Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms Authors A. Doelman Mathematisch Instituut Universiteit Utrecht V. Rottschäfer Mathematisch Instituut Universiteit Utrecht Article

Received: 18 March 1996 Accepted: 22 November 1996 DOI :
10.1007/BF02678142

Cite this article as: Doelman, A. & Rottschäfer, V. J Nonlinear Sci (1997) 7: 371. doi:10.1007/BF02678142
Summary Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small—as, for instance, can be the case in double-layer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the ‘Landau reduction’ to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called ‘localised structures’ in the underlying system: They connect simple periodic patterns atx → ± ∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.

Communicated by Phillip Holmes

References [1]

L. Arnold, C. Jones, K. Mischaikow, and G. Raugel (1994)Dynamical Systems , Lecture Notes in Mathematics1609 , Springer-Verlag, New York.

[2]

P. Bollerman, A. van Harten, and G. Schneider (1994) On the justification of the Ginzburg-Landau approximation, inNonlinear Dynamics and Pattern Formation in the Natural Environment (A. Doelman and A. van Harten, eds.),Pitman Res. Notes in Math.
335 , Longman, UK, 20–36.

[3]

P. Bollerman (1996)On the Theory of Validity of Amplitude Equations , thesis, Utrecht University, the Netherlands.

[4]

A. Doelman (1993) Traveling waves in the complex Ginzburg-Landau equation,

J. Nonlin. Sci.
3 225–266.

MATH CrossRef [5]

A. Doelman, R.A. Gardner, and C.K.R.T. Jones (1995) Instability of quasi-periodic solutions of the Ginzburg-Landau equation,Proc. Roy. Soc. Edinburg
125A 501–517.

[6]

A. Doelman and P. Holmes (1996) Homoclinic explosions and implosions,

Phil. Trans. Roy. Soc. London A
354 845–893.

MATH CrossRef [7]

J. Duan, H. V. Ly, and E.S. Titi (1996) The effects of nonlocal interactions on the dynamics of the Ginzburg-Landau equation,

Z. Angew. Math. Phys.
47 433–455.

CrossRef [8]

W. Eckhaus (1983) Relaxation oscillations including a standard chase on French ducks, in

Asymptotic Analysis II , Springer Lect. Notes Math.

985 449–494.

CrossRef [9]

W. Eckhaus (1992) On modulation equations of the Ginzburg-Landau type, inICIAM 91: Proc. 2nd Int. Conf. Ind. Appl. Math. (R.E. O’Malley, ed.), Society for Industrial and Applied Mathematics, Philadelphia, 83–98.

[10]

W. Eckhaus (1993) The Ginzburg-Landau manifold is an attractor,

J. Nonlin. Sci.
3 329–348.

MATH CrossRef [11]

N. Fenichel (1979) Geometric singular perturbation theory for ordinary differential equations,

J. Diff. Eq.
31 53–98.

MATH CrossRef [12]

A. van Harten (1991) On the validity of Ginzburg-Landau’s equation,

J. Nonlin. Sci.
1 397–422.

MATH CrossRef [13]

D.R. Jenkins (1985) Non-linear interaction of morphological and convective instabilities during solidification of a binary alloy,

I.M.A. J. Appl. Math.
35 145–157.

MATH CrossRef [14]

C.K.R.T. Jones and N. Kopell (1994) Tracking invariant manifolds with differential forms in singularly perturbed systems,

J. Diff. Eq.
108 64–88.

MATH CrossRef [15]

C.K.R.T. Jones, T. Kaper, and N. Kopell (1996) Tracking invariant manifolds up to exponentially small errors,

SIAM J. Math. An.
27 558–577.

MATH CrossRef [16]

T. Kapitula (1996) Existence and stability of singular heteroclinic orbits for the Ginzburg-Landau equation,

Nonlinearity
9 669–685.

MATH CrossRef [17]

E. Knobloch and J. De Luca (1990) Amplitude equations for travelling wave convection,

Nonlinearity
3 975–980.

MATH CrossRef [18]

G. Manogg and P. Metzener (1994) Interaction of modes with disparate scales in Rayleigh-Bénard convection, inNonlinear Dynamics and Pattern Formation in the Natural Environment (A. Doelman and A. van Harten, eds.),Pitman Res. Notes in Math.
335 , Longman, Harlow, Essex, UK, 188–205.

[19]

B.J. Matkovsky and V. Volpert (1992) Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion,Physica
54D 203–219.

[20]

B.J. Matkovsky and V. Volpert (1993) Stability of plane wave solutions of complex Ginzbrug-Landau equations,Quart. Appl. Math.
51 265–281.

[21]

G.J. Merchant and S.H. Davis (1990) Morphological instability in rapid directional solidification,

Acta Metall. Mater.
38 2683–2693.

CrossRef [22]

P. Metzener and M.R.E. Proctor (1992) Interaction of patterns with disparate scales,

Eur. J. Mech. B/Fluids
11 759–778.

MATH [23]

R.D. Pierce and C.E. Wayne (1995) On the validity of mean-field amplitude equations for counterpropagating wavetrains,

Nonlinearity
8 769–779.

MATH CrossRef [24]

M.R.E. Proctor and C.A. Jones (1988) The interaction of two spatially resonant patterns in thermal convection, Part 1. Exact 2:1 resonance,

J. Fluid Mech.
188 301–335.

MATH CrossRef [25]

S. Rasenat, F. Busse, and I. Rehberg (1989) A theoretical and experimental study of double-layer convection,

J. Fluid Mech.
199 519–540.

MATH CrossRef [26]

D.S. Riley and S.H. Davis (1990) Long-wave interaction in morphological and convective instabilities,

I.M.A. J. Appl. Math.
45 267–285.

MATH CrossRef [27]

C. Robinson (1983) Sustained resonance for a nonlinear system with slowly varying coefficients,

SIAM Math. An.
14 847–860.

MATH CrossRef [28]

W. van Saarloos and P.C. Hohenberg (1992) Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations,Physica
56D 303–367.

[29]

R.M.J. Schielen and A. Doelman (1996) Modulation equations for spatially periodic systems: Derivation and solutions, preprint.

[30]

J.T. Stuart (1958) On the non-linear mechanics of hydrodynamic stability,

J. Fluid Mech.
4 1–21.

MATH CrossRef [31]

G. Vittori and P. Blondeaux (1992) Sand ripples under sea waves, Part 3. Brick pattern ripple formation,

J. Fluid Mech.
239 23–45.

MATH CrossRef [32]

S. Wiggins (1988)

Global Bifurcations and Chaos , Springer-Verlag, New York.

MATH © Springer-Verlag New York Inc 1997