E-Prints Complutense

H. T. Grahn, R. J. Haug, W. M¨uller and K. Ploog, Phys.

Rev. Lett. 67, 1618 (1991); J. Kastrup, H. T. Grahn,

K. Ploog, F. Prengel, A. Wacker and E. Sch¨oll, Appl.

Phys. Lett. 65, 1808 (1994); J. Kastrup, F. Prengel, H.

T. Grahn, K. Ploog and E. Sch¨oll, Phys. Rev. B 53, 1502

(1996).

J. Kastrup, R. Klann, H.T. Grahn, K. Ploog, L.L.

Bonilla, J. Gal´an, M. Kindelan, M. Moscoso, and R.

Merlin, Phys. Rev. B 52, 13761 (1995); J. Kastrup, R.

Hey, K.H. Ploog, H.T. Grahn, L. L. Bonilla, M. Kindelan,

M. Moscoso, A. Wacker and J. Gal´an, Phys. Rev. B

55, 2476 (1997); E. Schomburg, K. Hofbeck, J. Grenzer,

T. Blomeier, A. A. Ignatov, K. F. Renk, D.G. Pavel’ev,

Yu. Koschurinov, V. Ustinov, A. Zukhov, S. Ivanov and

P.S. Kop’ev, Appl. Phys. Lett. 71, (1997); Y. Zhang, J.

Kastrup, R. Klann, K.H. Ploog and H. T. Grahn, Phys.

Rev. Lett. 77, 3001 (1996); K. J. Luo, H. T. Grahn, K.

H. Ploog and L. L. Bonilla, Phys. Rev. Lett. 81, 1290

(1998); J. C. Cao and X.L. Lei, Phys. Rev. B 60, 1871

(1999).

L.L. Bonilla, J. Gal´an, J.A. Cuesta, F.C. Mart´ınez and J.

M. Molera, Phys. Rev. B 50, 8644 (1994); A. Wacker, in

Theory and transport properties of semiconductor nanostructures,

edited by E. Sch¨oll (Chapman and Hall, New

York, 1998). Chapter 10.

M. B¨uttiker and H. Thomas, Phys. Rev. Lett. 38, 78

(1977); A. Sibille, J. F. Palmier, F. Mollot, H. Wang and

J.C. Esnault, Phys. Rev. B 39, 6272 (1989).

L. L. Bonilla, M. Kindelan, M. Moscoso and S. Venakides,

SIAM J. Appl. Math. 57, 1588 (1997).

A. Carpio, L. L. Bonilla, A. Wacker and E. Sch¨oll, Phys.

Rev. E 61, 4866 (2000).

J. P. Keener, SIAM J. Appl. Math. 47, 556 (1987).

J.P. Keener and J. Sneyd, Mathematical Physiology

(Springer, New York, 1998). Chapter 9.

A. E. Bugrim, A.M. Zhabotinsky and I.R. Epstein, Biophys.

J. 73, 2897 (1997).

G Gr¨uner, Rev. Mod. Phys. 60, 1129 (1988); A.A. Middleton,

Phys. Rev. Lett. 68, 670 (1992).

H. S. J. van der Zant, T. P. Orlando, S. Watanabe and

S. H. Strogatz, Phys. Rev. Lett. 74, 174 (1995); S. Flach

and M. Spicci, J. Phys. C 11, 321 (1999).

M. L¨ocher, G.A. Johnson and E.R. Hunt, Phys. Rev.

Lett. 77, 4698 (1996).

F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford

University Press, Oxford, 1967).

P. M. Chaikin and T. C. Lubensky, Principles of condensed

matter physics (Cambridge University Press,

Cambridge, 1995). Chapter 10.

A. Carpio and L.L. Bonilla, Phys. Rev. Lett. (2001), to

appear.

L. L. Bonilla, G. Platero and D. S´anchez, Phys. Rev. B

62, 2786 (2000).

L. L. Bonilla, in Nonlinear Dynamics and Pattern Formation

in Semiconductors and Devices, edited by F.-J.

Niedernostheide (Springer-Verlag, Berlin, 1995), page 1.

F. J. Higuera and L. L. Bonilla, Physica D 57, 161 (1992);

L. L. Bonilla, I. R. Cantalapiedra, G. Gomila and J. M.

Rub´ı, Phys. Rev. E 56, 1500 (1997).

Refs. [5] and [18] consider current self-oscillations mediated

by monopoles in SL with D = 0 or by dipoles in the

hiperbolic limit of the Gunn effect, respectively. We can

use these studies to describe self-oscillations in SL with

D 6= 0. The only thing we need to change is to substitute

our equal-area formulas (23) or (B2) instead of the

equal-area formulas considered in those papers.

If we have ln(0) ≤ un(0) such that u˙n ≥ D(un) (un+1 −

2un+un−1)/−v(un) (un−un−1)/+J−v(un), and ˙ ln ≤

D(ln) (ln+1−2ln+ln−1)−v(ln) (ln−ln−1)/+J −v(ln),

then ln(t) ≤ un(t) for all later times. ln(t) and un(t)

are called sub and supersolutions, respectively. See Ref.

[6]. The use of comparison principles is restricted to spatially

discrete or continuous differential equations which

are first order in time. For applications to parabolic partial

differential equations, see M.H. Protter and H.F.

Weinberger, Maximum principles in differential equations

(Springer, New York, 1984).

A. Carpio, S.J. Chapman, S. Hastings, J.B. McLeod, Eur.

J. Appl. Math. 11, 399 (2000).

S. Flach, Y. Zolotaryuk and K. Kladko, Phys. Rev. E 59,

6105 (1999). See paragraph below Eq. (27).

B. Zinner, J. Diff. Eqs. 96, 1 (1992); A.-M. Filip and S.

Venakides, Comm. Pure Appl. Math. 52, 693 (1999).

L. L. Bonilla, J. Statist. Phys. 46, 659 (1987).

B. W. Knight and G. A. Peterson, Phys. Rev. 147, 617

(1966); J. D. Murray, J. Fluid Mech. 44, 315 (1970).