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Nonlinear Dynamics of a System of Particle-Like Wavepackets

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Instability in Models Connected with Fluid Flows I

Part of the book series: International Mathematical Series ((IMAT,volume 6))

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References

  1. A. Babin and A. Figotin, Nonlinear Photonic Crystals: I. Quadratic nonlinearity, Waves Random Media 11 (2001), no. 2, R31-R102.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Babin and A. Figotin, Nonlinear Photonic Crystals: II. Interaction classification for quadratic nonlinearities, Waves Random Media 12 (2002), no. 4, R25-R52.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Babin and A. Figotin, Nonlinear Photonic Crystals: III. Cubic Nonlinearity, Waves Random Media 13 (2003), no. 4, R41-R69.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. Babin and A. Figotin, Nonlinear Maxwell Equations in Inhomogenious Media, Commun. Math. Phys. 241 (2003), 519-581.

    MATH  MathSciNet  Google Scholar 

  5. A. Babin and A. Figotin, Polylinear spectral decomposition for nonlinear Maxwell equations, In: Partial Differential Equations (M. S. Agranovich and M.A. Shubin, Eds.), Am. Math. Soc. Translations. Series 2 206, 2002, pp. 1-28.

    MathSciNet  Google Scholar 

  6. A. Babin and A. Figotin, Nonlinear Photonic Crystals: IV Nonlinear Schrödinger Equation Regime, Waves Random Complex Media 15 (2005), no. 2, 145-228.

    Article  MathSciNet  Google Scholar 

  7. A. Babin and A. Figotin, Wavepacket Preservation under Nonlinear Evolution, arXiv:math.AP/0607723.

    Google Scholar 

  8. A. Babin and A. Figotin, Linear superposition in nonlinear wave dynamics, Reviews Math. Phys. 18 (2006), no. 9, 971-1053.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Bambusi, Birkhoff normal form for some nonlinear PDEs, Commun. Math. Phys. 234 (2003), no. 2, 253-285.

    Article  MATH  MathSciNet  Google Scholar 

  10. W. Ben Youssef and D. Lannes, The long wave limit for a general class of 2D quasilinear hyperbolic problems, Commun. Partial Differ. Equations 27 (2002), no. 5-6, 979-1020.

    Google Scholar 

  11. N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Hindustan Publ. Corp. Delhi, 1961.

    Google Scholar 

  12. J. L. Bona, T. Colin, and D. Lannes, Long wave approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 373–410.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, Am. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  14. T. Cazenave, Semilinear Schrödinger Equations, Am. Math. Soc., Providence, RI, 2003.

    Google Scholar 

  15. T. Colin, Rigorous derivation of the nonlinear Schrödinger equation and Davey-Stewartson systems from quadratic hyperbolic systems, Asymp. Anal. 31 (2002), no. 1, 69–91.

    MATH  MathSciNet  Google Scholar 

  16. T. Colin and D. Lannes, Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems., Discr. Cont. Dyn. Syst. 11 (2004), no. 1, 83–100.

    MATH  MathSciNet  Google Scholar 

  17. W. Craig and M. D. Groves, Normal forms for wave motion in fluid interfaces, Wave Motion 31 (2000), no. 1, 21–41.

    Article  MathSciNet  Google Scholar 

  18. W. Craig, C. Sulem, and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497–522.

    Article  MATH  MathSciNet  Google Scholar 

  19. T. Gallay and C. E. Wayne, Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on R2, Arch. Ration. Mech. Anal. 163 (2002), no. 3, 209–258.

    Article  MathSciNet  Google Scholar 

  20. Zh. Gang and I. M. Zhou, On soliton dynamics in nonlinear Schrödinger equations, Geom. Funct. Anal. 16 (2006), no. 6, 1377–1390.

    Article  MATH  MathSciNet  Google Scholar 

  21. Zh. Gang and I. M. Zhou, Relaxation of Solitons in Nonlinear Schrodinger Equations with potential, arXiv:math-ph/0603060v1

    Google Scholar 

  22. J. Giannoulis and A. Mielke, The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities, Nonlinearity 17 (2004), no. 2, 551–565.

    Article  MATH  MathSciNet  Google Scholar 

  23. G. Iooss and E. Lombardi, Polynomial normal forms with exponentially small remainder for analytic vector fields. J. Differ. Equations 212 (2005), no. 1, 1–61.

    Article  MATH  MathSciNet  Google Scholar 

  24. J.-L. Joly, G. Metivier, and J. Rauch, Diffractive nonlinear geometric optics with rectification, Indiana Univ. Math. J. 47 (1998), no. 4, 1167–1241.

    Article  MATH  MathSciNet  Google Scholar 

  25. B. L. G. Jonsoon, J. Fröhlich, S. Gustafson, and I. M. Sigal, Long time motion of NLS solitary waves in aconfining potential, Ann. Henri Poincaré 7 (2006), no. 4, 621–660.

    Article  Google Scholar 

  26. L. A. Kalyakin, Long-wave asymptotics. Integrable equations as the asymptotic limit of nonlinear systems, Russian Math. Surv. 44 (1989), no. 1, 3–42.

    Article  MATH  MathSciNet  Google Scholar 

  27. L. A. Kalyakin, Asymptotic decay of a one-dimensional wave packet in a nonlinear dispersion medium, Math. USSR Sb. Surveys 60 (2) (1988) 457–483.

    Article  MathSciNet  Google Scholar 

  28. T. Kato, Perturbation Theory for Linear Operators, Springer, 1980.

    Google Scholar 

  29. S. B. Kuksin, Fifteen years of KAM for PDE, In: Geometry, Topology, and Mathematical Physics, Am. Math. Soc., Providence, RI, 2004, pp. 237–258.

    Google Scholar 

  30. P. Kirrmann, G. Schneider, and A. Mielke, The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85–91.

    MathSciNet  MATH  Google Scholar 

  31. J. Krieger and W. Schlag, Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension, J. Am. Math. Soc. 19 (2006), no. 4, 815–920 (electronic).

    Article  MATH  MathSciNet  Google Scholar 

  32. V. P. Maslov, Non-standard characteristics in asymptotic problems, Russian Math. Surv. 38 (1983), 6, 1–42.

    Article  MATH  Google Scholar 

  33. A. Mielke, G. Schneider, and A. Ziegra, Comparison of inertial manifolds and application to modulated systems, Math. Nachr. 214 (2000), 53–69.

    Article  MATH  MathSciNet  Google Scholar 

  34. R. D. Pierce and C. E. Wayne, On the validity of mean-field amplitude equations for counterpropagating wavetrains, Nonlinearity 8 (1995), no. 5, 769–779.

    Article  MATH  MathSciNet  Google Scholar 

  35. W. Schlag, Spectral theory and nonlinear partial differential equations: a survey, Discr. Cont. Dyn. Syst. 15 (2006), no. 3, 703–723.

    MATH  MathSciNet  Google Scholar 

  36. G. Schneider, Justification of modulation equations for hyperbolic systems via normal forms, NoDEA, Nonlinear Differ. Equ. Appl. 5 (1998), no. 1, 69–82.

    Article  MATH  Google Scholar 

  37. G. Schneider, Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances, J. Differ. Equations 216 (2005), no. 2, 354–386.

    Article  MATH  Google Scholar 

  38. G. Schneider and H. Uecker, Existence and stability of modulating pulse solutions in Maxwell’s equations describing nonlinear optics, Z. Angew. Math. Phys. 54 (2003), no. 4, 677–712.

    Article  MATH  MathSciNet  Google Scholar 

  39. C. Sulem and P.-L. Sulem, The Nonlinear Schrodinger Equation, Springer, 1999.

    Google Scholar 

  40. A. Soffer and M. I. Weinstein, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9–74.

    Article  MathSciNet  Google Scholar 

  41. G. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, 1974.

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of California at Irvine, USA

    Anatoli Babin & Alexander Figotin

Authors
  1. Anatoli Babin
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  2. Alexander Figotin
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Editor information

Editors and Affiliations

  1. Laboratoire J.-L. Lions, Université Denis Diderot, Université 6, Paris, France

    Claude Bardos

  2. Institute of Numerical Mathematics RAS, Moscow State University, Moscow, Russia

    Andrei Fursikov

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Babin, A., Figotin, A. (2008). Nonlinear Dynamics of a System of Particle-Like Wavepackets. In: Bardos, C., Fursikov, A. (eds) Instability in Models Connected with Fluid Flows I. International Mathematical Series, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75217-4_3

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