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Communications in Mathematical Physics

, Volume 302, Issue 3, pp 815–841 | Cite as

Breather Solutions in Periodic Media

  • Carsten Blank
  • Martina Chirilus-Bruckner
  • Vincent Lescarret
  • Guido SchneiderEmail author
Article

Abstract

For nonlinear wave equations existence proofs for breathers are very rare. In the spatially homogeneous case up to rescaling the sine-Gordon equation \({\partial^2_t u = \partial^2_x u - \sin (u)}\) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media no examples of breather solutions have been known so far. Using spatial dynamics, center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation
$$ s(x) \partial^2_t u(x,t) = \partial^2_x u(x,t) - q(x) u(x,t)+ r(x)u(x,t)^3, $$
with spatially periodic coefficients s, q, and r on the real axis. Such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage.

Keywords

Photonic Crystal Homoclinic Orbit Stable Manifold Center Manifold Nonlinear Wave Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bambusi, D., Paleari, S., Penati, T.: Existence and continuous approximation of small amplitude breathers in 1D and 2D Klein–Gordon lattices. Preprint 2009Google Scholar
  2. 2.
    Bambusi D., Penati T.: Continuous approximation of breathers in one and two dimensional DNLS lattices. Nonlinearity 23, 143–157 (2010)zbMATHCrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Birnir B., McKean H.P., Weinstein A.: The rigidity of sine-Gordon breathers. Comm. Pure Appl. Math. 47(8), 1043–1051 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Busch K., Schneider G., Tkeshelashvili L., Uecker H.: Justification of the Nonlinear Schrödinger equation in spatially periodic media. ZAMP 57, 1–35 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Busch K., von Freyman G., Linden S., Mingaleev S.F., Theshelashvili L., Wegener M.: Periodic nanostructures for photonics. Phys. Rep. 444, 101–202 (2007)CrossRefADSGoogle Scholar
  6. 6.
    Denzler J.: Nonpersistence of breather families for the perturbed sine Gordon equation. Commun. Math. Phys. 158(2), 397–430 (1993)zbMATHCrossRefMathSciNetADSGoogle Scholar
  7. 7.
    Eastham M.S.P.: The spectral theory of periodic differential equations. Scottish Academic Press, Edinburgh (1973)zbMATHGoogle Scholar
  8. 8.
    Eckmann J.-P., Wayne C.E.: The nonlinear stability of front solutions for parabolic partial differential equations. Commun. Math. Phys. 161(2), 323–334 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. 9.
    Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman lectures on physics. Vol. 2: Mainly electromagnetism and matter. Reading, MA-London: Addison-Wesley Publishing Co., Inc., 1964Google Scholar
  10. 10.
    Groves M.D., Mielke A.: A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. Roy. Soc. Edinburgh Sect. A 131, 83–136 (2001)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Groves M.D., Schneider G.: Modulating pulse solutions for a class of nonlinear wave equations. Commun. Math. Phys. 219(3), 489–522 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Groves M.D., Schneider G.: Modulating pulse solutions for quasilinear wave equations. J. Diff. Eq. 219(1), 221–258 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Groves M.D., Schneider G.: Modulating pulse solutions to quadratic quasilinear wave equations over exponentially long length scales. Commun. Math. Phys. 278(3), 567–625 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences, 42. New York: Springer-Verlag, 1983Google Scholar
  15. 15.
    Haragus M., Schneider G.: Bifurcating fronts for the Taylor-Couette problem in infinite cylinders. Z. Angew. Math. Phys. 50(1), 120–151 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer Lecture Notes in Mathematics Vol. 840, Berlin-Heidelberg-NewYork:Springer, 1981Google Scholar
  17. 17.
    James G., Noble P.: Breathers on diatomic Fermi-Pasta-Ulam lattices. Physica D 196(1–2), 124–171 (2004)zbMATHCrossRefMathSciNetADSGoogle Scholar
  18. 18.
    James, G., Sirr, Y.: Center manifold theory in the context of infinite one-dimensional lattices. The Fermi-Pasta-Ulam problem,Lecture Notes in Phys. Vol. 728, Berlin-Heidelberg-New York: Springer, 2008, pp. 208–238Google Scholar
  19. 19.
    James G., Sanchez-Rey B., Cuevas J.: Breathers in inhomogeneous nonlinear lattices: an analysis via center manifold reduction. Rev. Math. Phys. 21(1), 1–59 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    MacKay R.S., Aubry S.: Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7, 1623–1643 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Kirchgässner K.: Wave solutions of reversible systems and applications. J. Diff. Eq. 45, 113–127 (1982)zbMATHCrossRefGoogle Scholar
  22. 22.
    Lescarret V., Blank C., Chirilus-Bruckner M., Chong C., Schneider G.: Standing modulating pulse solutions for a nonlinear wave equation in periodic media. Nonlinearity 22(8), 1869–1898 (2009)zbMATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Ntinos A.A.: Lengths of instability intervals of second order periodic differential equations. Quart. J. Math. Oxford 27, 387–394 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Pankov A.: Periodic nonlinear Schrödinger equation with application to photonic crystals. Milan J. Math. 73, 259–287 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Pelinovsky D.E., Kevrekidis P.G., Frantzeskakis D.J.: Persistence and stability of discrete vortices in nonlinear Schrödinger lattices. Physica D 212, 20–53 (2005)zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Pelinovsky D., Schneider G.: Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential. Applicable Analysis 86(8), 1017–1036 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Pelinovsky D., Schneider G., MacKay R.S.: Justification of the lattice equation for a nonlinear elliptic problem with a periodic potential. Commun. Math. Phys. 284(3), 803–831 (2008)zbMATHCrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Reed M., Simon B.: Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London (1978)zbMATHGoogle Scholar
  29. 29.
    Vanderbauwhede, A., Iooss, G.: Center manifold theory in infinite dimensions. In: Dynamics reported: expositions in dynamical systems, Berlin: Springer, 1992, pp. 125–163Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Carsten Blank
    • 1
  • Martina Chirilus-Bruckner
    • 1
  • Vincent Lescarret
    • 1
  • Guido Schneider
    • 1
    Email author
  1. 1.Institut für Analysis, Dynamik und ModellierungUniversität StuttgartStuttgartGermany

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