Stable small-amplitude solutions in reaction-diffusion systems
Author:
G. Bard Ermentrout
Journal:
Quart. Appl. Math. 39 (1981), 61-86
MSC:
Primary 35K55; Secondary 35B32, 80A30
DOI:
https://doi.org/10.1090/qam/613952
MathSciNet review:
613952
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Abstract: Bifurcation and perturbation techniques are used to construct small-amplitude periodic wave-trains for general systems of reaction and diffusion. All solutions are characterized by the amplitude $a$ and the wavenumber $k$. For scalar diffusion, $k \sim a$, while for certain types of nonscalar diffusion, $k$ is bounded away from zero as $a \searrow 0$. For certain ranges of $a$ and $k$, linear stability of waves is demonstrated.
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D. S. Cohen, P. S. Hagan, and H. C. Simpson, Traveling waves in single and multi-species predator-prey communities, SIAM J. Appl. Math., to appear
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D. S. Cohen, F. C. Hoppensteadt, and R. M. Miura, Slowly modulated oscillations in nonlinear diffusion processes, SIAM J. Appl. Math. 33, 217–229 (1977)
D. S. Cohen, P. S. Hagan, and H. C. Simpson, Traveling waves in single and multi-species predator-prey communities, SIAM J. Appl. Math., to appear
P. C. Fife, Asymptotic states for equations of reaction and diffusion, Bull. AMS 84, 693–726 (1978)
L. N. Howard and N. Kopell, Slowly varying waves and shock structures in reaction-diffusion equations, Stud. Appl. Math. 56, 95–145 (1977)
N. Kopell and L. N. Howard, Plane wave solutions to reaction-diffusion equations, Stud. Appl. Math. 52, 291–328 (1973)
A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech. 38, 279–303 (1969)
P. Ortoleva and J. Ross, On a variety of wave phenomena in chemical oscillations, J. Chem. Phys. 60, 5090–5107 (1974)
J. Rinzel, Neutrally stable traveling waves of nerve conduction equations, J. Math. Biol. 2, 205–217 (1975)
D. H. Sattinger, Topics in stability theory and bifurcation theory, Lectures Notes in Mathematics No. 309, Springer-Verlag, Heidelberg, New York, 1973
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Article copyright:
© Copyright 1981
American Mathematical Society
