-
4
-
-
84931510837
-
-
An overview of these models and references to this literature can, e.g., be found in the three separate papers by D. G. Aronson, E. D. Conway, and H. F. Weinberger, in Partial Differential Equations and Dynamical Systems, edited by W. E. Fitzgibbon III (Pitman, Boston, 1984).
-
-
-
-
6
-
-
35949037602
-
-
Rev. Mod. Phys. 47, 487 (1975).
-
(1975)
Rev. Mod. Phys.
, vol.47
, pp. 487
-
-
-
9
-
-
84931510839
-
-
some aspects of the analysis in this paper are along the lines of those of Scott (see his book in Ref. 4).
-
-
-
-
11
-
-
0000686658
-
-
This way of creating front propagation is discussed in some more detail by
-
(1985)
J. Stat. Phys.
, vol.39
, pp. 705
-
-
Dee, G.1
-
24
-
-
84931510840
-
-
See, e.g., P. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Vol. 28 of Lecture Notes in Biomathematics, edited by S. Levin (Springer, New York, 1979); J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1983).
-
-
-
-
26
-
-
84931510842
-
-
See, e.g., Oscillations and Traveling Waves in Chemical Systems, edited by R. J. Field and M. Burger (Wiley, New York, 1985); P. Fife, in Nonequilibrium Cooperative Phenomena in Physics and Related Fields, edited by M. G. Velarde (Plenum, New York, 1984).
-
-
-
-
29
-
-
84931510829
-
-
cf. also B. A. Malomed and A. M. Zhabotinskii, in Nonlinear and Turbulent Processes, edited by R. Z. Sagdeev (Gordon and Breach, New York, 1984).
-
-
-
-
30
-
-
84931510832
-
-
I. M. Gel'fand, Usp. Mat. Nauk. 14, No. 2 (86), 87 (1959)
-
-
-
-
38
-
-
33645069615
-
-
Adv. Math. 30, 33 (1978)
-
(1978)
Adv. Math.
, vol.30
, pp. 33
-
-
-
39
-
-
84931510835
-
-
see also H. Weinberger, Ref. 3, and references therein.
-
-
-
-
43
-
-
84931510853
-
-
This concept of marginal stability was first discussed in the context of dendritic growth.
-
-
-
-
48
-
-
84931510851
-
-
Hence the ``natural'' asymptotic speed vstar is the slowest one possible. This seemingly counterintuitive result will be explained in Sec. II.
-
-
-
-
49
-
-
0000375409
-
-
The idea that the leading edge is the most relevant part of the profile for the type of front propagation discussed here already emerged in a paper by
-
(1983)
Phys. Rev. A
, vol.27
, pp. 499
-
-
Langer, J.S.1
Müller-Krumbhaar, H.2
-
50
-
-
84931510852
-
-
Dee (Ref. 8) discusses an example where the marginal stability result (1.5) also applies to the propagation of fronts into a periodic unstable state.
-
-
-
-
52
-
-
0001575917
-
-
the equation is referred to in Ref. 6 as the Swift-Hohenberg-Pomeau-Manneville equation since it was also studied in the context of pattern formation by
-
(1980)
Phys. Lett.
, vol.75 A
, pp. 296
-
-
Pomeau, Y.1
Manneville, P.2
-
54
-
-
84931510856
-
-
Note in this respect that the proof of Aronson and Weinberger (Ref. 22) relies on the use of a maximum principle for positive solutions of Eq. (1.3) (see also P. Fife, Ref. 13). For the Swift-Hohenberg equation, the pattern emerging behind the front is not positive everywhere and the maximum principle cannot be applied.
-
-
-
-
55
-
-
84931510857
-
-
In view of the present zeitgeist, it seems appropriate to stress that we advocate this connection here only for the well-defined class of problems where there is a continuous range of steady-state front solutions, parametrized by their (envelope) velocity v. This generally is the case for propagation into an unstable state but not for propagation into a metastable state. See Secs. III and IV for further discussion of this point.
-
-
-
-
57
-
-
84931510854
-
-
See in particular the work by Keener (Ref. 14), who discussed coupled equations of the form ε partial u/ partial t=D1del2u +f (u,v), partial v/ partial t= ε2D2del2v +g(u,v), ε <<1. Note also that the Swift-Hohenberg equation (1.6) can be written as partial u / partial t= del2v+v+ ε u-u3, del2u+v=0, indicating that it may be possible to analyze this equation following the methods discussed by Keener. I thank P. C. Hohenberg for pointing this out to me.
-
-
-
-
63
-
-
84931510855
-
-
This large friction regime is the regime γ W/v >>1, where W is the wall width and v the wall velocity.
-
-
-
-
64
-
-
84931510862
-
-
Although walls in liquid crystals are sometimes called solitons, it should be kept in mind that the term soliton usually refers to traveling waves which maintain their shape following interaction with other solitons. [See, e.g., G. L. Lamb, Jr., Elements of Soliton Theory (Wiley, New York, 1980).] In the high friction limit relevant for liquid crystals, such waves do not exist.
-
-
-
-
66
-
-
0012950341
-
-
The existence of a continuous family of steady-state solutions appears to be a general feature of front propagation into an unstable state, but we have not been able to prove this in general. For the Swift-Hohenberg equation (1.6), a two-parameter continuous family of traveling-wave solutions was proven to exist in a certain parameter range by
-
(1986)
Commun. Math. Phys.
, vol.107
, pp. 39
-
-
Collet, P.1
Eckmann, J.P.2
-
68
-
-
84931510863
-
-
Presumably, this implies that there is a one-parameter family of stable front solutions.
-
-
-
-
69
-
-
84931510859
-
-
See Sec. IV and Appendix A for further discussion.
-
-
-
-
70
-
-
84931510871
-
-
This analogy with crystal growth was emphasized to me by G. H. Gilmer and J. D. Weeks (private communication).
-
-
-
-
72
-
-
84931510873
-
-
Sharp shocks can and do of course occur in the first-order partial differential equations studied by Shraiman and Bensimon (Ref. 33). In fact, Fig. 3(b) was inspired by their Fig. 2.
-
-
-
-
73
-
-
84931510874
-
-
Clearly, steady-state solutions can also change stability if their structure changes at a certain point, so that, e.g., condition (2.2) below becomes violated. This will prevent the marginal stability point defined by Eq. (1.5) from being approached. This happens for uniformly translating front solutions for γ > case 1 over 12 in the extended FK equation to be discussed in Sec. III. The analysis of this equation suggests that such a change of stability is often accompanied by a change in structure at the marginal stability point.
-
-
-
-
75
-
-
84931510875
-
-
In fact, in the numerical studies of Ref. 33, the creation of fronts with a higher velocity via the judicious choice of initial conditions required careful programming since small errors tended to drive the velocity to the marginal stability value [D. Bensimon (private communication)].
-
-
-
-
78
-
-
84931510876
-
-
We have omitted odd derivatives φx, φxxx, etc., from F since in cases with spatial reflection symmetry these terms are usually absent from the linearized equation (they can occur in the form φx2/ φ2, though—see, e.g., the model studied in Ref. 33). The analysis in Sec. III is unaffected by the presence of such terms in F, but it should be kept in mind that according to the analysis presented Appendix A, there may not always be a continuous family of steady-state front solutions in the absence of reflection symmetry.
-
-
-
-
79
-
-
84931510877
-
-
For an analytic function ω (k), the Cauchy-Riemann equations express the fact that the derivative d ω /dk is unique and independent of the way in which the variation is taken. This implies Red ω /dk= partial ωr/ partial kr and also, at the points where (3.13) holds, partial ωr/ partial ki=- partial ωi/ partial kr=0. In considering v(kr) in (3.15), it is understood that ki is expressed as a function of kr with the aid of (3.13), so that ωr stands for ωr[kr,ki(kr)] and d ωr/dkr for partial ωr/ partial kr+ partial ωi/ partial kr(dki/dkr). Using the above two results based on the analyticity of ω (k), we then get in (3.15) d ωr/dkr= Re d ω /dk.
-
-
-
-
80
-
-
84931510879
-
-
Thus these considerations do not apply to Landau-Ginzburg–type equations with complex coefficients, which arise, e.g., in the theory of chemical waves [see, e.g., Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer, New York, 1984)].
-
-
-
-
81
-
-
84931510881
-
-
This generalizes the observation (see, e.g., Ref. 6) that the solutions of Eq. (1.3) become oscillatory in space for velocities below the marginal stability value.
-
-
-
-
82
-
-
84931510883
-
-
A similar independent variable transformation is also useful for the discussion of the stability of solitary waves.
-
-
-
-
85
-
-
84931510903
-
-
In Fig. 6(a), it is assumed that qr intersects the line qr=(kr)star. From the analysis in the text, it therefore would appear that solutions with qr>(kr)star initially for all ur would approach steady-state solutions with kr>(kr)star. This is generally not the case, however, since such solutions are unstable and since, as argued in Sec. IIIB, such k values generally do not represent the asymptotic decay of the profile.
-
-
-
-
86
-
-
84931510902
-
-
In connection with the possibility sketched in Fig. 6(b), we remark that there is a possibly important difference between first-order partial differential equation and higher-order partial differential equations. As noted by Bensimon and Shraiman (Ref. 33), if q (which is real for first-order equations) is less than kmax initially, then qkmax are generated if initially qr
-
-
-
-
87
-
-
84931510901
-
-
This subject will be reviewed by Hohenberg and Cross (Ref. 7); see also the article by Saul and Showalker, in the book by Field and Burger, Ref. 15.
-
-
-
-
95
-
-
84931510900
-
-
To some extent, these extensions are all more or less related. For instance, the problem of propagation into a periodic unstable state of the AE can, as shown by Dee (Ref. 8), be transformed into the problem of propagation into a homogeneous unstable state of a pair of equations for two fields.
-
-
-
-
96
-
-
84931510899
-
-
A detailed mathematical proof for the Swift-Hohenberg equation has recently been given by P. Collet and J.-P. Eckmann, Helv. Phys. Acta (to be published).
-
-
-
|
