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Title
Abstract
Introduction
Statement of Results
Conclusion 4.17
Examples of Wavepacket Data
References
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Physics
Statistical and Nonlinear Physics

Linear Superposition in Nonlinear Wave Dynamics

Profile image of Alexander FigotinAlexander Figotin

2006, Reviews in Mathematical Physics

https://doi.org/10.1142/S0129055X06002851
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Abstract

We study nonlinear dispersive wave systems described by hyperbolic PDE's in R d and difference equations on the lattice Z d . The systems involve two small parameters: one is the ratio of the slow and the fast time scales, and another one is the ratio of the small and the large space scales. We show that a wide class of such systems, including nonlinear Schrodinger and Maxwell equations, Fermi-Pasta-Ulam model and many other not completely integrable systems, satisfy a superposition principle. The principle essentially states that if a nonlinear evolution of a wave starts initially as a sum of generic wavepackets (defined as almost monochromatic waves), then this wave with a high accuracy remains a sum of separate wavepacket waves undergoing independent nonlinear evolution. The time intervals for which the evolution is considered are long enough to observe fully developed nonlinear phenomena for involved wavepackets. In particular, our approach provides a simple justification for numerically observed effect of almost non-interaction of solitons passing through each other without any recourse to the complete integrability. Our analysis does not rely on any ansatz or common asymptotic expansions with respect to the two small parameters but it uses rather explicit and constructive representation for solutions as functions of the initial data in the form of functional analytic series.

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References (45)

  1. Babin A. and Figotin A., Nonlinear Photonic Crystals: I. Quadratic nonlinearity, Waves in Random Media, 11, R31-R102, (2001).
  2. Babin A. and Figotin A., Nonlinear Photonic Crystals: II. Interaction classification for quadratic nonlinearities, Waves in Random Media, 12, R25-R52, (2002).
  3. Babin A. and Figotin A., Nonlinear Photonic Crystals: III. Cubic Nonlinearity, Waves in Random Media, 13, pp. R41-R69 (2003).
  4. Babin A. and Figotin A., Nonlinear Maxwell Equations in Inhomogenious Media, Com- mun. Math. Phys. 241, 519-581 (2003).
  5. Babin A. and Figotin A., Polylinear spectral decomposition for nonlinear Maxwell equa- tions, in Partial Differential Equations, M.S. Agranovich and M.A. Shubin eds, Advances in Mathematical Sciences, American Mathematical Society Translations -Series 2, Vol. 206, 2002, p. 1-28.
  6. Babin A. and Figotin A., Nonlinear Photonic Crystals: IV Nonlinear Schrodinger Equa- tion Regime, Waves in Random and Complex Media, Vol. 15, No. 2 (2005), pp. 145-228.
  7. Babin A. and Figotin A., Wavepacket preservation under nonlinear evolution, submit- ted; e-print available online at arxiv.org arXiv:math.AP/0607723
  8. Babin A., Mahalov A. and Nicolaenko B., Global regularity of 3D rotating Navier-Stokes equations for resonant domains, Indiana University Mathematics Journal vol. 48 no. 3 (1999), p.1133-1176.
  9. Babin A., Mahalov A. and Nicolaenko B, Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics, M2AN,v.34,no.2, 2000, p.201- 222.
  10. Bambusi, D., Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253-285.
  11. Ben Youssef, W.; Lannes, D., The long wave limit for a general class of 2D quasilinear hyperbolic problems, Comm. Partial Differential Equations 27 (2002), no. 5-6, 979-1020.
  12. Berman G.P, Izrailev F.M., The Fermi-Pasta-Ulam problem: 50 years of progress, arXiv:nlin.CD
  13. Bogoliubov N. N. and Mitropolsky Y. A., Asymptotic Methods In The Theory Of Non- Linear Oscillations, Delhi, Hindustan Pub. Corp., 1961.
  14. Bona, J. L.; Colin, T.; Lannes, D., Long wave approximations for water waves, Arch. Ration. Mech. Anal. 178 (2005), no. 3, 373-410.
  15. Bourgain, J., Global solutions of nonlinear Schrödinger equations. American Mathemat- ical Society Colloquium Publications, 46. American Mathematical Society, Providence, RI, 1999.
  16. Cazenave T., Semilinear Schrödinger equations. Courant Lecture Notes in Mathemat- ics, 10., New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
  17. Colin, T., Rigorous derivation of the nonlinear Schrödinger equation and Davey- Stewartson systems from quadratic hyperbolic systems, Asymptot. Anal. 31 (2002), no. 1, 69-91.
  18. Colin, T.; Lannes, D., Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems., Discrete Contin. Dyn. Syst. 11 (2004), no. 1, 83-100.
  19. Craig W.; Groves M. D., Normal forms for wave motion in fluid interfaces, Wave Motion 31 (2000), no. 1, 21-41.
  20. Craig, W.; Sulem, C.; Sulem, P.-L., Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity 5 (1992), no. 2, 497-522.
  21. Dineen S., Complex Analysis on Infinite Dimensional Spaces, Springer, 1999.
  22. Gallay T.; Wayne C. E., Invariant manifolds and the long-time asymptotics of the Navier- Stokes and vorticity equations on R 2 ., Arch. Ration. Mech. Anal. 163 (2002), no. 3, 209-258.
  23. Giannoulis, J.; Mielke, A.,The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities., Nonlinearity 17 (2004), no. 2, 551-565.
  24. N. Hayashi and P. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities. Int. J. Pure Appl. Math. 3 (2002), no. 3, 255-273.
  25. Hille E. and Phillips R. S., Functional Analysis and Semigroups, AMS, 1991.
  26. Infeld, E. and Rowlands, G. Nonlinear Waves, Solitons, and Chaos, 2nd ed. Cambridge, England: Cambridge University Press, 2000.
  27. Iooss, G.; Lombardi, E., Polynomial normal forms with exponentially small remainder for analytic vector fields. J. Differential Equations 212 (2005), no. 1, 1-61.
  28. Joly, J.-L.; Metivier, G.; Rauch, J., Diffractive nonlinear geometric optics with rectifi- cation, Indiana Univ. Math. J. 47 (1998), no. 4, 1167-1241.
  29. Kalyakin, L. A., Long-wave asymptotics. Integrable equations as the asymptotic limit of nonlinear systems., Uspekhi Mat. Nauk 44 (1989), no. 1(265), 5-34, 247; translation in Russian Math. Surveys 44 (1989), no. 1, 3-42.
  30. Kalyakin L.A., Asymptotic decay of a one-dimensional wave packet in a nonlinear dis- persive medium, Math. USSR Sb. Surveys 60 (2) (1988) 457-483.
  31. Kuksin S. B., Fifteen years of KAM for PDE. Geometry, topology, and mathematical physics, 237-258, Amer. Math. Soc. Transl. Ser. 2, 212, Amer. Math. Soc., Providence, RI, 2004.
  32. Kirrmann P.; Schneider G.; Mielke A., The validity of modulation equations for extended systems with cubic nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 85-91.
  33. Lax P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.
  34. Maslov V.P., Non-standard characteristics in asymptotic problems, Uspekhi Mat. Nauk 38:6 (1983), 3-36, translation in Russian Math. Surveys 38:6 (1983),1-42.
  35. Mielke A., Schneider G., Ziegra A., Comparison of inertial manifolds and application to modulated systems, Math. Nachr. 214 (2000), 53-69.
  36. Nayfeh, A. H., Perturbation Methods, New York: Wiley, 1973.
  37. A. Pankov, Travelling Waves And Periodic Oscillations In Fermi-Pasta-Ulam Lattices, Imperial College Press, 2005.
  38. Pierce R. D.; Wayne C. E., On the validity of mean-field amplitude equations for coun- terpropagating wavetrains, Nonlinearity 8 (1995), no. 5, 769-779.
  39. Schneider, G., Justification of modulation equations for hyperbolic systems via normal forms, NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 1, 69-82.
  40. Schneider, G., Justification and failure of the nonlinear Schrödinger equation in case of non-trivial quadratic resonances. J. Differential Equations 216 (2005), no. 2, 354-386.
  41. Schneider G., Uecker H. Existence and stability of modulating pulse solutions in Maxwell's equations describing nonlinear optics, Z. Angew. Math. Phys. 54 (2003), no. 4, 677-712.
  42. Sulem C. and Sulem P.-L. , The Nonlinear Schrodinger Equation, Springer , 1999.
  43. Soffer A., Weinstein M. I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), no. 1, 9-74.
  44. Weissert T.P.,The Genesis of Simulation in Dynamics: pursuing the Fermi-Pasta-Ulam problem, Springer-Verlag, New York, 1997.
  45. Whitham G., Linear and Nonlinear Waves, John Wiley & Sons, 1974.

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