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Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation

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Abstract.

Assuming that f is a potential having three minima at the same level of energy, we study for the conservative equation  

$$ u^{iv}-g(u)u”-\frac{1}{2}g'(u)u'^2+f'(u)=0,$$

the existence of a heteroclinic connection between the extremal equilibria. Our method consists in minimizing the functional

$$\int_{-\infty}^{+\infty}\left[\frac{1}{2}[(u”{}^2)+g(u)u'{}^2]+f(u)\right] dx$$

whose Euler-Lagrange equation is given by (1), in a suitable space of functions.

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References

  1. van den Berg, J. B.: The phase plane picture for a class of fourth order conservative differential equations. J. Differential Equations 161, 110-153 (2000)

    Article  MATH  Google Scholar 

  2. Gomper, G., Schick, M.: Phase transitions and critical phenomena. Academic Press 1994

  3. Habets, P., Sanchez, L., Tarallo, M., Terracini, S.: Heteroclinics for a class of fourth order conservative differential equations. Equadiff 10, Prague 2001, CD ROM Proceedings

  4. Kalies, W.D., Kwapisz, J., VanderVorst, R.C.A.M.: Homotopy classes for stable connections between Hamiltonian saddle-focus equilibria. Commun. Math. Phys. 193, 337-371 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Leitão, H.: Estrutura e Termodinâmica de Misturas Ternárias com Anfifí lico. Ph.D. Thesis, Universidade de Lisboa 1998

  6. Leitão, H., Telo da Gama, M.M.: Scaling of the interfacial tension of microemulsions: A Landau theory approach. J. of Chemical Physics 108, 4189-4198 (1998)

    Article  Google Scholar 

  7. Kalies, W.D., VanderVorst, R.C.A.M.: Multitransition homoclinic and heteroclinic solutions of the extended Fisher-Kolmogorov equation. J. Differential Equations 131, 209-228 (1996)

    Article  MATH  Google Scholar 

  8. Peletier, L.A., Troy, W.C.: A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation. Topol. Methods in Nonlinear Anal. 6, 331-355 (1995)

    MathSciNet  MATH  Google Scholar 

  9. Smets, D., van den Berg, J.B.: Homoclinic solutions for Swift-Hohenberg and suspension bridges type equations. To appear in J. Differential Equations

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Authors and Affiliations

  1. Institut de Mathématique Pure et Appliquée, Université Catholique de Louvain, Chemin du Cyclotron, 2, 1348, Louvain-la-Neuve, Belgium

    D. Bonheure, L. Sanchez, M. Tarallo & S. Terracini

Authors
  1. D. Bonheure
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  2. L. Sanchez
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  3. M. Tarallo
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  4. S. Terracini
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Corresponding author

Correspondence to D. Bonheure.

Additional information

Received: 15 March 2002, Accepted: 16 June 2002, Published online: 17 December 2002

Mathematics Subject Classification (2000):

34C37, 37J45

D. Bonheure: A part of this work was done during a stay of the first author at the Universidade de Lisboa. He would like to thank the CMAF for hospitality and support.

L. Sanchez: The second author was supported by Fundação para Ciência e a Tecnologia.

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Bonheure, D., Sanchez, L., Tarallo, M. et al. Heteroclinic connections between nonconsecutive equilibria of a fourth order differential equation. Cal Var 17, 341–356 (2003). https://doi.org/10.1007/s00526-002-0172-y

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  • DOI: https://doi.org/10.1007/s00526-002-0172-y

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