Geodesic
Instabilities and Dynamics of Weakly Subcritical Patterns
H.-C Kao 1   ; E. Knobloch 1
1 Department of Physics, University of California, Berkeley CA 94720, USA
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 131-154 Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

The bifurcation to one-dimensional weakly subcritical periodic patterns is described by the cubic-quintic Ginzburg-Landau equation       At = µA + Axx + i(a1|A|2Ax + a2A2Ax*) + b|A|2A - |A|4A. These periodic patterns may in turn become unstable through one of two different mechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamics near the instability threshold in each of these cases using the corresponding modulation equations and compare the results with those obtained from direct numerical simulation of the equation. We also study the stability properties and dynamical evolution of different types of fronts present in the protosnaking region of this equation. The results provide new predictions for the dynamical properties of generic systems in the weakly subcritical regime.
DOI : 10.1051/mmnp/20138509

H.-C Kao  1   ; E. Knobloch  1

1 Department of Physics, University of California, Berkeley CA 94720, USA 
@article{10_1051_mmnp_20138509,
     author = {H.-C Kao and E. Knobloch},
     title = {Instabilities and {Dynamics} of {Weakly} {Subcritical} {Patterns}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {131--154},
     year = {2013},
     volume = {8},
     number = {5},
     doi = {10.1051/mmnp/20138509},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138509/}
}
TY  - JOUR
AU  - H.-C Kao
AU  - E. Knobloch
TI  - Instabilities and Dynamics of Weakly Subcritical Patterns
JO  - Mathematical modelling of natural phenomena
PY  - 2013
SP  - 131
EP  - 154
VL  - 8
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138509/
DO  - 10.1051/mmnp/20138509
LA  - en
ID  - 10_1051_mmnp_20138509
ER  - 
%0 Journal Article
%A H.-C Kao
%A E. Knobloch
%T Instabilities and Dynamics of Weakly Subcritical Patterns
%J Mathematical modelling of natural phenomena
%D 2013
%P 131-154
%V 8
%N 5
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138509/
%R 10.1051/mmnp/20138509
%G en
%F 10_1051_mmnp_20138509
H.-C Kao; E. Knobloch. Instabilities and Dynamics of Weakly Subcritical Patterns. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 131-154. doi: 10.1051/mmnp/20138509

[1] H.-C. Kao, E. Knobloch Phys. Rev. E 2012 026207

[2] W. Eckhaus, G. Iooss Physica D 1989 124 146

[3] A. Shepeleva Math. Method Appl. Sci. 1997 1239 1256

[4] H. Brand, R. Deissler Phys. Rev. A 1992 3732 3736

[5] J. Burke, E. Knobloch Phys. Rev. E 2006 056211

[6] O. Batiste, E. Knobloch, A. Alonso, I. Mercader J. Fluid Mech. 2006 149 158

[7] C. Beaume, A. Bergeon, E. Knobloch Phys. Fluids 2011 094102

[8] A. Doelman, W. Eckhaus Physica D 1991 249 266

[9] J. Duan, P. Holmes Proc. Edinburgh Math. Soc. 1995 77 97

[10] A. Shepeleva Nonlinearity 1998 409 429

[11] D. Henry. Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, 1981.

[12] R. Hoyle. Pattern Formation, Cambridge University Press, Cambridge, 2006.

[13] J. Duan, P. Holmes, E. S. Titi Nonlinearity 1992 1303 1314

[14] S. M. Cox, P. C. Matthews J. Comp. Phys. 2002 430 455

[15] L. Kramer, W. Zimmermann Physica D 1985 221 232

[16] E. Ben-Jacob, H. Brand, G. Dee, L. Kramer, J. S. Langer Physica D 1985 348 364

[17] W. Van Saarloos Phys. Rev. A 1989 6367 6390

[18] J. Burke, J. H. P. Dawes SIAM J. Appl. Dyn. Syst. 2012 261 284

[19] L. J. Boya Eur. J. Phys. 1988 139 144

[20] C. M. Elliott, S. Zheng Arch. Rational Mech. Anal. 1986 339 357

[21] C. Beaume, A. Bergeon, H.-C. Kao, E. Knobloch J. Fluid Mech. 2013 417 448

[22] E. Knobloch, J. Deluca Nonlinearity 1990 975 980

Cité par Sources :