Equations Of Reaction-diffusion Type Research Articles

Polymers on disordered trees, spin glasses, and traveling waves

We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.

Tunneling Soliton in the Equations of Reaction‐Diffusion Type

Annals of the New York Academy of SciencesVolume 491, Issue 1 p. 149-156 Tunneling Soliton in the Equations of Reaction-Diffusion Type M. I. FREIDLIN, M. I. FREIDLINSearch for more papers by this author M. I. FREIDLIN, M. I. FREIDLINSearch for more papers by this author First published: April 1987 https://doi.org/10.1111/j.1749-6632.1987.tb30049.xCitations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume491, Issue1Reports from the Moscow Refusnik SeminarApril 1987Pages 149-156 RelatedInformation

Dissipative structure in the Characea: Spatial pattern of proton flux as a dissipative structure in characean cells

To clarify an essential mechanism on the formation of alternating alignment of acid and alkali bands along cell walls of Characean algae as Nitella and Chara, the behaviour of proton transport in a simplified model system is investigated theoretically from the viewpoint of nonequilibrium thermodynamics. The present model system is described by a linear diffusion equation for proton concentration with a nonlinear boundary condition on the flux continuity across the cell membrane under an integral constraint on the total flux connected with the light intensity. It is confirmed by numerical calculations that the present model can reproduce band-type spatial patterns of proton concentration above a critical light intensity, as has been observed experimentally. An expression for the pattern of proton flux with a characteristic M-shape is also derived analytically. It is shown that a formation of the spatial patterns can be described analogously to the occurrence of a phase separation if a local potential proposed in this paper is used in place of a chemical potential. An essential mechanism on the formation of spatial patterns in the Characean algae as a kind of dissipative structure is discussed with the aid of a kinetic equation of reaction-diffusion type derived from the local potential near the bifurcation point of the present system.

Nonlinear equations of reaction-diffusion type for neural populations

Several simplified differential equations are derived from the Wilson and Cowan model describing the dynamics of excitatory and inhibitory neurons. It is shown, by expansions of the convolution integrals and the input-output functions, that the basic integrodifferential equations can be reduced to two coupled nonlinear partial differential equations of reaction-diffusion type. Further simplification leads to the coupled partial differential equations mathematically equivalent to the FitzHugh-Nagumo equations for the nerve impulse. Through a brief stability analysis in relation to the existing investigations on the bifurcation phenomena, an attempt is made to clarify the consequence due to the approximations introduced in this paper.

Finite Wavetrains in a One-Dimensional Medium

A modified Rinzel-Keller equation of reaction-diffusion type is investigated analytically. Both on the ring and on the linear fiber, the possible existence of wavetrains containing an arbitrary finite number of impulses is demonstrated. The longest wavetrain shown explicitly consists of ten pulses. Unlike other approaches, the present method varies many parameters simultaneously. An implicit algebraic equation is formulated which contains all possible wave solutions of the system. This equation is then solved with a standard technique (Powell's algorithm). The results obtained extend the findings of other authors. The method can be applied, after appropriate modification, to other piecewise linear systems, that is, to other prototype reaction-diffusion systems