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High Frequency Solutions for the Singularly-Perturbed One-Dimensional Nonlinear Schrödinger Equation

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Abstract

This article is devoted to the nonlinear Schrödinger equation

when the parameter ε approaches zero. All possible asymptotic behaviors of bounded solutions can be described by means of envelopes, or alternatively by adiabatic profiles. We prove that for every envelope, there exists a family of solutions reaching that asymptotic behavior, in the case of bounded intervals. We use a combination of the Nehari finite dimensional reduction together with degree theory. Our main contribution is to compute the degree of each cluster, which is a key piece of information in order to glue them.

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References

  1. Ai, S.: Multi-pulse like orbits for a singularly perturbed nearly integrable system. J. Differential Equations 179, 384–432 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ai, S.: Multi-bump solutions to Carrier's problem. J. Math. Anal. Appl. 277, 405–422 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ai, S., Chen, X., Hastings, S.: Layers and spikes in Non homogeneous bistable reaction-diffusion equations. To appear Trans. Amer. Math. Soc

  4. Ai, S., Hastings, S.: A shooting approach to layers and chaos in a forced Duffing equation J. Differential Equations 185, 389–436 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alessio, F., Montecchiari, P.: Multibump solutions for a class of Lagrangian systems slowly oscillating at infinity. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 107–135 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Alikakos, N., Bates, P.W., Fusco, G.: Solutions to the nonautonomous bistable equation with specified Morse index. I. Existence. Trans. Amer. Math. Soc. 340, 614–654 (1993)

    MathSciNet  Google Scholar 

  7. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd ed. Springer-Verlag, Berlin, 1989

  8. Berestycki, H.: Le nombre de solutions de certains problèmes semi-linéaires elliptiques. J. Funct. Anal. 40, 1–29 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourland, F., Haberman, R.: Separatrix crossing: Time potentials with dissipation. SIAM J. Appl. Math. 50, 1716–1744 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Chow, S.-N., Wang, D.: On the monotonicity of the period function of some second order equations. Caposis Pest. Mat. 111, 14–25 (1986)

    MATH  ADS  Google Scholar 

  11. Coti Zelati, V., Rabinowitz, P.: Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc. 4, 693–727 (1992)

    Article  Google Scholar 

  12. del Pino, M., Felmer, P., Tanaka, K.: An elementary construction of complex patterns in nonlinear Schrödinger equations. Nonlinearity 15, 1653–1671 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Felmer, P., Torres, J.J.: Semi classical limits for the nonlinear Schrödinger equation. Commun. Contemp. Math. 4, 481–512 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Felmer, P., Martínez, S.: High energy solutions for a phase transition problem. J. Differential Equations 194, 198–220 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Felmer, P., Martínez, S., Tanaka, K.: Multi-clustered high energy solutions for a phase transition problem. Proc. Roy. Soc. Edinburgh Sect. A 135, 731–765 (2005)

    MATH  MathSciNet  Google Scholar 

  16. Felmer, P., Martínez, S., Tanaka, K.: High frequency chaotic solutions for a slowly varying dynamical system. To appear in Ergodic Theory Dynam Systems

  17. Gedeon, T., Kokubu, H., Mischaikow, K., Oka, H.: Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing. J. Dynam. Differential Equations 14, 63–84 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hastings, S., Mc Leod, K.: On the periodic solutions of a forced second order equation. J. Nonlinear Sci. 1, 225–245 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kang, X., Wei, J.: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differential Equations 5, 899–928 (2000)

    MATH  MathSciNet  Google Scholar 

  20. Kurland, H.: Monotone and oscillating equilibrium solutions of a problem arising in population genetics. Nonlinear Partial Differential Equations (Durham, N.H., 1982), 323–342, Contemporary Mathematics, Volume 17, Providence, R.I.: American Mathematical Society, 1983

  21. Nakashima, K.: Multi-layered stationary solutions for a spatially inhomogeneous Allen-Cahn equation. J. Differential Equations 191, 234–276 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Nakashima, K.: Stable transition layers in a balanced bistable equation. Differ. Integral Equ. Appl. 13(7–9), 1025–1038 (2000)

  23. Nakashima, K., Tanaka, K.: Clustering layers and boundary layers in spatially in-homogeneous phase transition problems. Ann. Inst. H. Poincaré Anal Non Linéaire, 20, 107–143 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  24. Nehari, Z.: Characteristic values associated with a class of nonlinear second-order differential equations. Acta Math. 105, 141–175 (1961)

    MathSciNet  Google Scholar 

  25. Neishtadt, A.: Passage through a separatrix in a resonance problem with a slowly-varying parameter. PPM 39, 621–632 (1975)

    MATH  MathSciNet  Google Scholar 

  26. Neishtadt, A.: About changes of adiabatic invariant at passage through separatrix. Plasma Physics 12, 992–1001 (1986)

    Google Scholar 

  27. Séré, E.: Existence of infinitely many homoclinic orbits in Hamiltonian systems. Math. Z. 209, 27–42 (1991)

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMR2071 CNRS-UChile Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Patricio Felmer & Salomé Martínez

  2. Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo, 169-8555, Japan

    Kazunaga Tanaka

Authors
  1. Patricio Felmer
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  2. Salomé Martínez
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  3. Kazunaga Tanaka
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Correspondence to Patricio Felmer.

Additional information

Communicated by P. Rabinowitz

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Felmer, P., Martínez, S. & Tanaka, K. High Frequency Solutions for the Singularly-Perturbed One-Dimensional Nonlinear Schrödinger Equation. Arch Rational Mech Anal 182, 333–366 (2006). https://doi.org/10.1007/s00205-006-0431-8

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  • DOI: https://doi.org/10.1007/s00205-006-0431-8

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