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Spectral and Dynamical Stability of Nonlinear Waves pp 75–115Cite as

Asymptotic Stability of Waves in Dissipative Systems

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Part of the book series: Applied Mathematical Sciences ((AMS,volume 185))

Abstract

A key motivation for investigating the spectrum of linear operators is to understand the stability of equilibria of nonlinear evolution equations, as well as to describe the flow in a neighborhood of manifolds of approximate equilibria.

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References

  1. In P. Kevrekidis, D. Frantzeskakis, and R. Carretero-González, editors, Emergent Nonlinear Phenomena in Bose–Einstein Condensates, volume 45 of Springer Series in Atomic, Molecular and Optical Physics. Springer-Verlag, New York, 2008.

    Google Scholar 

  2. M. Ablowitz, B. Prinari, and A. Trubatch. Discrete and Continuous Nonlinear Schrödinger Systems, volume 302 of London Math. Soc. Lecture Note Series. Cambridge University Press, Cambridge, 2004.

    Google Scholar 

  3. W. Arendt, C. Batty, M. Hieber, and F. Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems. Springer-Verlag, New York, 2011.

    Book  MATH  Google Scholar 

  4. P. Bates and C.K.R.T. Jones. Invariant manifolds for semilinear partial differential equations. Dynamics Reported, 2:1–38, 1989.

    Article  MathSciNet  Google Scholar 

  5. M. Beck, H. Hupkes, B. Sandstede, and K. Zumbrun. Nonlinear stability of semidiscrete shocks for two-sided schemes. SIAM J. Math. Anal., 42: 857–903, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Bellsky, A. Doelman, T. Kaper, and K. Promislow. Adiabatic stability under semi-strong interactions: the weakly damped regime. to appear in Indiana U. Math. J., 2013.

    Google Scholar 

  7. P. Bressloff and S. Folias. Front bifurcations in an excitatory neural network. SIAM J. Appl. Math., 65:131–151, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Chicone and Y. Latushkin. Evolution Semigroups in Dynamical Systems and Differential Equations, volume 70 of Math. Surv. Monogr. American Mathematical Society, Providence, RI, 1999.

    Google Scholar 

  9. E. Cytrynbaum and J. Keener. Stability conditions for the traveling pulse: Modifying the restitution hypothesis. Chaos, 12(3):788–789, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Doelman and T. Kaper. Semi-strong pulse interactions in a class of coupled reaction–diffusion equations. SIAM J. Appl. Dyn. Sys., 2(1):53–96, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Doelman, T. Kaper, and K. Promislow. Nonlinear asymptotic stability of the semi-strong pulse dynamics in a regularized Gierer–Meinhardt model. SIAM J. Math. Anal., 38(6):1760–1787, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  12. K. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, New York, 2000.

    MATH  Google Scholar 

  13. L. Evans. Partial Differential Equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.

    Google Scholar 

  14. S. Folias and P. Bressloff. Breathing pulses in an excitatory neural network. SIAM J. Appl. Dyn. Sys., 3:378–407, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Folias and P. Bressloff. Stimulus-locked waves and breathers in an excitatory neural network. SIAM J. Appl. Math., 65:2067–2092, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Gardner and C.K.R.T. Jones. Stability of travelling wave solutions of diffusive predator–prey systems. Trans. AMS, 327(2):465–524, 1991b.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Gardner and K. Zumbrun. The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math., 51(7): 797–855, 1998.

    Article  MathSciNet  Google Scholar 

  18. F. Gesztesy, C.K.R.T. Jones, Y. Latushkin, and M. Stanislavova. A spectral mapping theorem and invariant manifolds for nonlinear Schrödinger equations. Indiana U. Math. J., 49(1):221–243, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Ghazaryan, Y. Latushkin, S. Schechter, and A. de Souza. Stability of gasless combustion fronts in one-dimensional solids. Arch. Rat. Mech. Anal., 198:981–1030, 2010a.

    Article  MATH  Google Scholar 

  20. A. Ghazaryan, Y. Latushkin, and S. Schecter. Stability of traveling waves for a class of reaction–diffusion systems that arise in chemical reaction models. SIAM J. Math. Anal., 42:2434–2472, 2010b.

    Article  MathSciNet  MATH  Google Scholar 

  21. A. Ghazaryan, Y. Latushkin, and S. Schechter. Stability of traveling waves for degenerate systems of reaction –diffusion equations. Indiana U. Math. J., 60:443–472, 2011.

    Article  MATH  Google Scholar 

  22. J. Goldstein. Semigroups of Linear Operators and Applications. Oxford University Press, New York, 1985.

    MATH  Google Scholar 

  23. B. Helffer and J. Sjöstrand. From resolvent bounds to semigroup bounds. arXiv1001.4171v1, 2013.

    Google Scholar 

  24. P. Howard. Pointwise estimates for the stability for scalar conservation laws. PhD thesis, Indiana University, 1998.

    Google Scholar 

  25. P. Howard and K. Zumbrun. Stability of undercompressive shock profiles. J. Diff. Eq., 225(1):308–360, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Hupkes and B. Sandstede. Stability of pulse solutions for the discrete Fitzhugh–Nagumo system. Trans. Amer. Math. Soc., 365:251–301, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  27. C.K.R.T. Jones and M. Romeo. Stability of neuronal pulses composed of two concatenated unstable kinks. Phys. Rev. E, 63:011904, 2001.

    Google Scholar 

  28. C.K.R.T. Jones, R. Gardner, and T. Kapitula. Stability of travelling waves for nonconvex scalar conservation laws. Comm. Pure Appl. Math., 46: 505–526, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Kapitula. On the stability of travelling waves in weighted L ∞ spaces. J. Diff. Eq., 112(1):179–215, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  30. T. Kapitula and B. Sandstede. Stability of bright solitary wave solutions to perturbed nonlinear Schrödinger equations. Physica D, 124(1–3):58–103, 1998c.

    Article  MathSciNet  MATH  Google Scholar 

  31. T. Kapitula, P. Kevrekidis, and Z. Chen. Three is a crowd: Solitary waves in photorefractive media with three potential wells. SIAM J. Appl. Dyn. Sys., 5(4):598–633, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Kawashima and A. Matsumura. Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion. Comm. Math. Phys., 101:97–127, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Kevrekidis. The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computation, and Physical Perspectives, volume 232 of Springer Tracts in Modern Physics. Springer-Verlag, New York, 2009.

    Google Scholar 

  34. P. Kevrekidis and M. Weinstein. Dynamics of lattice kinks. Physica D, 142 (1–2):113–152, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. Kevrekidis, R. Carretero-González, G. Theocharis, D. Frantzeskakis, and B. Malomed. Stability of dark solitons in a Bose–Einstein condensate trapped in an optical lattice. Phys. Rev. A, 68(3):035602, 2003.

    Google Scholar 

  36. Y. Latushkin and V. Yurov. Stability estimates for semigroups on Banach spaces. to appear in Disc. Cont. Dyn. Sys. Ser. B, 2013.

    Google Scholar 

  37. R. Moore and K. Promislow. Renormalization group reduction of pulse dynamics in thermally loaded optical parametric oscillators. Physica D, 206:62–81, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  38. R. Moore, W. Kath, B. Sandstede, C.K.R.T. Jones, and J. Alexander. Stability of multiple pulses in optical fibers with phase-sensitive amplification and noise. Optics Comm., 195:1–28, 2001.

    Article  Google Scholar 

  39. A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  40. D. Pelinovsky. Localization in Periodic Potentials. Cambridge University Press, New York, 2011.

    Book  MATH  Google Scholar 

  41. D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis. Persistence and stability of discrete vortices in nonlinear Schrödinger lattices. Physica D, 212:20–53, 2005a.

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Pelinovsky, P. Kevrekidis, and D. Frantzeskakis. Stability of discrete solitons in nonlinear Schrödinger lattices. Physica D, 212:1–19, 2005b.

    Article  MathSciNet  MATH  Google Scholar 

  43. K. Promislow. A renormalization method for modulational stability of quasi-steady patterns in dispersive systems. SIAM J. Math. Anal., 33(6): 1455–1482, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  44. K. Promislow and J. Kutz. Bifurcation and asymptotic stability in the large detuning limit of the optical parametric oscillator. Nonlinearity, 13: 675–698, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. Prüss. On the spectrum of C 0-semigroups. Trans. Amer. Math. Soc., 284:847–857, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Reed and B. Simon. Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, New York, 1980.

    MATH  Google Scholar 

  47. G. Samaey and B. Sandstede. Determining stability of pulses for partial differential equations with time delays. Dyn. Sys., 20:201–222, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  48. B. Sandstede. Center manifolds for homoclinic solutions. J. Dyn. Diff. Eq., 12: 449–510, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  49. B. Sandstede. Evans functions and nonlinear stability of travelling waves in neuronal network models. Int. J. Bif. Chaos, 17:2693–2704, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  50. D. Sattinger. On the stability of waves of nonlinear parabolic systems. Advances in Math., 22:312–355, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Smoller. Shock Waves and Reaction Diffusion Equations. Springer-Verlag, New York, 1983.

    Book  MATH  Google Scholar 

  52. E. Titchmarsh. The Theory of Functions. Oxford University Press, Oxford, UK, second edition, 1976.

    Google Scholar 

  53. J. Yang. Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, 2010.

    Google Scholar 

  54. K. Zumbrun. Multidimensional stability of planar viscous shock waves. In Advances in the theory of shock waves, volume 47 of Prog. Nonlinear Diff. Eq. Appl., pp. 307–516. Birkhäuser Boston, Boston, 2001.

    Google Scholar 

  55. K. Zumbrun. Instantaneous shock location and one-dimensional nonlinear stability of viscous shock waves. Quart. Appl. Math., 69(1):177–202, 2011a.

    MathSciNet  MATH  Google Scholar 

  56. K. Zumbrun. Stability and dynamics of viscous shock waves. In Nonlinear Conservation Laws and Applications, volume 153 of IMA Vol. Math. Appl., pp. 123–167. Springer, New York, 2011b.

    Google Scholar 

  57. K. Zumbrun and P. Howard. Pointwise semigroup methods and stability of viscous shock waves. Indiana U. Math. J., 47:741–781, 1998.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Calvin College, Grand Rapids, Michigan, USA

    Todd Kapitula

  2. Department of Mathematics, Michigan State University, East Lansing, Michigan, USA

    Keith Promislow

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  1. Todd Kapitula
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Kapitula, T., Promislow, K. (2013). Asymptotic Stability of Waves in Dissipative Systems. In: Spectral and Dynamical Stability of Nonlinear Waves. Applied Mathematical Sciences, vol 185. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6995-7_4

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