Abundance of stable stationary localized solutions to the generalized 1D Swift-Hohenberg equation

Computers and Mathematics with Applications

Volumn 34, Issue 2-4, 1997, Pages 253-266

  Belyakov, L A ^a   Glebsky, L Yu ^b   Lerman, L M ^b  

^a Res Inst Appl Math and Cybernet   (Russian Federation)

^b INSTITUTE OF APPLIED PHYSICS   (Russian Federation)

Author keywords

Gradient PDE; Homoclinic; Localized; Stability

Indexed keywords

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; MATHEMATICAL OPERATORS;

SADDLE FOCUS EQUILIBRIUM; SWIFT HOHENBERG EQUATION;

PARTIAL DIFFERENTIAL EQUATIONS;

EID: 0031192046     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0898-1221(97)00127-2     Document Type: Article

References (27)

1. L.Yu. Glebsky and L.M. Lerman, On small stationary localized solutions for the generalized 1D Swift-Hohenberg equation, Chaos: Internat. J. Nonlin. Sci. 5 (2) (1995).
2. J. Swift and P.S. Hohenberg, Hydrodynamic fluctuations ar the convective instability, Phys. Rev. A15, 319 (1977).
3. Y. Pomeau and S. Zaleski, J. de Physique 42 (4), 515 (1981).
4. P. Collet and J.-P. Eckmann, Instabilities and Fronts in Extended Systems, Princeton Univ. Press, Princeton NJ, (1990).
5. I.S. Aranson, K.A. Gorshkov, A.S. Lomov and M.I. Rabinovich, Physica D43, 435 (1990).
6. M. Tlidi, M. Georgiou and P. Mandel, Phys. Rev. A 48 (2) (1993).
7. J. Lega, J.V. Moloney and A.C. Newell, Swift-Hohenberg equation for lasers, Phys. Rev. Lett. 73 (22), 2978 (1994).
8. W. Eckhaus, Studies in Nonlinear Stability Theory, Springer, New York, (1965).
9. C. van der Meer, Hamiltonian Hopf Bifurcation. Springer Lect. Notes in Math., p. 1160, Springer-Verlag, New York, (1985).
10. L.Yu. Glebsky and L.M. Lerman, Instability of small stationary localized solutions to a class of reversible 1+1 PDEs, Nonlinearity 10, 389-407 (1997).
11. L.N. Korzinov, M.I. Rabinovich and L.Sh. Tsimring, Bearing of spatial disorder in gradient models, (in Russian), 17 pages, Preprint N19, Institute of Appl. Phys. of RAS, (1992).
12. R. Devaney, J. Diff. Equat. 21, 431 (1976).
13. L.A. Belyakov and L.P. Shil'nikov, Selecta Math. Sov. 9 (3) (1990), (translation of the paper in Russian of 1984).
14. L.M. Lerman, Chaos: Interdisc. J. Nonlin. Sci. 1, 174 (1991).
15. S. Wiggins, Global Bifurcations and Chaos: Analytical Methods, Springer-Verlag, New York, (1988).
16. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math., N. 840, Springer-Verlag, Berlin, (1981).
17. F. Hille and R.S. Phillips, Functional Analysis and Semigroups, American Mathematical Society, Colloquium Publications, Providence, RI, (1957).
18. O.Yu. Koltsova and L.M. Lerman, Int. J. Bifurcation and Chaos 5 (3) (1995).
19. G. Iooss and M.C. Pérouéme, J. Diff. Equal. 102, 62-88 (1993).
20. J. Moser, Comm. Pure Appl. Math. 11, 257 (1958).
21. H. Rüssmann, Math. Ann. 154, 285 (1964).
22. V.V. Lychagin, Matem. Sbornik 104, 248 (1977).
23. N.K. Gavrilov and L.P. Shil'nikov, Part I, Math. USSR Sb. 17, 446 (1973).
24. N.K. Gavrilov and L.P. Shil'nikov, Part II, Math. USSR Sb. 19, 1 (1973).
25. M.A. Naimark, Linear Differential Operators, Nauka, Moscow, (1969).
26. T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, (1966).
27. R.W. Hamming, Numerical Methods for Scientists and Engineers, McGraw-Hill, New York, (1962).