Pattern propagation in nonlinear dissipative systems

Abstract We discuss the problem of pattern selection in situations where a stable, nonuniform state of a nonlinear dissipative system propagates into an initially unstable, homogeneous region. Our strategy is to consider this process as a generalization of front propagation in a nonlinear diffusion problem for which rigorous results are known; and we point out that these known properties are consistent with a marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. We then describe a more general interpretation of the marginal-stability hypothesis and, finally, present numerical evidence for its validity from three different pattern-forming models.

[1]  P. Ortoleva,et al.  Spontaneous pattern formation in precipitating systems , 1978 .

[2]  H. Müller-Krumbhaar,et al.  Mode selection in a dendritelike nonlinear system , 1983 .

[3]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[4]  J. Murray.  Lectures on nonlinear-differential-equation models in biology , 1977 .

[5]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[6]  J. Langer.  Instabilities and pattern formation in crystal growth , 1980 .

[7]  J. Swift,et al.  Hydrodynamic fluctuations at the convective instability , 1977 .

[8]  E. Ben-Jacob,et al.  Wavelength Selection in Systems Far from Equilibrium , 1982 .

[9]  G. Ahlers,et al.  Vortex-Front Propagation in Rotating Couette-Taylor Flow , 1983 .

[10]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[11]  Harry L. Swinney,et al.  Hydrodynamic instabilities and the transition to turbulence , 1981 .

[12]  Y. Pomeau,et al.  Wavelength selection in axisymmetric cellular structures , 1981 .

[13]  M. Cross,et al.  Phase-winding solutions in a finite container above the convective threshold , 1983, Journal of Fluid Mechanics.

[14]  Paul Manneville,et al.  Stability and fluctuations of a spatially periodic convective flow , 1979 .

[15]  James S. Langer,et al.  Propagating pattern selection , 1983 .

[16]  J. Ross,et al.  Mechanism of chemical instability for periodic precipitation phenomena , 1974 .

[17]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[18]  P. Hagan.  The Instability of Nonmonotonic Wave Solutions of Parabolic Equations , 1981 .

[19]  A. N. Stokes.  On two types of moving front in quasilinear diffusion , 1976 .

[20]  M. Cross,et al.  Effect of Distant Sidewalls on Wave-Number Selection in Rayleigh-Bénard Convection , 1980 .

[21]  E. Magyari.  Travelling kinks in Schlogl's second model for non-equilibrium phase transitions , 1982 .