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Stability of multiple-pulse solutions


Author: Björn Sandstede
Journal: Trans. Amer. Math. Soc. 350 (1998), 429-472
MSC (1991): Primary 35B35, 58F14, 34C37
DOI: https://doi.org/10.1090/S0002-9947-98-01673-0
MathSciNet review: 1360230
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Abstract: In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of $N$-pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the $N$-pulses.

As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many $N$-pulses bifurcate for any fixed $N>1$. Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and $N-1$ in the right half plane can be prescribed.


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  • [AGJ90] J. C. Alexander, R. A. Gardner, and C. K. R. T. Jones, A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410 (1990), 167-212. MR 92d:58028
  • [AJ93] J. C. Alexander and C. K. R. T. Jones, Existence and stability of asymptotically oscillatory triple pulses, Z. Angew. Math. Phys. 44 (1993), 189-200. MR 94k:35042
  • [AJ94] -, Existence and stability of asymptotically oscillatory double pulses, J. Reine Angew. Math. 446 (1994), 49-79. MR 94m:35152
  • [BCT94] B. Buffoni, A. R. Champneys, and J. F. Toland, Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system, J. Dynam. Differential Equations 8 (1996), 221-279. CMP 96:12
  • [BJ89] P. W. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations. In Dynamics Reported (U. Kirchgraber and H.-O. Walther, editors), volume 2, pages 1-38, John Wiley & Sons and Teubner, 1989. MR 90g:58017
  • [CDF90] S.-N. Chow, B. Deng, and B. Fiedler, Homoclinic bifurcation at resonant eigenvalues, J. Dyn. Diff. Eqs. 2 (1990), 177-244. MR 91g:58199
  • [Cha94] A. R. Champneys, Subsidiary homoclinic orbits to a saddle-focus for reversible systems, Int. J. Bifurcation and Chaos 4 (1994), 1447-1482. MR 96e:34074
  • [Dev76] R. L. Devaney, Homoclinic orbits in Hamiltonian systems, J. Diff. Eq. 21 (1976), 431-438. MR 56:1365
  • [EFF82] J. W. Evans, N. Fenichel, and J. A. Feroe, Double impulse solutions in nerve axon equations, SIAM J. Appl. Math. 42 (1982), 219-234. MR 83h:92018a
  • [EMS90] C. Elphick, E. Meron, and E. A. Spiegel, Patterns of propagating pulses, SIAM J. Appl. Math. 50 (1990), 490-503. MR 91k:35033
  • [Fer86] J. A. Feroe, Existence of travelling wave trains in nerve axon equations, SIAM J. Appl. Math. 46 (1986), 1079-1097. MR 88e:92008
  • [Gas83] P. Gaspard, Generation of a countable set of homoclinic flows through bifurcation, Phys. Lett. A 97 (1983), 1-4. MR 84k:58186
  • [GJ90] R. A. Gardner and C. K. R. T. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Indiana Univ. Math. J. 39 (1990), 1197-1222. MR 92d:35039
  • [Gle89] P. Glendinning, Subsidiary bifurcations near bifocal homoclinic orbits, Math. Proc. Cambridge Phil. Soc. 105 (1989), 597-605. MR 90e:58118
  • [Här93] J. Härterich, Kaskaden homokliner Orbits in reversiblen dynamischen Systemen, Diploma thesis, University of Stuttgart, 1993.
  • [Hen81] D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer, New York, Berlin, Heidelberg, 1981. MR 83j:35084
  • [HKK94] A. J. Homburg, H. Kokubu, and M. Krupa, The cusp horseshoe and its bifurcations from inclination-flip homoclinic orbits, Ergodic Theory and Dynamical Systems 14 (1994), 667-693. MR 96a:58134
  • [Kat66] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1966. MR 34:3324
  • [KKO93] M. Kisaka, H. Kokubu, and H. Oka, Bifurcation to $N$-homoclinic orbits and $N$-periodic orbits in vector fields, J. Dyn. Diff. Eqs. 5 (1993), 305-358. MR 94m:58158
  • [Lin90] X.-B. Lin, Using Melnikov's method to solve Silnikov's problems, Proc. Royal Soc. Edinburgh 116A (1990), 295-325. MR 92b:58195
  • [Mer92] E. Meron, Pattern formation in excitable media, Physics Reports 218 (1992), 1-66. CMP 93:01
  • [Nii95a] S. Nii, An extension of the stability index for travelling wave solutions and its application for bifurcations, SIAM J. Math. Anal. 28 (1997), 402-433. CMP 97:08
  • [Nii95b] -, Stability of the travelling $N$-front (N-back) wave solutions of the FitzHugh-Nagumo equations, SIAM J. Math. Anal. 28 (1997), 1094-1112.
  • [Nis94] Y. Nishiura, Coexistence of infinitely many stable solutions to reaction-diffusion systems in the singular limit, In Dynamics Reported (C. K. R. T. Jones, U. Kirchgraber, and H.-O. Walther, editors), volume 3, pages 25-103, Springer, New Series, 1994.
  • [PW92] R. L. Pego and M. I. Weinstein, A class of eigenvalue problems, with applications to instabilities of solitary waves, Phil. Trans. Roy. Soc. London A 340 (1992), 47-94. MR 93g:35115
  • [San93] B. Sandstede, Verzweigungstheorie homokliner Verdopplungen, Doctoral thesis, University of Stuttgart, 1993.
  • [SNI94] H. Suzuki, Y. Nishiura, and H. Ikeda, Stability of traveling waves and a relation between the Evans function and the SLEP method, J. Reine Angew. Math. 475 (1996), 1-37. MR 97f:35014
  • [Wil65] J. H. Wilkinson, The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965. MR 89j:65031
  • [Yan87] E. Yanagida, Branching of double pulse solutions from single pulse solutions in nerve axon equations, J. Diff. Eq. 66 (1987), 243-262. MR 88c:35078
  • [YM89] E. Yanagida and K. Maginu, Stability of double-pulse solutions in nerve axon equations, SIAM J. Appl. Math. 49 (1989), 1158-1173. MR 91d:35110

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Additional Information

Björn Sandstede
Affiliation: Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany
Address at time of publication: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210-1174
Email: sandstede@wias-berlin.de

DOI: https://doi.org/10.1090/S0002-9947-98-01673-0
Keywords: Partial differential equations, solitary waves, homoclinic orbits, stability
Received by editor(s): April 25, 1995
Received by editor(s) in revised form: September 19, 1995
Article copyright: © Copyright 1998 American Mathematical Society

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