Stability of traveling waves: Dichotomies and eigenvalue conditions on finite intervals

Numerical Functional Analysis and Optimization

Volumn 20, Issue 3, 1999, Pages 201-244

  Beyn, Wolf Jürgen ^a   Lorenz, Jens ^b  

^a BIELEFELD UNIVERSITY   (Germany)

^b UNIVERSITY OF NEW MEXICO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; EIGENVALUES AND EIGENFUNCTIONS; LINEARIZATION; MATHEMATICAL OPERATORS; WAVES;

EXPONENTIAL DICHOTOMIES; TRAVELING WAVES;

SPECTRUM ANALYSIS;

EID: 0032678491     PISSN: 01630563     EISSN: None     Source Type: Journal    
DOI: 10.1080/01630569908816889     Document Type: Article

References (33)

1. W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems,IMA J. Numer. Anal. 10, 379-405, 1990.
2. W.-J. Beyn, On well-posed problems for connecting orbits in dynamical systems AMS Series in Contemporary Mathematics 172 (P. Kloeden, K.J. Palmer Eds.), 1994.
3. P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, pp. 1-38 in Dynamics Reported, Vol.2 (U. Kirchgraber, H.-O. Walther Eds.), Teubner, 1989.
4. W.A. Coppel, Dichotomies in Stability Theory, Springer Lecture Notes in Mathematics 629, 1978.
5. F. de Hoog and R. Weiss, An approximation method for boundary value problems on infinite intervals, Computing 24 (1980), 333-351.
6. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445-466, 1961.
7. I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, 1982
8. R. D. Grigorieff, Diskrete Approximation von Eigenwertproblemen, I Qualitative Konvergenz, II Konvergenzordnung, Numer. Math. 24, 355-374, 1975 and ibid. 415-433.
9. R. D. Grigorieff, Diskrete Approximation von Eigenwertproblemen, I Qualitative Konvergenz, II Konvergenzordnung, Numer. Math. 24, 355-374, 1975 and ibid. 415-433.
10. T. Hagstrom and H. B. Keller, The numerical calculation of traveling wave solutions of nonlinear parabolic equations SIAM J. Sci. Stat. Comput. 7 (1986), 978-988.
11. T. Hagstrom and H. B. Keller, Exact boundary conditions at an artificial boundary for partial differential equations in cylinders, SIAM J. Math. Anal. 17 (1986), 322-341.
12. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, Berlin, 1981.
13. L.M. Hocking, K. Stewartson, On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance. Proc. Roy. Soc. London Ser. A, 326, 289-313, 1972.
14. M.C. Irwin Smooth dynamical systems. Academic Press, 1980.
15. T. Kato, Perturbation Theory for Linear Operators, Grundlehren der mathematischen Wissenschaften 132, Springer, Berlin, 1980.
16. H. B. Keller and M. Lentini, Boundary value problems over semi-infinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980), 577-604.
17. J.-M. Kleinkauf, Numerische Analyse tangentialer homokliner Orbits, PhD Thesis, University of Bielefeld, 1998.
18. H.-O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2 (1962), 153-181.
19. G. Kreiss and H.-O. Kreiss, Stability of systems of viscous conservation laws, UCLA, CAM Report 97-54, 1997.
20. G. Kreiss, H.-O. Kreiss and A. Petersson, On the convergence to steady state oj solutions of nonlinear hyperbolic-parabolic systems, SIAM J. Numer. Anal. 31 (1994), 1577-1604.
21. H.-O. Kreiss and J. Lorenz, Stability for time dependent differential equations, Acta Numerica 1998, pp. 203-285, Cambridge University Press, 1998.
22. C.D. Levermore, M. Oliver, The complex Ginzburg-Landau equation as a model problem, pp.141-190 in Dynamical systems and probabilistic methods in partial differential equations (P. Delft, C.D. Levermore, C.E. Wayne Eds.), Lectures in Appl. Math. 31, AMS, 1996.
23. P. Markowich, A theory for the approximation of solutions of boundary value problems on infinite intervals, SIAM J. Math. Anal. 13 (1982), 484-513.
24. A. Mielke, The complex Ginzburg-Landau equation on large and unbounded domains: sharper bounds and attractors, Nonlinearity 10, 199-222, 1997.
25. A. Mielke, Pulses in cGL, private communication, 1997.
26. R.M. Miura, Accurate computation of the stable solitary waves for the FitzHugh-Nagumo equations, J. Math. Biol. 13, 247-269, 1982.
27. J. D. Murray, Mathematical Biology (2nd ed.), Springer, 1993.
28. K. J. Palmer, Exponential dichotomies and transversal homoclinic points. J. Differential Equations 55, 225-256, 1984.
29. D. Peterhof, B. Sandstede, A. Scheel, Exponential dichotomies for solitary-wave solutions of semilinear elliptic equations on infinite cylinders. J. Differential Equations 140, 266-308, 1997.
30. B. Sandstede, Verzweigungstheorie homokliner Verdoppelungen. PhD thesis, University of Stuttgart, 1993.
31. V. Thümmler, Numerische Stabilitätsindikatoren für wandernde Wellen, Diploma thesis, University of Bielefeld, 1998.
32. G. Vainikko, Über die Konvergenz und Divergenz von Näherungsmethoden bei Eigenwert-problemen, Math. Nachr. 78, 145-164, 1977.
33. A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Math. Monographs, Vol. 140, AMS, 1994.