Propagating fronts near a Lifshitz point

Abstract

Dee and Saarloos 1 found in an extended Fisher-Kolmogorov (EFK) equation the transition from a uniformly translating front to a pattern-generating envelope front. A new 6 3/4 scaling of the front velocity is described and electrohydrodynamic convection (EHC) in planar nematic liquid crystals is proposed as an appropriate system for investigating some aspect of this front type.In EHC one expects for some nematic materials as a function of the frequency co of the applied voltage a continuous transition form oblique to normal-oriented convection rolls. 2 " 6 Near the transition frequency co z (Lifshitz point) the weakly nonlinear behavior is described by the amplitude equation 3 " 5 r 0 d t A=[e + ^d 2 -iZ^d x d 2 + Wtidf-tfd*-\A\ 2 ]A; (1) e = (V 2 -V 2 )/V 2 measures the distance from the normal-roll threshold V c . Numerical values for the relaxation time To and the coherence lengths §/ and Z are given elsewhere. 3 ' 5,6 W is proportional to co -co z . For periodic solutions A ^EexpliiQx+Py)], with2 and e c =0 at gc =^c =0 for normal rolls. The stability of these solutions as well as zigzag and undulated solutions have been investigated. 4,5 Equation (1) is more general then the EFK equation in Ref. 1. It is an equation for a complex field A instead of a real field and includes two space dimensions. For real ^-dependent functions Eq. (1) reduces to the EFK equation in Ref. 1, however, the scaling used there is restricted to W > 0 and 6 = 1. Using the ansatz A =exp[/(gx + 0)]Z?(y), with B a real function, then the marginal-stability analysis, described, for example, in Refs. 1 and 7, gives the velocity for fronts propagating in the y direction (parallel to the convection rolls). For U> Vl2f and U = W+QZ the uniformly translating fronts move with the velocity v\ l [36eU+U 3 -(U 2 -12e) 3/2 V nl (2) V54 TO and for U < Vl27 the envelope fronts move with velocity 2 24e + 2U 2 -FU £3 v 2 s (3) V54 V4C/ + F T o with F = (7U 2 + 24€) l/2 -3U. The wavelength of the periodic patterns behind the front for U < Vl2f is, according to the arguments used in Refs. 1 and 7, 1/2 . _ 8;r *~ 3 24e + 2U 2 -~FU \2e-U 2 -FU This wavelength X diverges for U-£3. + Vl2l.For W<0 and Q -Q c at 6> e c , however, P = 2;rA reaches the critical wave number P c of the oblique periodic solution.One could call U == -\/\2e the Lifshitz point for the transient periodic pattern the behind fronts, similar to W= § for the stationary periodic pattern. For U=0 the first nonvanishing derivative with respect to y is of fourth order in Eq. (1) and then v 2 = f x(i) 3/ V /4 £ 3 /r 0 . Very remarkable is the e 3/4 scaling, which is different to the well-known front velocity v x = 2£\y/e/Vo perpendicular to the convection rolls and to v 2 -(e-e c ) U2 for t/»0 and l/«;0 parallel to the rolls. The measurement of this new e 3/4 scaling seems more easily feasible than the transient periodicity behind the front. A possibility to detect this e law is to measure the growth velocities (v x ,v y -V2) of nuclei of convection rolls, as done in Ref. 8 apart from the Lif...

Multitransition Homoclinic and Heteroclinic Solutions of the Extended Fisher–Kolmogorov Equation

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Existence and Stability of Traveling Fronts in the Extended Fisher–Kolmogorov Equation

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A Geometric Approach to Stationary Defect Solutions in One Space Dimension

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“…see [11,12,53,65]. By the simple rescaling ξ → h 1/4 ξ [53], the stationary problem associated to (4.1) can be written as…”