Probing a subcritical instability with an amplitude expansion: An exploration of how far one can get

Physica D: Nonlinear Phenomena

Volumn 238, Issue 18, 2009, Pages 1827-1840

  Becherer, Paul ^a   Morozov, Alexander N ^b   van Saarloos, Wim ^a  

^a UNIVERSITEIT LEIDEN   (Netherlands)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Amplitude expansion; Subcritical instabilities

Indexed keywords

PARTIAL DIFFERENTIAL EQUATIONS;

AMPLITUDE EXPANSION; ASYMPTOTIC APPROXIMATION; KURAMOTO-SIVASHINSKY EQUATIONS; PATTERN-FORMING SYSTEM; RADIUS OF CONVERGENCE; RELIABILITY AND ROBUSTNESS; SUBCRITICAL INSTABILITY; SWIFT-HOHENBERG EQUATIONS;

NONLINEAR EQUATIONS;

EID: 68749097202     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2009.03.009     Document Type: Article

References (64)

1. Cross M.C., and Hohenberg P.C. Pattern formation outside of equilibrium. Rev. Modern Phys. 65 (1993) 851-1112
2. Newell A.C., Passot T., and Lega J. Order parameter equations for patterns. Annu. Rev. Fluid Mech. 25 (1993) 399-453
3. van Hecke M., Hohenberg P.C., and van Saarloos W. Amplitude equations for pattern forming systems. In: van Beijeren H., and Ernst M.H. (Eds). Fundamental Problems in Statistical Mechanics vol. VIII (1994), North-Holland, Amsterdam
4. Nicolis G. Introduction to Nonlinear Science (1995), Cambridge University Press, Cambridge, UK
5. Walgraef D. Spatio-Temporal Pattern Formation (1997), Springer, New York
6. Fauve S. Pattern forming instabilities. In: Godrèche C., and Manneville P. (Eds). Hydrodynamics and Nonlinear Instabilities (1998), Cambridge University Press, Cambridge, UK
7. Nishiura Y. Far-from-Equilibrium Dynamics. Translations of Mathematical Monographs vol. 209 (2002), American Mathematical Society, Providence, R.I.
8. Hoyle R. Pattern Formation, An Introduction to Methods (2006), Cambridge, Cambridge
9. Bestehorn M., and Haken H. Transient patterns of the convection instability - a model-calculation. Z. Phys. B 57 (1984) 329-333
10. Turing A.M. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237 (1952) 37-72
11. Rosenblat S., and Davis S.H. Bifurcation from infinity. SIAM J. Appl. Math. 37 (1979) 1-19
12. Kerswell R.R. Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (2005) 17-44
13. Eckhardt B., Schneider T.M., Hof B., and Westerweel J. Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (2007) 447-468
14. Morozov A.N., and van Saarloos W. Subcritical finite-amplitude solutions in plane Couette flow of viscoelastic fluids. Phys. Rev. lett. 95 (2005) 024501
15. Morozov A.N., and van Saarloos W. An introductory essay on subcritical instabilities and the transition to turbulence in visco-elastic parallel shear flows. Phys. Rep. 447 (2007) 112-143
16. Herbert T. Periodic secondary motions in a plane channel. Proc. 5th Int. Conf. on Numerical Methods in Fluid Dynamics (1976), Springer
17. Orszag S.A., and Patera A.T. Subcritical transition to turbulence in planar shear flows. Transition and Turbulence (1981), Academic Press
18. Meulenbroek B., Storm C., Morozov A.N., and van Saarloos W. Weakly nonlinear subcritical instability of visco-elastic Poiseuille flow. J. Non-Newtonian Fluid Mech. 116 (2004) 235-268
19. Sivashinsky G.I. Non-linear analysis of hydrodynamic instability in laminar flames. 1. Derivation of basic equations. Acta Astronaut. 4 (1977) 1177-1206
20. Sivashinsky G.I. Instabilities, pattern-formation and turbulence in flames. Annu. Rev. Fluid Mech. 15 (1983) 179-199
21. Kuramoto Y., and Tsuzuki T. Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Progr. Theor. Phys. 55 (1976) 356-369
22. Nagata M. 3-dimensional finite-amplitude solutions in plane Couette-flow - bifurcation from infinity. J. Fluid Mech. 217 (1990) 519-527
23. Waleffe F. Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 6 (2003) 1517
24. Wedin H., and Kerswell R.R. Exact coherent structures in pipe flow: Travelling wave solutions. J. Fluid Mech. 508 (2004) 333-371
25. Waleffe F. On a self-sustaining process in shear flows. Phys. Fluids 9 (1997) 883
26. Fraser S.J. The steady state and equilibrium approximations: A geometrical picture. J. Chem. Phys. 88 (1988) 4732-4738
27. Heineken F.G., Tsuchiya H.M., and Aris R. On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math. Biosci. 1 (1967) 95-113
28. Landau L.D. On the problem of turbulence. Dokl. Akad. Nauk SSSR 44 (1944) 311-314
29. Stuart J.T. On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. J. Fluid Mech. 9 (1960) 353-370
30. Watson J. On the non-linear mechanics of wave disturbances in stable and unstable parallel flows. J. Fluid Mech. 9 (1960) 371-389
31. Herbert T. Nonlinear stability of parallel flows by high-order amplitude expansions. AIAA J. 18 (1980) 243-248
32. Eckhaus W. Studies in Non-Linear Stability Theory (1965), Springer-Verlag, Berlin
33. Newell A.C., and Whitehead J.A. Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38 (1969) 279-303
34. Weinstein M. Nonlinear instability in plane Poiseuille flow: A quantitative comparison between the methods of amplitude expansions and the method of multiple scales. Proc. R. Soc. Lond. A 375 (1981) 155-167
35. Fujimura K. Methods of centre manifold reduction and multiple scales in the theory of weakly nonlinear stability for fluid motions. Proc. R. Soc. Lond. A 434 (1991) 719-733
36. Fujimura K. Centre manifold reduction and the Stuart-Landau equation for fluid motions. Proc. R. Soc. Lond. A 453 (1997) 181-203
37. Swift J., and Hohenberg P.C. Hydrodynamic fluctuations at convective instability. Phys. Rev. A 15 (1977) 319-328
38. Haken H. Advanced Synergetics (1983), Springer, Berlin
39. Hilali M., Métens S., Borckmans P., and Dewel G. Pattern selection in the generalized Swift-Hohenberg equation. Phys. Rev. E 51 (1995) 2046-2052
40. van Saarloos W. Front propagation into unstable states. II. Linear versus nonlinear marginal stability and rate of convergence. Phys. Rev. A 39 (1989) 6367-6390
41. Haken H. Synergetics-An Introduction (1977), Springer, Berlin
42. Foias C., Sell G.R., and Temam R. Inertial manifolds for nonlinear evolutionary equations. J. Differential Equations 73 (1988) 309-353
43. Arnol'd V.I. Geometrical Methods in the Theory of Ordinary Differential Equations (1983), Springer, New York
44. Guckenheimer J., and Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), Springer, New York
45. Wasow W. Asymptotic Expansions for Ordinary Differential Equations (1965), Interscience, New York
46. Levin J.J., and Levinson N. Singular perturbations of non-linear systems of differential equations and an associated boundary layer problem. J. Ration. Mech. Anal. 3 (1954) 247-270
47. Pomeau Y., and Manneville P. Wavelength selection in cellular flows. Phys. Lett. 75A (1980) 296-298
48. Peletier L.A., and Troy W.C. Spatial Patterns: Higher Order Models in Physics and Mechanics (2001), Birkhäuser, Boston
49. Van den Berg G.J.B., Peletier L.A., and Troy W.C. Global branches of multi-bump periodic solutions of the Swift-Hohenberg equation. Arch. Ration. Mech. Anal. 158 (2001) 91-153
50. Peletier L.A., and Rottschäfer V. Pattern selection of solutions of the Swift-Hohenberg equation. Physica D 194 (2001) 95-126
51. Press W.H., Teukolsky S.A., Vetterling W.T., and Flannery B.P. Numerical Recipes in Fortran 77. 2nd ed. (1992), Cambridge University Press, Cambridge, UK
52. Fornberg B. A Practical Guide to Pseudospectral Methods (1996), Cambridge University Press, Cambridge, UK
53. Kuramoto Y. Waves, Chemical Oscillations, and Turbulence (1984), Springer, Berlin
54. Greene J.M., and Kim J.-S. The steady states of the Kuramoto-Sivashinsky equation. Physica D 33 (1988) 99-120
55. Elder K.R., Gunton J.D., and Goldenfeld N. Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinsky equation. Phys. Rev. E 56 (1997) 1631-1634
56. Misbah C., and Valance A. Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation. Phys. Rev. E 49 (1994) 166-183
57. Chaté H., and Manneville P. Transition to turbulence via spatiotemporal intermittency. Phys. Rev. Lett. 58 (1987) 112-115
58. Malomed B.A., Feng B.-F., and Kawahara T. Stabilized Kuramoto-Sivashinksy system. Phys. Rev. Lett. 64 (2001) 046304
59. van Hecke M. Building blocks of spatiotemporal chaos. Phys. Rev. Lett. 80 (1998) 1896-1899
60. Arnéodo A., Argoul F., Elezgaray J., and Richetti P. Homoclinic chaos in chemical systems. Physica D 62 (1993) 134-169
61. Balmforth N.J. Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech. 27 (1995) 335-373
62. van Hecke M., and Howard M. Ordered and self-disordered dynamics of holes and defects in the one-dimensional complex Ginzburg-Landau equation. Phys. Rev. Lett. 86 (2001) 218-221
63. Bender C.M., and Orszag S.A. Advanced Mathematical Methods for Scientists and Engineers (1978), McGraw-Hill, New York
64. van Saarloos W., and Hohenberg P.C. Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations. Physica D 56 (1992) 303-367