Pattern propagation in nonlinear dissipative systems

Generation of Defects and Disorder from Deeply Quenching a Liquid to Form a Solid

“…Thus k r determines the wavelength of the density modulations in the front. More importantly, if no phase slips take place, then the wavenumber of the density modulations left behind by the front is [7,8,16,24]:…”

“…In the case where the liquid is unstable the properties of the solidification front can often be determined from the marginal stability hypothesis [7,8,16,23,24]: Consider the leading edge of such a front, where the growing density modulations are still small in amplitude and suppose this front is advancing with velocity v. In a reference frame that moves with the front, Eq. (5) becomes…”

Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

“…Front selection has been discussed in the literature primarily in terms of what has come to be known as "linear marginal stability" [9][10][11]. This involves a "linear front" with velocity v* and decay rate KS, which are simply obtained by use of a stationary phase argument [10,13,23] from the dispersion relation of the starting equations linearized about the A = 0 state.…”

“…(1.1) and extensions thereof, by formulating a set of conjectures which generalize the "marginal stability" [9][10][11] and "pinch-point" [12][13][14] hypotheses of earlier authors for front dynamics, and reduce to these in appropriate special cases. Moreover, we verify our conjectures by carrying out detailed numerical computations as well as by an analytic perturbation expansion near the dissipationless limit of eq.…”

Marginal stability analysis of the phase field crystal model in one spatial dimension

“…In this analysis the periodicity and velocity at the propagative front is selected by the mode that is marginally stable and can be used to formulate analytical conditions for dynamic selection rules. Such analysis has been applied to the dynamics described by the Kolmogorov-Fisher and Swift-Hohenberg equations [21][22][23] . In the present paper this is applied to the PFC model in one dimension for the case in which a stable periodic ("solid") state invades an unstable uniform state.…”