Hostname: page-component-745bb68f8f-grxwn Total loading time: 0 Render date: 2025-01-13T08:38:45.281Z Has data issue: false hasContentIssue false
  • English
  • Français

Travelling waves in radially symmetric parabolic systems

Published online by Cambridge University Press:  14 November 2011

Peter W. Bates
Peter W. Bates
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

It is shown that all travelling or standing plane-wave solutions to certain radially symmetric parabolic systems may be found by solving a related scalar ordinary differential equation (ODE). The radially symmetric systems considered here are those whose reaction term is radially directed and points inward near infinity. The stability of these waves is also discussed. Many systems arising in the physical sciences are included in the class studied and so the classification and stability of the travelling waves has physical significance.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 3-4 , 1985 , pp. 269 - 275
Copyright
Copyright © Royal Society of Edinburgh 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Aronson, D. G. and Weinberger, H. F.. Nonlinear diffusion in population genetics, combustion, and nerve propagation. In Partial Differential Equations, ed. Goldstein, J.. Lecture Notes in Mathematics 446, 5–49 (New York: Springer, 1975).Google Scholar
2Barrow, D. L. and Bates, P. W.. Bifurcation and stability of periodic travelling waves for a reaction-diffusion system. J. Differential Equations 50 (1983), 218–233.Google Scholar
3Barrow, D. L. and Bates, P. W.. Bifurcation of periodic travelling waves for a reaction-diffusion system. Lecture Notes in Mathematics 964, 69–76 (New York: Springer, 1982).Google Scholar
4Ben-Jacob, E., Brand, H., Dee, G., Kramer, L. and Langer, J. S.. Pattern propagation in nonlinear dissipative systems. Physica D, to appear.Google Scholar
5Brown, K. J., Dunne, P. C. and Gardner, R. A.. A semilinear parabolic system arising in the theory of superconductivity. J. Differential Equations 40 (1981), 232–252.CrossRefGoogle Scholar
6Coleman, S.. Classical lumps and their quantum descendants. In New Phenomena in Subnuclear Physics, ed. Zichichi, A. (New York: Plenum, 1977).Google Scholar
7Dee, G. and Langer, J. S.. Propagating pattern selection. Phys. Rev. Lett. 50 (1983), 383–386.Google Scholar
8Ginibre, J. and Velo, G.. On a class of nonlinear Schrödinger equations. I and II. J. Funct. Anal. 32 (1979), 1–71.Google Scholar
9Henry, D.. Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics 840 (New York: Springer, 1981).Google Scholar
10Jaffe, A. and Taubes, C.. Vortices and monopoles (Progress in Physics 2) (Boston: Birkhäuser, 1980).Google Scholar
11Kuramoto, Y. and Yamada, T.. Turbulent state in chemical reactions. Progr. Theoret. Phys. 56 (1976), 679–681.CrossRefGoogle Scholar
12Kuramoto, Y. and Yamada, T.. Pattern formation in oscillatory chemical reactions. Progr. Theoret. Phys. 56 (1976), 724–740.CrossRefGoogle Scholar
13Lopes, J. Leite. Gauge field theories an introduction (New York: Pergamon Press, 1981).Google Scholar
14Rajaraman, R.. Some non-perturbative semiclassical methods in quantum field theory (a pedagogical review). Phys. Rep. 21 (1975), 227–313.CrossRefGoogle Scholar
15Scott, A. C., Chu, F. Y. F. and McLaughlin, D. W.. The soliton: a new concept in applied science. Proc. IEEE 61 (1973), 1143–1183.CrossRefGoogle Scholar
16Skyrme, T. H. R.. A non-linear theory of strong interactions. Proc. Roy. Soc. London Sect. A 24 (1958), 260–278.Google Scholar
17Terman, D.. Travelling wave solutions of multistable reaction-diffusion equations (Contemporary Math. 17), pp. 361–378 (Providence, R.I.: 1983).Google Scholar