Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms. (English) Zbl 0885.58075

From the authors’ abstract: “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, destabilizing mechanisms…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. 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…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”.
Reviewer: J.Andres (Olomouc)


37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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[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. · Zbl 0836.35147
[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. · Zbl 0805.35050 · doi:10.1007/BF02429865
[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. · Zbl 0840.35104
[6] A. Doelman and P. Holmes (1996) Homoclinic explosions and implosions,Phil. Trans. Roy. Soc. London A 354 845–893. · Zbl 0881.58044 · doi:10.1098/rsta.1996.0035
[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. · Zbl 0861.35112 · doi:10.1007/BF00916648
[8] W. Eckhaus (1983) Relaxation oscillations including a standard chase on French ducks, inAsymptotic Analysis II, Springer Lect. Notes Math.985 449–494. · Zbl 0509.34037 · doi:10.1007/BFb0062381
[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. · Zbl 0797.35070 · doi:10.1007/BF02429869
[11] N. Fenichel (1979) Geometric singular perturbation theory for ordinary differential equations,J. Diff. Eq. 31 53–98. · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9
[12] A. van Harten (1991) On the validity of Ginzburg-Landau’s equation,J. Nonlin. Sci. 1 397–422. · Zbl 0795.35112 · doi:10.1007/BF02429847
[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. · Zbl 0588.76095
[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. · Zbl 0796.34038 · doi:10.1006/jdeq.1994.1025
[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. · Zbl 0871.58072 · doi:10.1137/S003614109325966X
[16] T. Kapitula (1996) Existence and stability of singular heteroclinic orbits for the Ginzburg-Landau equation,Nonlinearity 9 669–685. · Zbl 0895.34042 · doi:10.1088/0951-7715/9/3/004
[17] E. Knobloch and J. De Luca (1990) Amplitude equations for travelling wave convection,Nonlinearity 3 975–980. · Zbl 0717.35070 · doi:10.1088/0951-7715/3/4/001
[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. · Zbl 0836.35132
[19] B.J. Matkovsky and V. Volpert (1992) Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion,Physica 54D 203–219. · Zbl 0742.76093
[20] B.J. Matkovsky and V. Volpert (1993) Stability of plane wave solutions of complex Ginzbrug-Landau equations,Quart. Appl. Math. 51 265–281. · Zbl 0778.35107
[21] G.J. Merchant and S.H. Davis (1990) Morphological instability in rapid directional solidification,Acta Metall. Mater. 38 2683–2693. · doi:10.1016/0956-7151(90)90282-L
[22] P. Metzener and M.R.E. Proctor (1992) Interaction of patterns with disparate scales,Eur. J. Mech. B/Fluids 11 759–778. · Zbl 0762.76038
[23] R.D. Pierce and C.E. Wayne (1995) On the validity of mean-field amplitude equations for counterpropagating wavetrains,Nonlinearity 8 769–779. · Zbl 0833.35128 · doi:10.1088/0951-7715/8/5/007
[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. · Zbl 0649.76018 · doi:10.1017/S0022112088000746
[25] S. Rasenat, F. Busse, and I. Rehberg (1989) A theoretical and experimental study of double-layer convection,J. Fluid Mech. 199 519–540. · Zbl 0659.76115 · doi:10.1017/S0022112089000467
[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. · Zbl 0714.76048
[27] C. Robinson (1983) Sustained resonance for a nonlinear system with slowly varying coefficients,SIAM Math. An. 14 847–860. · Zbl 0523.34035 · doi:10.1137/0514066
[28] W. van Saarloos and P.C. Hohenberg (1992) Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations,Physica 56D 303–367. · Zbl 0763.35088
[29] R.M.J. Schielen and A. Doelman (1996) Modulation equations for spatially periodic systems: Derivation and solutions, preprint. · Zbl 0908.35009
[30] J.T. Stuart (1958) On the non-linear mechanics of hydrodynamic stability,J. Fluid Mech. 4 1–21. · Zbl 0081.41001 · doi:10.1017/S0022112058000276
[31] G. Vittori and P. Blondeaux (1992) Sand ripples under sea waves, Part 3. Brick pattern ripple formation,J. Fluid Mech. 239 23–45. · Zbl 0754.76021 · doi:10.1017/S0022112092004300
[32] S. Wiggins (1988)Global Bifurcations and Chaos, Springer-Verlag, New York. · Zbl 0661.58001
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