002.MATHMathSciNet A. Carpio and L.L. Bonilla. Pulse propagation in discrete systems of coupled excitable cells. SIAM J. Appl. Math., 63(2):619–635, 2002.MATHMathSciNet
[Cho03]
Zurück zum Zitat S.-N. Chow. Lattice dynamical systems. In J.W. Macki and P. Zecca, editors, Dynamical Systems, pages 1–102. Springer, 2003. S.-N. Chow. Lattice dynamical systems. In J.W. Macki and P. Zecca, editors, Dynamical Systems, pages 1–102. Springer, 2003.
[CL10]
Zurück zum Zitat G.A. Cassatella Contra and D. Levi. Discrete multiscale analysis: a biatomic lattice system. J. Nonlinear Math. Phys., 17:357–377, 2010.MATHMathSciNet G.A. Cassatella Contra and D. Levi. Discrete multiscale analysis: a biatomic lattice system. J. Nonlinear Math. Phys., 17:357–377, 2010.MATHMathSciNet
[CMJ09]
Zurück zum Zitat N. Costanzino, V. Manukian, and C.K.R.T. Jones. Solitary waves of the regularized short pulse and Ostrovsky equations. SIAM J. Math. Anal., 41(5):2088–2106, 2009.MATHMathSciNet N. Costanzino, V. Manukian, and C.K.R.T. Jones. Solitary waves of the regularized short pulse and Ostrovsky equations. SIAM J. Math. Anal., 41(5):2088–2106, 2009.MATHMathSciNet
[CML02]
Zurück zum Zitat D. Cai, D.W. McLaughlin, and K.T.R. Mc Laughlin. The nonlinear Schrödinger equation as both a PDE and a dynamical system. In B. Fiedler, editor, Handbook of Dynamical Systems 2, pages 599–675. Elsevier, 2002. D. Cai, D.W. McLaughlin, and K.T.R. Mc Laughlin. The nonlinear Schrödinger equation as both a PDE and a dynamical system. In B. Fiedler, editor, Handbook of Dynamical Systems 2, pages 599–675. Elsevier, 2002.
[Coo05]
Zurück zum Zitat S. Coombes. Waves, bumps, and patterns in neural field theories. Biol. Cybern., 93:91–108, 2005.MATHMathSciNet S. Coombes. Waves, bumps, and patterns in neural field theories. Biol. Cybern., 93:91–108, 2005.MATHMathSciNet
[Daf10]
Zurück zum Zitat C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer, 2010. C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics. Springer, 2010.
[DDvGV03]
Zurück zum Zitat G. Derks, A. Doelman, S.A. van Gils, and T. Visser. Travelling waves in a singularly perturbed sine-Gordon equation. Physica D, 180:40–70, 2003.MATHMathSciNet G. Derks, A. Doelman, S.A. van Gils, and T. Visser. Travelling waves in a singularly perturbed sine-Gordon equation. Physica D, 180:40–70, 2003.MATHMathSciNet
[DEK00]
Zurück zum Zitat A. Doelman, W. Eckhaus, and T.J. Kaper. Slowly modulated two-pulse solutions in the Gray-Scott Model I: asymptotic construction and stability. SIAM J. Appl. Math., 61(3):1080–1102, 2000.MATHMathSciNet A. Doelman, W. Eckhaus, and T.J. Kaper. Slowly modulated two-pulse solutions in the Gray-Scott Model I: asymptotic construction and stability. SIAM J. Appl. Math., 61(3):1080–1102, 2000.MATHMathSciNet
[DEK06]
Zurück zum Zitat A. Doelman, W. Eckhaus, and T.J. Kaper. Slowly modulated two-pulse solutions in the Gray–Scott Model II: geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math., 61(6): 2036–2062, 2006. A. Doelman, W. Eckhaus, and T.J. Kaper. Slowly modulated two-pulse solutions in the Gray–Scott Model II: geometric theory, bifurcations, and splitting dynamics. SIAM J. Appl. Math., 61(6): 2036–2062, 2006.
[DGK01]
Zurück zum Zitat A. Doelman, R.A. Gardner, and T.J. Kaper. Large stable pulse solutions in reaction–diffusion equations. Indiana Univ. Math. J., 50(1):443–507, 2001.MATHMathSciNet A. Doelman, R.A. Gardner, and T.J. Kaper. Large stable pulse solutions in reaction–diffusion equations. Indiana Univ. Math. J., 50(1):443–507, 2001.MATHMathSciNet
[DGK02]
Zurück zum Zitat A. Doelman, R.A. Gardner, and T.J. Kaper. A Stability Index Analysis of 1-D Patterns of the Gray-Scott Model, volume 737 of Mem. Amer. Math. Soc. AMS, 2002. A. Doelman, R.A. Gardner, and T.J. Kaper. A Stability Index Analysis of 1-D Patterns of the Gray-Scott Model, volume 737 of Mem. Amer. Math. Soc. AMS, 2002.
[DHV04]
Zurück zum Zitat A. Doelman, G. Hek, and N. Valkhoff. Stabilization by slow diffusion in a real Ginzburg–Landau system. J. Nonlinear Sci., 14(3):237–278, 2004.MATHMathSciNet A. Doelman, G. Hek, and N. Valkhoff. Stabilization by slow diffusion in a real Ginzburg–Landau system. J. Nonlinear Sci., 14(3):237–278, 2004.MATHMathSciNet
[DHV07]
Zurück zum Zitat A. Doelman, G. Hek, and N. Valkhoff. Algebraically decaying pulses in a Ginzburg–Landau system with a neutrally stable mode. Nonlinearity, 20:357–389, 2007.MATHMathSciNet A. Doelman, G. Hek, and N. Valkhoff. Algebraically decaying pulses in a Ginzburg–Landau system with a neutrally stable mode. Nonlinearity, 20:357–389, 2007.MATHMathSciNet
[DIN04]
Zurück zum Zitat A. Doelman, D. Iron, and Y. Nishiura. Destabilization of fronts in a class of bistable systems. SIAM J. Math. Anal., 35(6):1420–1450, 2004.MATHMathSciNet A. Doelman, D. Iron, and Y. Nishiura. Destabilization of fronts in a class of bistable systems. SIAM J. Math. Anal., 35(6):1420–1450, 2004.MATHMathSciNet
[DK03]
Zurück zum Zitat A. Doelman and T.J. Kaper. Semistrong pulse interactions in a class of coupled reaction–diffusion equations. SIAM J. Appl. Dyn. Syst., 2(1):53–96, 2003.MATHMathSciNet A. Doelman and T.J. Kaper. Semistrong pulse interactions in a class of coupled reaction–diffusion equations. SIAM J. Appl. Dyn. Syst., 2(1):53–96, 2003.MATHMathSciNet
[DKP07]
Zurück zum Zitat A. Doelman, T.J. Kaper, and K. Promislow. Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer–Meinhardt model. SIAM J. Math. Anal., 38(6):1760–1787, 2007.MATHMathSciNet A. Doelman, T.J. Kaper, and K. Promislow. Nonlinear asymptotic stability of the semistrong pulse dynamics in a regularized Gierer–Meinhardt model. SIAM J. Math. Anal., 38(6):1760–1787, 2007.MATHMathSciNet
[DMS03]
Zurück zum Zitat C.R. Doering, C. Mueller, and P. Smereka. Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A, 325:243–259, 2003.MATHMathSciNet C.R. Doering, C. Mueller, and P. Smereka. Interacting particles, the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A, 325:243–259, 2003.MATHMathSciNet
[Doc92]
Zurück zum Zitat J.D. Dockery. Existence of standing pulse solutions for an excitable activator-inhibitory system. J. Dyn. Diff. Eq., 4(2):231–257, 1992.MATHMathSciNet J.D. Dockery. Existence of standing pulse solutions for an excitable activator-inhibitory system. J. Dyn. Diff. Eq., 4(2):231–257, 1992.MATHMathSciNet
[Doe93]
Zurück zum Zitat A. Doelman. Traveling waves in the complex Ginzburg–Landau equation. J. Nonlinear Sci., 3(1): 225–266, 1993.MATHMathSciNet A. Doelman. Traveling waves in the complex Ginzburg–Landau equation. J. Nonlinear Sci., 3(1): 225–266, 1993.MATHMathSciNet
[Doe96]
Zurück zum Zitat A. Doelman. Breaking the hidden symmetry in the Ginzburg–Landau equation. Physica D, 97(4): 398–428, 1996.MATHMathSciNet A. Doelman. Breaking the hidden symmetry in the Ginzburg–Landau equation. Physica D, 97(4): 398–428, 1996.MATHMathSciNet
[DPK07a]
Zurück zum Zitat F. Dumortier, N. Popovic, and T.J. Kaper. The asymptotic critical wave speed in a family of scalar reaction–diffusion equations. J. Math. Anal. Appl., 326(2):1007–1023, 2007.MATHMathSciNet F. Dumortier, N. Popovic, and T.J. Kaper. The asymptotic critical wave speed in a family of scalar reaction–diffusion equations. J. Math. Anal. Appl., 326(2):1007–1023, 2007.MATHMathSciNet
[DPK07b]
Zurück zum Zitat F. Dumortier, N. Popovic, and T.J. Kaper. The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off. Nonlinearity, 20(4):855–877, 2007.MATHMathSciNet F. Dumortier, N. Popovic, and T.J. Kaper. The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off. Nonlinearity, 20(4):855–877, 2007.MATHMathSciNet
[DPK10]
Zurück zum Zitat F. Dumortier, N. Popovic, and T.J. Kaper. A geometric approach to bistable front propagation in scalar reaction–diffusion equations with cut-off. Physica D, 239(20):1984–1999, 2010.MATHMathSciNet F. Dumortier, N. Popovic, and T.J. Kaper. A geometric approach to bistable front propagation in scalar reaction–diffusion equations with cut-off. Physica D, 239(20):1984–1999, 2010.MATHMathSciNet
[DSSS09]
Zurück zum Zitat A. Doelman, B. Sandstede, A. Scheel, and G. Schneider. The dynamics of modulated wavetrains. Memoirs of the AMS, 199(934):1–105, 2009.MathSciNet A. Doelman, B. Sandstede, A. Scheel, and G. Schneider. The dynamics of modulated wavetrains. Memoirs of the AMS, 199(934):1–105, 2009.MathSciNet
[DvHK09]
Zurück zum Zitat A. Doelman, P. van Heijster, and T.J. Kaper. Pulse dynamics in a three-component system: existence analysis. J. Dyn. Diff. Eq., 21:73–115, 2009.MATH A. Doelman, P. van Heijster, and T.J. Kaper. Pulse dynamics in a three-component system: existence analysis. J. Dyn. Diff. Eq., 21:73–115, 2009.MATH
[DvHK13]
Zurück zum Zitat A. Doelman, P. van Heijster, and T.J. Kaper. An explicit theory for pulses in two component, singularly perturbed, reaction–diffusion equations. J. Dyn. Diff. Eq., pages 1–42, 2013. accepted, to appear. A. Doelman, P. van Heijster, and T.J. Kaper. An explicit theory for pulses in two component, singularly perturbed, reaction–diffusion equations. J. Dyn. Diff. Eq., pages 1–42, 2013. accepted, to appear.
[EF77]
Zurück zum Zitat J.W. Evans and J.A. Feroe. Local stability of the nerve impulse. Math. Biosci., 37(1):23–50, 1977.MATH J.W. Evans and J.A. Feroe. Local stability of the nerve impulse. Math. Biosci., 37(1):23–50, 1977.MATH
[EM93]
Zurück zum Zitat G.B. Ermentrout and J.B. McLeod. Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinburgh A, 123(3):461–478, 1993.MATHMathSciNet G.B. Ermentrout and J.B. McLeod. Existence and uniqueness of travelling waves for a neural network. Proc. R. Soc. Edinburgh A, 123(3):461–478, 1993.MATHMathSciNet
[EN93]
Zurück zum Zitat T. Erneux and G. Nicolis. Propagating waves in discrete bistable reaction–diffusion systems. Physica D, 67(1):237–244, 1993.MATHMathSciNet T. Erneux and G. Nicolis. Propagating waves in discrete bistable reaction–diffusion systems. Physica D, 67(1):237–244, 1993.MATHMathSciNet
[EV05]
Zurück zum Zitat C. Elmer and E.S. Van Vleck. Spatially discrete FitzHugh–Nagumo equations. SIAM J. Appl. Math., 65(4):1153–1174, 2005.MATHMathSciNet C. Elmer and E.S. Van Vleck. Spatially discrete FitzHugh–Nagumo equations. SIAM J. Appl. Math., 65(4):1153–1174, 2005.MATHMathSciNet
[Eva72]
Zurück zum Zitat J. Evans. Nerve axon equations III: stability of nerve impulses. Indiana U. Math. J., 22:577–594, 1972.MATH J. Evans. Nerve axon equations III: stability of nerve impulses. Indiana U. Math. J., 22:577–594, 1972.MATH
[Eva02]
Zurück zum Zitat L.C. Evans. Partial Differential Equations. AMS, 2002. L.C. Evans. Partial Differential Equations. AMS, 2002.
[EvS00]
Zurück zum Zitat U. Ebert and W. van Saarloos. Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D, 146:1–99, 2000.MATHMathSciNet U. Ebert and W. van Saarloos. Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D, 146:1–99, 2000.MATHMathSciNet
[Fif88]
Zurück zum Zitat P.C. Fife. Dynamics of internal layers and diffusive interfaces. SIAM, 1988. P.C. Fife. Dynamics of internal layers and diffusive interfaces. SIAM, 1988.
[Fis37]
Zurück zum Zitat R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:353–369, 1937. R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:353–369, 1937.
[FL12]
Zurück zum Zitat H. Fan and X.-B. Lin. Standing waves for phase transitions in a spherically symmetric nozzle. SIAM J. Math. Anal., 44(1):405–436, 2012.MATHMathSciNet H. Fan and X.-B. Lin. Standing waves for phase transitions in a spherically symmetric nozzle. SIAM J. Math. Anal., 44(1):405–436, 2012.MATHMathSciNet
[Flo91]
Zurück zum Zitat G. Flores. Stability analysis for the slow traveling pulse of the FitzHugh–Nagumo system. SIAM J. Math. Anal., 22(2):392–399, 1991.MATHMathSciNet G. Flores. Stability analysis for the slow traveling pulse of the FitzHugh–Nagumo system. SIAM J. Math. Anal., 22(2):392–399, 1991.MATHMathSciNet
[FM77]
Zurück zum Zitat P. Fife and J.B. McLeod. The approach of solutions nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal., 65:335–361, 1977.MATHMathSciNet P. Fife and J.B. McLeod. The approach of solutions nonlinear diffusion equations to travelling front solutions. Arch. Rational Mech. Anal., 65:335–361, 1977.MATHMathSciNet
[Fri92]
Zurück zum Zitat A. Friedman. Partial Differential Equations of Parabolic Type. Dover, 1992. A. Friedman. Partial Differential Equations of Parabolic Type. Dover, 1992.
[FS95]
Zurück zum Zitat H. Freistühler and P. Szmolyan. Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves. SIAM J. Math. Anal., 26(1):112–128, 1995.MATHMathSciNet H. Freistühler and P. Szmolyan. Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves. SIAM J. Math. Anal., 26(1):112–128, 1995.MATHMathSciNet
[FS02a]
Zurück zum Zitat H. Freistühler and P. Szmolyan. Spectral stability of small shock waves. Arch. Rational Mech. Anal., 164:287–309, 2002.MATH H. Freistühler and P. Szmolyan. Spectral stability of small shock waves. Arch. Rational Mech. Anal., 164:287–309, 2002.MATH
[FS10a]
Zurück zum Zitat H. Freistühler and P. Szmolyan. Spectral stability of small-amplitude viscous shock waves in several space dimensions. Arch. Rational Mech. Anal., 195(2):353–373, 2010.MATH H. Freistühler and P. Szmolyan. Spectral stability of small-amplitude viscous shock waves in several space dimensions. Arch. Rational Mech. Anal., 195(2):353–373, 2010.MATH
[Gar84]
Zurück zum Zitat R. Gardner. Existence of travelling wave solutions of predator–prey systems via the connection index. SIAM J. Appl. Math., 44(1):56–79, 1984.MATHMathSciNet R. Gardner. Existence of travelling wave solutions of predator–prey systems via the connection index. SIAM J. Appl. Math., 44(1):56–79, 1984.MATHMathSciNet
[GC02]
Zurück zum Zitat S.A. Gourley and M.A.J. Chaplain. Travelling fronts in a food-limited population model with time delay. Proc. Roy. Soc. Edinburgh Sect. A, 132:75–89, 2002.MATHMathSciNet S.A. Gourley and M.A.J. Chaplain. Travelling fronts in a food-limited population model with time delay. Proc. Roy. Soc. Edinburgh Sect. A, 132:75–89, 2002.MATHMathSciNet
[GGJ07]
Zurück zum Zitat A. Ghazaryan, P. Gordon, and C.K.R.T. Jones. Traveling waves in porous media combustion: uniqueness of waves for small thermal diffusivity. J. Dyn. Diff. Eq., 19(4):951–966, 2007.MATHMathSciNet A. Ghazaryan, P. Gordon, and C.K.R.T. Jones. Traveling waves in porous media combustion: uniqueness of waves for small thermal diffusivity. J. Dyn. Diff. Eq., 19(4):951–966, 2007.MATHMathSciNet
[Gha09a]
Zurück zum Zitat A. Ghazaryan. Nonlinear stability of high Lewis number combustion fronts. Indiana Univ. Math. J., 58:181–212, 2009.MATHMathSciNet A. Ghazaryan. Nonlinear stability of high Lewis number combustion fronts. Indiana Univ. Math. J., 58:181–212, 2009.MATHMathSciNet
[Gha09b]
Zurück zum Zitat A. Ghazaryan. On the stability of high Lewis number combustion fronts. Discrete Contin. Dyn. Syst. A, 24:809–826, 2009.MATHMathSciNet A. Ghazaryan. On the stability of high Lewis number combustion fronts. Discrete Contin. Dyn. Syst. A, 24:809–826, 2009.MATHMathSciNet
[Gha10]
Zurück zum Zitat A. Ghazaryan. On the existence of high Lewis number combustion fronts. Math. Comput. Simul., 82(6):1133–1141, 2010.MathSciNet A. Ghazaryan. On the existence of high Lewis number combustion fronts. Math. Comput. Simul., 82(6):1133–1141, 2010.MathSciNet
[GHL13]
Zurück zum Zitat A. Ghazaryan, J. Humphreys, and J. Lytle. Spectral behavior of combustion fronts with high exothermicity. SIAM J. Appl. Math., 73(1):422–437, 2013.MATHMathSciNet A. Ghazaryan, J. Humphreys, and J. Lytle. Spectral behavior of combustion fronts with high exothermicity. SIAM J. Appl. Math., 73(1):422–437, 2013.MATHMathSciNet
[GJ91]
Zurück zum Zitat R. Gardner and C.K.R.T. Jones. Stability of travelling wave solutions of diffusive predator–prey systems. Trans. Amer. Math. Soc., 327(2):465–524, 1991.MATHMathSciNet R. Gardner and C.K.R.T. Jones. Stability of travelling wave solutions of diffusive predator–prey systems. Trans. Amer. Math. Soc., 327(2):465–524, 1991.MATHMathSciNet
[GK98]
Zurück zum Zitat P.P.N. De Groen and G.E. Karadzhov. Exponentially slow travelling waves on a finite interval for Burgers’-type equation. Electron. J. Differential Equat., 1980(30):1–38, 1998. P.P.N. De Groen and G.E. Karadzhov. Exponentially slow travelling waves on a finite interval for Burgers’-type equation. Electron. J. Differential Equat., 1980(30):1–38, 1998.
[GK01]
Zurück zum Zitat P.P.N. De Groen and G.E. Karadzhov. Slow travelling waves on a finite interval for Burgers’-type equations. J. Comput. Appl. Math., 132:155–189, 2001.MATHMathSciNet P.P.N. De Groen and G.E. Karadzhov. Slow travelling waves on a finite interval for Burgers’-type equations. J. Comput. Appl. Math., 132:155–189, 2001.MATHMathSciNet
[Gou00]
Zurück zum Zitat S.A. Gourley. Travelling fronts in the diffusive Nicholson’s blowflies equation with distributed delay. Math. Comput. Model., 32(7):843–853, 2000.MATHMathSciNet S.A. Gourley. Travelling fronts in the diffusive Nicholson’s blowflies equation with distributed delay. Math. Comput. Model., 32(7):843–853, 2000.MATHMathSciNet
[GR03]
Zurück zum Zitat S.A. Gourley and S. Ruan. Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J. Math. Anal., 35(3):806–822, 2003.MATHMathSciNet S.A. Gourley and S. Ruan. Convergence and travelling fronts in functional differential equations with nonlocal terms: a competition model. SIAM J. Math. Anal., 35(3):806–822, 2003.MATHMathSciNet
[GSS13]
Zurück zum Zitat A. Ghazaryan, S. Schecter, and P.L. Simon. Gasless combustion fronts with heat loss. SIAM J. Appl. Math., 73(3):1303–1326, 2013.MATHMathSciNet A. Ghazaryan, S. Schecter, and P.L. Simon. Gasless combustion fronts with heat loss. SIAM J. Appl. Math., 73(3):1303–1326, 2013.MATHMathSciNet
[Hal99]
Zurück zum Zitat G. Haller. Homoclinic jumping in the perturbed nonlinear Schrödinger equation. Comm. Pure Appl. Math., 152(1):1–47, 1999.MathSciNet G. Haller. Homoclinic jumping in the perturbed nonlinear Schrödinger equation. Comm. Pure Appl. Math., 152(1):1–47, 1999.MathSciNet
[HDK13]
Zurück zum Zitat M. Holzer, A. Doelman, and T.J. Kaper. Existence and stability of traveling pulses in a reaction–diffusion-mechanics system. J. Nonlinear Sci., 23(1):129–177, 2013.MATHMathSciNet M. Holzer, A. Doelman, and T.J. Kaper. Existence and stability of traveling pulses in a reaction–diffusion-mechanics system. J. Nonlinear Sci., 23(1):129–177, 2013.MATHMathSciNet
[Hek01]
Zurück zum Zitat G. Hek. Fronts and pulses in a class of reaction–diffusion equations: a geometric singular perturbation approach. Nonlinearity, 14(1):35–72, 2001.MATHMathSciNet G. Hek. Fronts and pulses in a class of reaction–diffusion equations: a geometric singular perturbation approach. Nonlinearity, 14(1):35–72, 2001.MATHMathSciNet
[Hen81]
Zurück zum Zitat D. Henry. Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin Heidelberg, Germany, 1981.MATH D. Henry. Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin Heidelberg, Germany, 1981.MATH
[HHH13]
Zurück zum Zitat J.M. Hong, C.-H. Hsu, and B.-C. Huang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Comm. Pure Appl. Anal., 12(3):1501–1526, 2013.MATHMathSciNet J.M. Hong, C.-H. Hsu, and B.-C. Huang. Geometric singular perturbation approach to the existence and instability of stationary waves for viscous traffic flow models. Comm. Pure Appl. Anal., 12(3):1501–1526, 2013.MATHMathSciNet
[HL99]
Zurück zum Zitat J.K. Hale and X.-B. Lin. Multiple internal layer solutions generated by spatially oscillatory perturbations. J. Differential Equat., 154(2):364–418, 1999.MATHMathSciNet J.K. Hale and X.-B. Lin. Multiple internal layer solutions generated by spatially oscillatory perturbations. J. Differential Equat., 154(2):364–418, 1999.MATHMathSciNet
[HM09a]
Zurück zum Zitat M. Hairer and J.C. Mattingly. Slow energy dissipation in anharmonic oscillator chains. Comm. Pure. Appl. Math., 62(8):999–1032, 2009.MATHMathSciNet M. Hairer and J.C. Mattingly. Slow energy dissipation in anharmonic oscillator chains. Comm. Pure. Appl. Math., 62(8):999–1032, 2009.MATHMathSciNet
[HN01]
Zurück zum Zitat F. Hamel and N. Nadirashvili. Travelling fronts and entire solutions of the Fisher–KPP equation in \(\mathbb{R}^{N}\). Arch. Ration. Mech. Anal., 157:91–163, 2001.MATHMathSciNet F. Hamel and N. Nadirashvili. Travelling fronts and entire solutions of the Fisher–KPP equation in \(\mathbb{R}^{N}\). Arch. Ration. Mech. Anal., 157:91–163, 2001.MATHMathSciNet
[HPS11]
Zurück zum Zitat H.J. Hupkes, D. Pelinovsky, and B. Sandstede. Propagation failure in the discrete Nagumo equation. Proc. Amer. Math. Soc., 139:3537–3551, 2011.MATHMathSciNet H.J. Hupkes, D. Pelinovsky, and B. Sandstede. Propagation failure in the discrete Nagumo equation. Proc. Amer. Math. Soc., 139:3537–3551, 2011.MATHMathSciNet
[HPT00]
Zurück zum Zitat J.K. Hale, L.A. Peletier, and W.C. Troy. Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis. SIAM J. Appl. Math., 61(1):102–130, 2000.MATHMathSciNet J.K. Hale, L.A. Peletier, and W.C. Troy. Exact homoclinic and heteroclinic solutions of the Gray–Scott model for autocatalysis. SIAM J. Appl. Math., 61(1):102–130, 2000.MATHMathSciNet
[HS10c]
Zurück zum Zitat H.J. Hupkes and B. Sandstede. Traveling pulse solutions for the discrete FitzHugh–Nagumo system. SIAM J. Appl. Dyn. Sys., 9(3):827–882, 2010.MATHMathSciNet H.J. Hupkes and B. Sandstede. Traveling pulse solutions for the discrete FitzHugh–Nagumo system. SIAM J. Appl. Dyn. Sys., 9(3):827–882, 2010.MATHMathSciNet
[HS12]
Zurück zum Zitat M. Holzer and A. Scheel. A slow pushed front in a Lotka–Volterra competition model. Nonlinearity, 25(7):2151–2179, 2012.MATHMathSciNet M. Holzer and A. Scheel. A slow pushed front in a Lotka–Volterra competition model. Nonlinearity, 25(7):2151–2179, 2012.MATHMathSciNet
[HS13]
Zurück zum Zitat H.J. Hupkes and B. Sandstede. Stability of traveling pulse solutions for the discrete FitzHugh–Nagumo system. Trans. Amer. Math. Soc., 365:251–301, 2013.MATHMathSciNet H.J. Hupkes and B. Sandstede. Stability of traveling pulse solutions for the discrete FitzHugh–Nagumo system. Trans. Amer. Math. Soc., 365:251–301, 2013.MATHMathSciNet
[HV07]
Zurück zum Zitat G. Hek and N. Valkhoff. Pulses in a complex Ginzburg–Landau system: persistence under coupling with slow diffusion. Physica D, 232(1):62–85, 2007.MATHMathSciNet G. Hek and N. Valkhoff. Pulses in a complex Ginzburg–Landau system: persistence under coupling with slow diffusion. Physica D, 232(1):62–85, 2007.MATHMathSciNet
[Jal04]
Zurück zum Zitat J. Jalics. Slow waves in mutually inhibitory neuronal networks. Physica D, 192:95–122, 2004.MATHMathSciNet J. Jalics. Slow waves in mutually inhibitory neuronal networks. Physica D, 192:95–122, 2004.MATHMathSciNet
[Jon84]
Zurück zum Zitat C.K.R.T. Jones. Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Amer. Math. Soc., 286(2): 431–469, 1984.MATHMathSciNet C.K.R.T. Jones. Stability of the travelling wave solution of the FitzHugh–Nagumo system. Trans. Amer. Math. Soc., 286(2): 431–469, 1984.MATHMathSciNet
[Kap05]
Zurück zum Zitat T. Kapitula. Stability analysis of pulses via the Evans function: dissipative systems. In Dissipative Solitons, volume 661 of Lecture Notes in Physics, pages 407–427. Springer, 2005. T. Kapitula. Stability analysis of pulses via the Evans function: dissipative systems. In Dissipative Solitons, volume 661 of Lecture Notes in Physics, pages 407–427. Springer, 2005.
[Kar93]
Zurück zum Zitat A. Karma. Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys. Rev. Lett., 71:1103–1106, 1993.MATHMathSciNet A. Karma. Spiral breakup in model equations of action potential propagation in cardiac tissue. Phys. Rev. Lett., 71:1103–1106, 1993.MATHMathSciNet
[Kat80]
Zurück zum Zitat T. Kato. Perturbation Theory for Linear Operators. Springer, 1980. T. Kato. Perturbation Theory for Linear Operators. Springer, 1980.
[KB09b]
Zurück zum Zitat Y.N. Kyrychko and K.B. Blyuss. Persistence of travelling waves in a generalized Fisher equation. Phys. Lett. A, 373(6):668–674, 2009.MATH Y.N. Kyrychko and K.B. Blyuss. Persistence of travelling waves in a generalized Fisher equation. Phys. Lett. A, 373(6):668–674, 2009.MATH
[KBB05]
Zurück zum Zitat Y.N. Kyrychko, M.V. Bartuccelli, and K.B. Blyuss. Persistence of travelling wave solutions of a fourth order diffusion system. J. Comp. Appl. Math., 176(2):433–443, 2005.MATHMathSciNet Y.N. Kyrychko, M.V. Bartuccelli, and K.B. Blyuss. Persistence of travelling wave solutions of a fourth order diffusion system. J. Comp. Appl. Math., 176(2):433–443, 2005.MATHMathSciNet
[Kee80]
[Kee87]
Zurück zum Zitat J.P. Keener. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math., 47:556–572, 1987.MATHMathSciNet J.P. Keener. Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math., 47:556–572, 1987.MATHMathSciNet
[KEW06]
Zurück zum Zitat T. Kolokolnikov, T. Erneux, and J. Wei. Mesa-type patterns in the one-dimensional Brusselator and their stability. Physica D, 214(1):63–77, 2006.MATHMathSciNet T. Kolokolnikov, T. Erneux, and J. Wei. Mesa-type patterns in the one-dimensional Brusselator and their stability. Physica D, 214(1):63–77, 2006.MATHMathSciNet
[Kin13]
Zurück zum Zitat J.R. King. Wavespeed selection in the heterogeneous Fisher equation: slowly varying inhomogeneity. Networks and Heterogeneous Media, 8(1):343–378, 2013.MATHMathSciNet J.R. King. Wavespeed selection in the heterogeneous Fisher equation: slowly varying inhomogeneity. Networks and Heterogeneous Media, 8(1):343–378, 2013.MATHMathSciNet
[KKS00a]
Zurück zum Zitat B. Katzengruber, M. Krupa, and P. Szmolyan. Bifurcation of traveling waves in extrinsic semiconductors. Physica D, 144(1): 1–19, 2000.MATHMathSciNet B. Katzengruber, M. Krupa, and P. Szmolyan. Bifurcation of traveling waves in extrinsic semiconductors. Physica D, 144(1): 1–19, 2000.MATHMathSciNet
[KKS04]
Zurück zum Zitat T. Kapitula, J.N. Kutz, and B. Sandstede. The Evans function for nonlocal equations. Indiana U. Math. J., 53(4):1095–1126, 2004.MATHMathSciNet T. Kapitula, J.N. Kutz, and B. Sandstede. The Evans function for nonlocal equations. Indiana U. Math. J., 53(4):1095–1126, 2004.MATHMathSciNet
[Kno00]
Zurück zum Zitat R. Knobel. An Introduction to the Mathematical Theory of Waves. AMS, 2000. R. Knobel. An Introduction to the Mathematical Theory of Waves. AMS, 2000.
[KP13]
Zurück zum Zitat T. Kapitula and K. Promislow. Spectral and Dynamical Stability of Nonlinear Waves. Springer, 2013. T. Kapitula and K. Promislow. Spectral and Dynamical Stability of Nonlinear Waves. Springer, 2013.
[KPP91]
Zurück zum Zitat A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V.M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248–270. Kluwer, 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937. A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V.M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248–270. Kluwer, 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937.
[KR00]
Zurück zum Zitat T. Kapitula and J. Rubin. Existence and stability of standing hole solutions to complex Ginzburg–Landau equations. Nonlinearity, 13(1):77–112, 2000.MATHMathSciNet T. Kapitula and J. Rubin. Existence and stability of standing hole solutions to complex Ginzburg–Landau equations. Nonlinearity, 13(1):77–112, 2000.MATHMathSciNet
[KR14]
Zurück zum Zitat C. Kuehn and M.G. Riedler. Large deviations for nonlocal stochastic neural fields. J. Math. Neurosci., 4(1):1–33, 2014.MathSciNet C. Kuehn and M.G. Riedler. Large deviations for nonlocal stochastic neural fields. J. Math. Neurosci., 4(1):1–33, 2014.MathSciNet
[KSWW06]
Zurück zum Zitat T. Kolokolnikov, W. Sun, M.J. Ward, and J. Wei. The stability of a stripe for the Gierer–Meinhardt model and the effect of saturation. SIAM J. Appl. Dyn. Syst., 5(2):313–363, 2006.MATHMathSciNet T. Kolokolnikov, W. Sun, M.J. Ward, and J. Wei. The stability of a stripe for the Gierer–Meinhardt model and the effect of saturation. SIAM J. Appl. Dyn. Syst., 5(2):313–363, 2006.MATHMathSciNet
[KT12]
Zurück zum Zitat B.L. Keyfitz and C. Tsikkou. Conserving the wrong variables in gas dynamics: a Riemann solution with singular shocks. Quart. Appl. Math., 70:407–436, 2012.MATHMathSciNet B.L. Keyfitz and C. Tsikkou. Conserving the wrong variables in gas dynamics: a Riemann solution with singular shocks. Quart. Appl. Math., 70:407–436, 2012.MATHMathSciNet
[LE92]
Zurück zum Zitat J.P. Laplante and T. Erneux. Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem., 96(12): 4931–4934, 1992. J.P. Laplante and T. Erneux. Propagation failure in arrays of coupled bistable chemical reactors. J. Phys. Chem., 96(12): 4931–4934, 1992.
[LeV92]
Zurück zum Zitat R.J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser, 1992. R.J. LeVeque. Numerical Methods for Conservation Laws. Birkhäuser, 1992.
[Li03]
Zurück zum Zitat Y. Li. Homoclinic tubes in discrete nonlinear Schrödinger equation under Hamiltonian perturbations. Nonlinear Dyn., 31(4):393–434, 2003. Y. Li. Homoclinic tubes in discrete nonlinear Schrödinger equation under Hamiltonian perturbations. Nonlinear Dyn., 31(4):393–434, 2003.
[Lin01]
Zurück zum Zitat X.-B. Lin. Construction and asymptotic stability of structurally stable internal layer solutions. Trans. Amer. Math. Soc., 353:2983–3043, 2001.MATHMathSciNet X.-B. Lin. Construction and asymptotic stability of structurally stable internal layer solutions. Trans. Amer. Math. Soc., 353:2983–3043, 2001.MATHMathSciNet
[Lin06]
Zurück zum Zitat X.-B. Lin. Slow eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws: an analytic approach. J. Dyn. Diff. Eq., 18(1):1–52, 2006.MATH X.-B. Lin. Slow eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws: an analytic approach. J. Dyn. Diff. Eq., 18(1):1–52, 2006.MATH
[Liu04]
Zurück zum Zitat W. Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discr. Cont. Dyn. Syst., 10(4):871–884, 2004.MATH W. Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discr. Cont. Dyn. Syst., 10(4):871–884, 2004.MATH
[LO93]
Zurück zum Zitat J.G. Laforgue and R.E. O’Malley. Supersensitive boundary value problems. In Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, pages 215–223. Springer, 1993. J.G. Laforgue and R.E. O’Malley. Supersensitive boundary value problems. In Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, pages 215–223. Springer, 1993.
[LO13a]
Zurück zum Zitat D.J.B. Lloyd and H. O’Farrell. On localised hotspots of an urban crime model. Physica D, 253:23–39, 2013.MATHMathSciNet D.J.B. Lloyd and H. O’Farrell. On localised hotspots of an urban crime model. Physica D, 253:23–39, 2013.MATHMathSciNet
[LS03]
Zurück zum Zitat X.-B. Lin and S. Schecter. Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws. SIAM J. Math. Anal., 35(4):884–921, 2003.MATHMathSciNet X.-B. Lin and S. Schecter. Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws. SIAM J. Math. Anal., 35(4):884–921, 2003.MATHMathSciNet
[LT03]
Zurück zum Zitat C.R. Laing and W.C. Troy. PDE methods for nonlocal models. SIAM J. Appl. Dyn. Syst., 2(3):487–516, 2003.MATHMathSciNet C.R. Laing and W.C. Troy. PDE methods for nonlocal models. SIAM J. Appl. Dyn. Syst., 2(3):487–516, 2003.MATHMathSciNet
[LTGE02]
Zurück zum Zitat C.R. Laing, W.C. Troy, B. Gutkin, and B. Ermentrout. Multiple bumps in a neuronal model of working memory. SIAM J. Appl. Math., 63(1):62–97, 2002.MATHMathSciNet C.R. Laing, W.C. Troy, B. Gutkin, and B. Ermentrout. Multiple bumps in a neuronal model of working memory. SIAM J. Appl. Math., 63(1):62–97, 2002.MATHMathSciNet
[LW10b]
Zurück zum Zitat G. Lv and M. Wang. Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. Nonl. Anal.: Real World Appl., 11(3):2035–2043, 2010. G. Lv and M. Wang. Existence, uniqueness and asymptotic behavior of traveling wave fronts for a vector disease model. Nonl. Anal.: Real World Appl., 11(3):2035–2043, 2010.
[Mag78]
Zurück zum Zitat K. Maginu. Stability of periodic travelling wave solutions of a nerve conduction equation. J. Math. Biol., 6(1):49–57, 1978.MATHMathSciNet K. Maginu. Stability of periodic travelling wave solutions of a nerve conduction equation. J. Math. Biol., 6(1):49–57, 1978.MATHMathSciNet
[Mag80]
Zurück zum Zitat K. Maginu. Existence and stability of periodic travelling wave solutions to Nagumo’s nerve equation. J. Math. Biol., 10(2):133–153, 1980.MATHMathSciNet K. Maginu. Existence and stability of periodic travelling wave solutions to Nagumo’s nerve equation. J. Math. Biol., 10(2):133–153, 1980.MATHMathSciNet
[Mag85]
Zurück zum Zitat K. Maginu. Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction–diffusion systems. SIAM J. Appl. Math., 45(5):750–774, 1985.MATHMathSciNet K. Maginu. Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction–diffusion systems. SIAM J. Appl. Math., 45(5):750–774, 1985.MATHMathSciNet
[Man06]
Zurück zum Zitat M.B.A. Mansour. Existence of traveling wave solutions in a hyperbolic-elliptic system of equations. Comm. Math. Sci., 4(4):731–739, 2006.MATH M.B.A. Mansour. Existence of traveling wave solutions in a hyperbolic-elliptic system of equations. Comm. Math. Sci., 4(4):731–739, 2006.MATH
[Man09a]
Zurück zum Zitat M.B.A. Mansour. Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation. Appl. Math. Mech., 30(4):513–516, 2009.MATHMathSciNet M.B.A. Mansour. Existence of traveling wave solutions for a nonlinear dissipative-dispersive equation. Appl. Math. Mech., 30(4):513–516, 2009.MATHMathSciNet
[Man09b]
Zurück zum Zitat M.B.A. Mansour. Travelling wave solutions for a singularly perturbed Burgers-KdV equation. Pramana, 73(5):799–806, 2009. M.B.A. Mansour. Travelling wave solutions for a singularly perturbed Burgers-KdV equation. Pramana, 73(5):799–806, 2009.
[Man13]
Zurück zum Zitat M.B.A. Mansour. A geometric construction of traveling waves in a generalized nonlinear dispersive–dissipative equation. J. Geom. Phys., pages 1–11, 2013. in press. M.B.A. Mansour. A geometric construction of traveling waves in a generalized nonlinear dispersive–dissipative equation. J. Geom. Phys., pages 1–11, 2013. in press.
[MCJS09]
Zurück zum Zitat V. Manukian, N. Costanzino, C.K.R.T. Jones, and B. Sandstede. Existence of multi-pulses of the regularized short-pulse and Ostrovsky equations. J. Dyn. Diff. Eq., 21:607–622, 2009.MATHMathSciNet V. Manukian, N. Costanzino, C.K.R.T. Jones, and B. Sandstede. Existence of multi-pulses of the regularized short-pulse and Ostrovsky equations. J. Dyn. Diff. Eq., 21:607–622, 2009.MATHMathSciNet
[MDK00]
Zurück zum Zitat D.S. Morgan, A. Doelman, and T.J. Kaper. Stationary periodic patterns in the 1D Gray-Scott model. Math. Appl. Anal., 7(1):105–150, 2000.MATHMathSciNet D.S. Morgan, A. Doelman, and T.J. Kaper. Stationary periodic patterns in the 1D Gray-Scott model. Math. Appl. Anal., 7(1):105–150, 2000.MATHMathSciNet
[MKK00]
Zurück zum Zitat G.S. Medvedev, T.J. Kaper, and N. Kopell. A reaction–diffusion system with periodic front dynamics. SIAM J. Appl. Math., 60(5):1601–1638, 2000.MATHMathSciNet G.S. Medvedev, T.J. Kaper, and N. Kopell. A reaction–diffusion system with periodic front dynamics. SIAM J. Appl. Math., 60(5):1601–1638, 2000.MATHMathSciNet
[MLR11]
Zurück zum Zitat K. Manktelow, M.J. Leamy, and M. Ruzzene. Multiple scales analysis of wave-wave interactions in a cubically nonlinear atomic chain. Nonlinear Dyn., 63:193–203, 2011.MATHMathSciNet K. Manktelow, M.J. Leamy, and M. Ruzzene. Multiple scales analysis of wave-wave interactions in a cubically nonlinear atomic chain. Nonlinear Dyn., 63:193–203, 2011.MATHMathSciNet
[MP99]
Zurück zum Zitat J. Mallet-Paret. The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Diff. Eq., 8:49–128, 1999.MathSciNet J. Mallet-Paret. The global structure of traveling waves in spatially discrete dynamical systems. J. Dyn. Diff. Eq., 8:49–128, 1999.MathSciNet
[MP12]
Zurück zum Zitat P. De Maesschalck and N. Popovic. Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations. J. Math. Anal. Appl., 386(2):542–558, 2012.MATHMathSciNet P. De Maesschalck and N. Popovic. Gevrey properties of the asymptotic critical wave speed in a family of scalar reaction–diffusion equations. J. Math. Anal. Appl., 386(2):542–558, 2012.MATHMathSciNet
[MR13]
Zurück zum Zitat C. Melcher and J.D.M. Rademacher. Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations. arXiv:1309.5523, pages 1–27, 2013. C. Melcher and J.D.M. Rademacher. Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations. arXiv:1309.5523, pages 1–27, 2013.
[MS74]
Zurück zum Zitat J.W. Milnor and J.D. Stasheff. Characteristic Classes. Princeton University Press, 1974. J.W. Milnor and J.D. Stasheff. Characteristic Classes. Princeton University Press, 1974.
[MS03b]
Zurück zum Zitat D. Marchesin and S. Schecter. Oxidation heat pulses in two-phase expansive flow in porous media. Z. Angew. Math. Phys., 54(1):48–83, 2003.MATHMathSciNet D. Marchesin and S. Schecter. Oxidation heat pulses in two-phase expansive flow in porous media. Z. Angew. Math. Phys., 54(1):48–83, 2003.MATHMathSciNet
[MS03e]
Zurück zum Zitat C.C. Mitchell and D.G. Schaeffer. A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol., 65:767–793, 2003. C.C. Mitchell and D.G. Schaeffer. A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol., 65:767–793, 2003.
[MS06]
Zurück zum Zitat J.C. Da Mota and S. Schecter. Combustion fronts in a porous medium with two layers. J. Dyn. Diff. Eq., 18(3):615–665, 2006.MATH J.C. Da Mota and S. Schecter. Combustion fronts in a porous medium with two layers. J. Dyn. Diff. Eq., 18(3):615–665, 2006.MATH
[MS09]
Zurück zum Zitat V. Manukian and S. Schecter. Travelling waves for a thin liquid film with surfactant on an inclined plane. Nonlinearity, 22(1):85–122, 2009.MATHMathSciNet V. Manukian and S. Schecter. Travelling waves for a thin liquid film with surfactant on an inclined plane. Nonlinearity, 22(1):85–122, 2009.MATHMathSciNet
[NF87]
Zurück zum Zitat Y. Nishiura and H. Fujii. Stability of singularly perturbed solutions to systems of reaction–diffusion equations. SIAM J. Math. Anal., 18(6):1726–1770, 1987.MATHMathSciNet Y. Nishiura and H. Fujii. Stability of singularly perturbed solutions to systems of reaction–diffusion equations. SIAM J. Math. Anal., 18(6):1726–1770, 1987.MATHMathSciNet
[Nii97]
Zurück zum Zitat S. Nii. Stability of travelling multiple-front (multiple-back) wave solutions of the FitzHugh–Nagumo equations. SIAM J. Math. Anal., 28(5):1094–1112, 1997.MATHMathSciNet S. Nii. Stability of travelling multiple-front (multiple-back) wave solutions of the FitzHugh–Nagumo equations. SIAM J. Math. Anal., 28(5):1094–1112, 1997.MATHMathSciNet
[Nis94]
Zurück zum Zitat Y. Nishiura. Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit. In Dynamics Reported, pages 25–103. Springer, 1994. Y. Nishiura. Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit. In Dynamics Reported, pages 25–103. Springer, 1994.
[NMIF90]
Zurück zum Zitat Y. Nishiura, M. Mimura, H. Ikeda, and H. Fujii. Singular limit analysis of stability of traveling wave solutions in bistable reaction–diffusion systems. SIAM J. Math. Anal., 21(1):85–122, 1990.MATHMathSciNet Y. Nishiura, M. Mimura, H. Ikeda, and H. Fujii. Singular limit analysis of stability of traveling wave solutions in bistable reaction–diffusion systems. SIAM J. Math. Anal., 21(1):85–122, 1990.MATHMathSciNet
[OHLM03]
Zurück zum Zitat J. Ockendon, S. Howison, A. Lacey, and A. Movchan. Applied Partial Differential Equations. OUP, 2003. J. Ockendon, S. Howison, A. Lacey, and A. Movchan. Applied Partial Differential Equations. OUP, 2003.
[OR75]
Zurück zum Zitat P. Ortoleva and J. Ross. Theory of propagation of discontinuities in kinetic systems with multiple time scales: fronts, front multiplicity, and pulses. J. Chem. Phys., 63:3398–3408, 1975. P. Ortoleva and J. Ross. Theory of propagation of discontinuities in kinetic systems with multiple time scales: fronts, front multiplicity, and pulses. J. Chem. Phys., 63:3398–3408, 1975.
[PE01a]
Zurück zum Zitat D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J. Appl. Math., 62(1):206–225, 2001. D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: I. Traveling fronts and pulses. SIAM J. Appl. Math., 62(1):206–225, 2001.
[PE01b]
Zurück zum Zitat D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses. SIAM J. Appl. Math., 62(1):226–243, 2001. D.J. Pinto and G.B. Ermentrout. Spatially structured activity in synaptically coupled neuronal networks: II. Lateral inhibition and standing pulses. SIAM J. Appl. Math., 62(1):226–243, 2001.
[PK06]
Zurück zum Zitat N. Popović and T.J. Kaper. Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction–diffusion equations. J. Dyn. Diff. Eq., 18(1):103–139, 2006.MATH N. Popović and T.J. Kaper. Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction–diffusion equations. J. Dyn. Diff. Eq., 18(1):103–139, 2006.MATH
[Pop11]
Zurück zum Zitat N. Popovic. A geometric analysis of front propagation in a family of degenerate reaction–diffusion equations with cut-off. Z. Angew. Math. Phys., 62(3):405–437, 2011.MATHMathSciNet N. Popovic. A geometric analysis of front propagation in a family of degenerate reaction–diffusion equations with cut-off. Z. Angew. Math. Phys., 62(3):405–437, 2011.MATHMathSciNet
[Pop12]
Zurück zum Zitat N. Popovic. A geometric analysis of front propagation in an integrable Nagumo equation with a linear cut-off. Physica D, 241:1976–1984, 2012.MathSciNet N. Popovic. A geometric analysis of front propagation in an integrable Nagumo equation with a linear cut-off. Physica D, 241:1976–1984, 2012.MathSciNet
[Rad13]
Zurück zum Zitat J.D.M. Rademacher. First and second order semistrong interaction in reaction–diffusion systems. SIAM J. Appl. Dyn. Syst., 12(1):175–203, 2013.MATHMathSciNet J.D.M. Rademacher. First and second order semistrong interaction in reaction–diffusion systems. SIAM J. Appl. Dyn. Syst., 12(1):175–203, 2013.MATHMathSciNet
[RP12]
Zurück zum Zitat I. Ratas and K. Pyragas. Pulse propagation and failure in the discrete FitzHugh–Nagumo model subject to high-frequency stimulation. Phys. Rev. E, 86:046211, 2012. I. Ratas and K. Pyragas. Pulse propagation and failure in the discrete FitzHugh–Nagumo model subject to high-frequency stimulation. Phys. Rev. E, 86:046211, 2012.
[RR04]
Zurück zum Zitat M. Renardy and R.C. Rogers. An Introduction to Partial Differential Equations. Springer, 2004. M. Renardy and R.C. Rogers. An Introduction to Partial Differential Equations. Springer, 2004.
[RV98]
Zurück zum Zitat J.-M. Roquejoffre and J.-P. Vila. Stability of ZND detonation waves in the Majda combustion model. Asymp. Anal., 18(3):329–348, 1998.MATHMathSciNet J.-M. Roquejoffre and J.-P. Vila. Stability of ZND detonation waves in the Majda combustion model. Asymp. Anal., 18(3):329–348, 1998.MATHMathSciNet
[RW95a]
Zurück zum Zitat L.G. Reyna and M.J. Ward. Metastable internal layer dynamics for the viscous Cahn–Hilliard equation. Meth. Appl. Anal., 2:285–306, 1995.MATHMathSciNet L.G. Reyna and M.J. Ward. Metastable internal layer dynamics for the viscous Cahn–Hilliard equation. Meth. Appl. Anal., 2:285–306, 1995.MATHMathSciNet
[RW95b]
Zurück zum Zitat L.G. Reyna and M.J. Ward. On the exponentially slow motion of a viscous shock. Comm. Pure Appl. Math., 48(2):79–120, 1995.MATHMathSciNet L.G. Reyna and M.J. Ward. On the exponentially slow motion of a viscous shock. Comm. Pure Appl. Math., 48(2):79–120, 1995.MATHMathSciNet
[RW01]
Zurück zum Zitat V. Rottschäfer and C.E. Wayne. Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation. J. Differential Equat., 176:532–560, 2001.MATH V. Rottschäfer and C.E. Wayne. Existence and stability of traveling fronts in the extended Fisher-Kolmogorov equation. J. Differential Equat., 176:532–560, 2001.MATH
[RX04]
Zurück zum Zitat S. Ruan and D. Xiao. Stability of steady states and existence of travelling waves in a vector-disease model. Proc. R. Soc. Edinburgh A, 134(5):991–1011, 2004.MATHMathSciNet S. Ruan and D. Xiao. Stability of steady states and existence of travelling waves in a vector-disease model. Proc. R. Soc. Edinburgh A, 134(5):991–1011, 2004.MATHMathSciNet
[San01]
Zurück zum Zitat B. Sandstede. Stability of travelling waves. In B. Fiedler, editor, Handbook of Dynamical Systems, volume 2, pages 983–1055. Elsevier, 2001. B. Sandstede. Stability of travelling waves. In B. Fiedler, editor, Handbook of Dynamical Systems, volume 2, pages 983–1055. Elsevier, 2001.
[Sch02]
Zurück zum Zitat S. Schecter. Undercompressive shock waves and the Dafermos regularization. Nonlinearity, 15(4): 1361–1377, 2002.MATHMathSciNet S. Schecter. Undercompressive shock waves and the Dafermos regularization. Nonlinearity, 15(4): 1361–1377, 2002.MATHMathSciNet
[Sch04]
Zurück zum Zitat S. Schecter. Existence of Dafermos profiles for singular shocks. J. Differential Equat., 205(1):185–210, 2004.MATHMathSciNet S. Schecter. Existence of Dafermos profiles for singular shocks. J. Differential Equat., 205(1):185–210, 2004.MATHMathSciNet
[Sch06]
Zurück zum Zitat S. Schecter. Eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws via geometric singular perturbation theory. J. Dyn. Diff. Eq., 18(1):53–101, 2006.MATHMathSciNet S. Schecter. Eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws via geometric singular perturbation theory. J. Dyn. Diff. Eq., 18(1):53–101, 2006.MATHMathSciNet
[SM01]
Zurück zum Zitat S. Schecter and D. Marchesin. Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media. Bull. Braz. Math. Soc., 32(3):237–270, 2001.MATHMathSciNet S. Schecter and D. Marchesin. Geometric singular perturbation analysis of oxidation heat pulses for two-phase flow in porous media. Bull. Braz. Math. Soc., 32(3):237–270, 2001.MATHMathSciNet
[Smo94]
Zurück zum Zitat J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer, 1994. J. Smoller. Shock Waves and Reaction-Diffusion Equations. Springer, 1994.
[SPM04]
Zurück zum Zitat S. Schecter, B.J. Plohr, and D. Marchesin. Computation of Riemann solutions using the Dafermos regularization and continuation. Discr. Cont. Dyn. Syst., 10:965–986, 2004.MATHMathSciNet S. Schecter, B.J. Plohr, and D. Marchesin. Computation of Riemann solutions using the Dafermos regularization and continuation. Discr. Cont. Dyn. Syst., 10:965–986, 2004.MATHMathSciNet
[SS04b]
Zurück zum Zitat S. Schecter and P. Szmolyan. Composite waves in the Dafermos regularization. J. Dyn. Diff. Eq., 16(3):847–867, 2004.MATHMathSciNet S. Schecter and P. Szmolyan. Composite waves in the Dafermos regularization. J. Dyn. Diff. Eq., 16(3):847–867, 2004.MATHMathSciNet
[SS09]
Zurück zum Zitat S. Schecter and P. Szmolyan. Persistence of rarefactions under Dafermos regularization: blow-up and an exchange lemma for gain-of-stability turning points. SIAM J. Applied Dynamical Systems, 8(3):822–853, 2009.MATHMathSciNet S. Schecter and P. Szmolyan. Persistence of rarefactions under Dafermos regularization: blow-up and an exchange lemma for gain-of-stability turning points. SIAM J. Applied Dynamical Systems, 8(3):822–853, 2009.MATHMathSciNet
[ST97]
Zurück zum Zitat H. Suzuki and O. Toshiyuki. On the spectra of pulses in a nearly integrable system. SIAM J. Appl. Math., 57(2):485–500, 1997.MATHMathSciNet H. Suzuki and O. Toshiyuki. On the spectra of pulses in a nearly integrable system. SIAM J. Appl. Math., 57(2):485–500, 1997.MATHMathSciNet
[Str08]
Zurück zum Zitat W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons, 2008. W.A. Strauss. Partial Differential Equations: An Introduction. John Wiley & Sons, 2008.
[SW00c]
Zurück zum Zitat X. Sun and M.J. Ward. Dynamics and coarsening of interfaces for the viscous Cahn–Hilliard equation in one spatial dimension. Stud. Appl. Math., 105(3):203–234, 2000.MATHMathSciNet X. Sun and M.J. Ward. Dynamics and coarsening of interfaces for the viscous Cahn–Hilliard equation in one spatial dimension. Stud. Appl. Math., 105(3):203–234, 2000.MATHMathSciNet
[SWR05]
Zurück zum Zitat X. Sun, M.J. Ward, and R. Russell. The slow dynamics of two-spike solutions for the Gray-Scott and Gierer–Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Syst., 4(4):904–953, 2005.MATHMathSciNet X. Sun, M.J. Ward, and R. Russell. The slow dynamics of two-spike solutions for the Gray-Scott and Gierer–Meinhardt systems: competition and oscillatory instabilities. SIAM J. Appl. Dyn. Syst., 4(4):904–953, 2005.MATHMathSciNet
[SZ04]
Zurück zum Zitat H.L. Smith and X.-Q. Zhao. Traveling waves in a bio-reactor model. Nonl. Anal. Real World Appl., 5(5):895–909, 2004.MATHMathSciNet H.L. Smith and X.-Q. Zhao. Traveling waves in a bio-reactor model. Nonl. Anal. Real World Appl., 5(5):895–909, 2004.MATHMathSciNet
[Szm89a]
Zurück zum Zitat P. Szmolyan. A singular perturbation analysis of the transient semiconductor-device equations. SIAM J. Appl. Math., 49(4):1122–1135, 1989.MATHMathSciNet P. Szmolyan. A singular perturbation analysis of the transient semiconductor-device equations. SIAM J. Appl. Math., 49(4):1122–1135, 1989.MATHMathSciNet
[Szm89b]
[TN94]
Zurück zum Zitat M. Taniguchi and Y. Nishiura. Instability of planar interfaces in reaction–diffusion systems. SIAM J. Math. Anal., 25(1):99–134, 1994.MATHMathSciNet M. Taniguchi and Y. Nishiura. Instability of planar interfaces in reaction–diffusion systems. SIAM J. Math. Anal., 25(1):99–134, 1994.MATHMathSciNet
[Vak10]
Zurück zum Zitat A.F. Vakakis. Relaxation oscillations, subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment. Nonlinear Dyn., 61:443–463, 2010.MATHMathSciNet A.F. Vakakis. Relaxation oscillations, subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment. Nonlinear Dyn., 61:443–463, 2010.MATHMathSciNet
[VD13]
Zurück zum Zitat F. Veerman and A. Doelman. Pulses in a Gierer–Meinhardt equation with a slow nonlinearity. SIAM J. Appl. Dyn. Syst., 12(1):28–60, 2013.MATHMathSciNet F. Veerman and A. Doelman. Pulses in a Gierer–Meinhardt equation with a slow nonlinearity. SIAM J. Appl. Dyn. Syst., 12(1):28–60, 2013.MATHMathSciNet
[VH08]
Zurück zum Zitat P. Várkonyi and P. Holmes. On synchronization and traveling waves in chains of relaxation oscillators with an application to Lamprey CPG. SIAM J. Appl. Dyn. Syst., 7(3):766–794, 2008.MATHMathSciNet P. Várkonyi and P. Holmes. On synchronization and traveling waves in chains of relaxation oscillators with an application to Lamprey CPG. SIAM J. Appl. Dyn. Syst., 7(3):766–794, 2008.MATHMathSciNet
[vHDK08]
Zurück zum Zitat P. van Heijster, A. Doelman, and T.J. Kaper. Pulse dynamics in a three-component system: stability and bifurcations. Physica D, 237(24):3335–3368, 2008.MATHMathSciNet P. van Heijster, A. Doelman, and T.J. Kaper. Pulse dynamics in a three-component system: stability and bifurcations. Physica D, 237(24):3335–3368, 2008.MATHMathSciNet
[vHDK+11]
Zurück zum Zitat P. van Heijster, A. Doelman, T.J. Kaper, Y. Nishiura, and K.-I. Ueda. Pinned fronts in heterogeneous media of jump type. Nonlinearity, 24:127–157, 2011.MATHMathSciNet P. van Heijster, A. Doelman, T.J. Kaper, Y. Nishiura, and K.-I. Ueda. Pinned fronts in heterogeneous media of jump type. Nonlinearity, 24:127–157, 2011.MATHMathSciNet
[vHDKP10]
Zurück zum Zitat P. van Heijster, A. Doelman, T.J. Kaper, and K. Promislow. Front interactions in a three-component system. SIAM J. Appl. Dyn. Syst., 9(2):292–332, 2010.MATHMathSciNet P. van Heijster, A. Doelman, T.J. Kaper, and K. Promislow. Front interactions in a three-component system. SIAM J. Appl. Dyn. Syst., 9(2):292–332, 2010.MATHMathSciNet
[vHS11]
Zurück zum Zitat P. van Heijster and B. Sandstede. Planar radial spots in a three-component FitzHugh–Nagumo system. J. Nonlinear Sci., 21:705–745, 2011.MATHMathSciNet P. van Heijster and B. Sandstede. Planar radial spots in a three-component FitzHugh–Nagumo system. J. Nonlinear Sci., 21:705–745, 2011.MATHMathSciNet
[vS03]
Zurück zum Zitat W. van Saarloos. Front propagation into unstable states. Physics Reports, 386:29–222, 2003.MATH W. van Saarloos. Front propagation into unstable states. Physics Reports, 386:29–222, 2003.MATH
[VVV94]
Zurück zum Zitat A.I. Volpert, V. Volpert, and V.A. Volpert. Traveling Wave Solutions of Parabolic Systems. Amer. Math. Soc., 1994. A.I. Volpert, V. Volpert, and V.A. Volpert. Traveling Wave Solutions of Parabolic Systems. Amer. Math. Soc., 1994.
[WB13]
Zurück zum Zitat M.A. Webber and P.C. Bressloff. The effects of noise on binocular rivalry waves: a stochastic neural field model. J. Stat. Mech., 2013:P03001, 2013.MathSciNet M.A. Webber and P.C. Bressloff. The effects of noise on binocular rivalry waves: a stochastic neural field model. J. Stat. Mech., 2013:P03001, 2013.MathSciNet
[WC73]
Zurück zum Zitat H. Wilson and J. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern., 13(2):55–80, 1973.MATH H. Wilson and J. Cowan. A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue. Biol. Cybern., 13(2):55–80, 1973.MATH
[WWL11]
Zurück zum Zitat L. Wang, Y. Wu, and T. Li. Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion. Physica D, 240(11):971–983, 2011.MATHMathSciNet L. Wang, Y. Wu, and T. Li. Exponential stability of large-amplitude traveling fronts for quasi-linear relaxation systems with diffusion. Physica D, 240(11):971–983, 2011.MATHMathSciNet
[Wol94]
Zurück zum Zitat G. Wolansky. On the slow evolution of quasi-stationary shock waves. J. Dyn. Diff. Eq., 6(2):247–276, 1994.MATHMathSciNet G. Wolansky. On the slow evolution of quasi-stationary shock waves. J. Dyn. Diff. Eq., 6(2):247–276, 1994.MATHMathSciNet
[WP10]
Zurück zum Zitat M. Wechselberger and G.J. Pettet. Folds, canards and shocks in advection–reaction–diffusion models. Nonlinearity, 23(8):1949–1969, 2010.MATHMathSciNet M. Wechselberger and G.J. Pettet. Folds, canards and shocks in advection–reaction–diffusion models. Nonlinearity, 23(8):1949–1969, 2010.MATHMathSciNet
[WR95]
Zurück zum Zitat M.J. Ward and L.G. Reyna. Internal layers, small eigenvalues, and the sensitivity of metastable motion. SIAM J. Appl. Math., 55(2):425–445, 1995.MATHMathSciNet M.J. Ward and L.G. Reyna. Internal layers, small eigenvalues, and the sensitivity of metastable motion. SIAM J. Appl. Math., 55(2):425–445, 1995.MATHMathSciNet
[WW07]
Zurück zum Zitat K. Wang and W. Wang. Propagation of HBV with spatial dependence. Math. Biosci., 210(1):78–95, 2007.MATHMathSciNet K. Wang and W. Wang. Propagation of HBV with spatial dependence. Math. Biosci., 210(1):78–95, 2007.MATHMathSciNet
[WZ05]
Zurück zum Zitat Y. Wu and X. Zhao. The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems. Physica D, 200(3):325–358, 2005.MATHMathSciNet Y. Wu and X. Zhao. The existence and stability of travelling waves with transition layers for some singular cross-diffusion systems. Physica D, 200(3):325–358, 2005.MATHMathSciNet
[YTE01]
Zurück zum Zitat A.C. Yew, D.H. Terman, and G.B. Ermentrout. Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math., 61(5):1578–1604, 2001.MATHMathSciNet A.C. Yew, D.H. Terman, and G.B. Ermentrout. Propagating activity patterns in thalamic neuronal networks. SIAM J. Appl. Math., 61(5):1578–1604, 2001.MATHMathSciNet
[ZP08]
Zurück zum Zitat J. Zhang and Y. Peng. Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect. Nonl. Anal.: Theor. Meth. Appl., 68(5):1263–1270, 2008. J. Zhang and Y. Peng. Travelling waves of the diffusive Nicholson’s blowflies equation with strong generic delay kernel and non-local effect. Nonl. Anal.: Theor. Meth. Appl., 68(5):1263–1270, 2008.
[ZT88]
Zurück zum Zitat E.C. Zachmanoglu and D.W. Thoe. Introduction to Partial Differential Equations with Applications. Dover, 1988. E.C. Zachmanoglu and D.W. Thoe. Introduction to Partial Differential Equations with Applications. Dover, 1988.
Metadaten
Titel
Spatial Dynamics
verfasst von
Christian Kuehn
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-12316-5_17