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Spatially localized free vibrations of certain semilinear wave equations on ℝ2: Recent results and open problems

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Differential Equations and Mathematical Physics

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1285))

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Bibliographical references

  1. Dodd, R.K., Eilbeck, T.C., Gibbon, T.D., Morris, H.C., "Solitons and Nonlinear Wave Equations". Academic Press (1982).

    Google Scholar 

  2. Lamb, G.L., "Elements of Soliton Theory", Wiley-Interscience Series in Pure and Applied Mathematics, J. Wiley and Sons, New York (1980).

    MATH  Google Scholar 

  3. Brézis, H., "Periodic Solutions of Nonlinear Vibrating Strings and Duality Principles", Bull. Am. Math. Soc. 8, 409–426 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  4. Coron, J.M., "Période Minimale pour une Corde Vibrante de Longueur Infinie", C.R. Acad. Sci. Paris, A294, 127–129 (1982).

    MathSciNet  Google Scholar 

  5. Weinstein, A., "Periodic Nonlinear Waves on a Half-Line", Commun. Math. Phys. 99, 385–388 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  6. Vuillermot, P.A., Nonexistence of Spatially Localized Free Vibrations for a Class of Nonlinear Wave Equations", Comment. Math. Helv., to appear (1987).

    Google Scholar 

  7. Rabinowitz, P., "Free Vibrations for a Semilinear Wave Equation", Commun. Pure Appl. Math. 31, 31–68 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  8. Brézis, H., Coron, J.M., Nirenberg, L., "Free Vibrations for a Nonlinear Wave Equation and a Theorem of P. Rabinowitz", Commun. Pure Appl. Math. 33, 667–689.

    Google Scholar 

  9. Dashen, R., Hasslacher, B., Neveu, A., "Particle Spectrum in Model Field Theories from Semiclassical Functional Integral Techniques", Phys. Rev. D11, 3424–3450 (1975).

    MathSciNet  Google Scholar 

  10. Eleonskii, V.M., Kulagin, N.E., Novoshilova, H.S., Silin, V.P., "Asymptotic Expansions and Qualitative Analysis of Finite-Dimensional Models in Nonlinear Field Theory", Teoreticheskaya i Mathematichskaya Fizika, 60, 3, 395–403 (1984).

    MathSciNet  Google Scholar 

  11. Eleonskii, V.M., Kulagin, N.E., Novoshilova, N.S., Silin, V.P., "The Asymptotic Expansion Method and a Qualitative Analysis of Finite-Dimensional Models in non-linear Field Theory", in "Nonlinear and Turbulent Processes in Physics", Gordon and Breach Press, Harwood Acad. Publ., 1333–1336 (1984).

    Google Scholar 

  12. Vuillermot, P.A., "A Class of Orlicz-Sobolev Space with Applications to Variational Problems Involving Nonlinear Hill's Equations", Jour. Math. Anal. Appl. 89, 1, 327–349 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  13. Vuilleemot, P.A., "A Class of Elliptic Partial Differential Equations with Exponential Nonlinearities", Math. Ann. 268, 497–518 (1984).

    Article  MathSciNet  Google Scholar 

  14. Vuillermot, P.A., "Hölder-Regularity for the Eigenfunctions of strongly nonlinear Eigenvalue Problems on Orlicz-Sobolev Spaces", Houston Journal of Mathematics, to appear (1986).

    Google Scholar 

  15. Vuillermot, P.A., "Elliptic Regularization for a Class of Strongly Nonlinear Degenerate Eigenvalue Problems on Orlicz-Sobolev Spaces. I: The ODE Case", Houston Journal of Mathematics, to appear (1986).

    Google Scholar 

  16. Vuillermot, P.A., "Existence and Regularity Theory for Isoperimetric Variational Problems on Orlicz-Sobolev Spaces: A Review", in "Nonlinear Systems of Partial Differential Equations in Applied Mathematics", AMS Series "Lectures in Applied Mathematics, part 2," 23, 109–122, (1986).

    MathSciNet  MATH  Google Scholar 

  17. Mikhailov, A.V., "Integrability of a Two-Dimensional Generalization of the Toda Chain", JETP Lett. 30, 7, 414–418 (1979).

    Google Scholar 

  18. Campbell, D.K., Private communications concerning work in preparation by Campbell, D.K., Negele, J., Peyrard, M. and their collaborators, (1986).

    Google Scholar 

  19. Vuillermot, P., "A Critical Point Theory for a Class of Nonlinear Wave Equations with Applications to the Breather Problem", in preparation.

    Google Scholar 

  20. Kruskal, M., Segur, H., "φ4 has no Small Breathers", Preprint, (1986).

    Google Scholar 

  21. Vuillermot, P., "Non-Existence de Vibrations Libres Localisées pour Certaines Equations des Ondes Semilineaires sur ℝ2", C.R. Acad.Sci.Paris, to appear,(1986).

    Google Scholar 

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Ian W. Knowles Yoshimi Saitō

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© 1987 Springer-Verlag

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Vuillermot, PA. (1987). Spatially localized free vibrations of certain semilinear wave equations on ℝ2: Recent results and open problems. In: Knowles, I.W., Saitō, Y. (eds) Differential Equations and Mathematical Physics. Lecture Notes in Mathematics, vol 1285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0080627

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  • DOI: https://doi.org/10.1007/BFb0080627

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  • Print ISBN: 978-3-540-18479-9

  • Online ISBN: 978-3-540-47983-3

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