The Stability of Traveling Wave Front Solutions of a Reaction-Diffusion System
We investigate the behavior of solutions of the problem $\frac{\partial x}{\partial t} = F(x, y) + \frac{D \partial^2x}{\partial \zeta^2},\quad \frac{\partial y}{\partial t} = G(x, y),$ $x(\zeta, 0) = \varphi (\zeta), \quad y(\zeta, 0) = \psi (\zeta),$ where t ≥ 0 and $-\infty < \zeta < \infty...
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| Published in: | SIAM journal on applied mathematics Vol. 41; no. 1; pp. 145 - 167 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Society for Industrial and Applied Mathematics
01-08-1981
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| Abstract | We investigate the behavior of solutions of the problem $\frac{\partial x}{\partial t} = F(x, y) + \frac{D \partial^2x}{\partial \zeta^2},\quad \frac{\partial y}{\partial t} = G(x, y),$ $x(\zeta, 0) = \varphi (\zeta), \quad y(\zeta, 0) = \psi (\zeta),$ where t ≥ 0 and $-\infty < \zeta < \infty$. Under appropriate assumptions on F and G this system is a model similar to the degenerate forms (i.e., recovery variables kept at equilibrium values) of the Hodgkin-Huxley nerve conduction equations and the Field-Noyes model of the Belousov-Zhabotinskii chemical reaction. We prove the existence and uniqueness of a traveling wave front solution. Secondly, we demonstrate the stability of the traveling wave solution for a general class of initial data. |
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| AbstractList | We investigate the behavior of solutions of the problem\[\begin{gathered} \frac{{\partial x}} {{\partial t}} = F( x,y ) + \frac{{D\partial ^2x }} {{\partial \zeta ^2 }},\quad \frac{{\partial y}} {{\partial t}} = G( x,y ), \hfill \\ x( \zeta ,0 ) = \varphi ( \zeta ),\quad y( \zeta ,0 ) = \psi ( \zeta ) \hfill \\ \end{gathered} \], where $t\geqq 0$ and $ - \infty < \zeta < \infty $. Under appropriate assumptions on F and G this system is a model similar to the degenerate forms (i.e., recovery variables kept at equilibrium values) of the Hodgkin-Huxley nerve conduction equations and the Field-Noyes model of the Belousov-Zhabotinskii chemical reaction. We prove the existence and uniqueness of a traveling wave front solution. Secondly, we demonstrate the stability of the traveling wave solution for a general class of initial data. [PUBLICATION ABSTRACT] We investigate the behavior of solutions of the problem\[\begin{gathered} \frac{{\partial x}} {{\partial t}} = F( x,y ) + \frac{{D\partial ^2x }} {{\partial \zeta ^2 }},\quad \frac{{\partial y}} {{\partial t}} = G( x,y ), \hfill \\ x( \zeta ,0 ) = \varphi ( \zeta ),\quad y( \zeta ,0 ) = \psi ( \zeta ) \hfill \\ \end{gathered} \], where $t\geqq 0$ and $ - \infty < \zeta < \infty $. Under appropriate assumptions on $F$ and $G$ this system is a model similar to the degenerate forms (i.e., recovery variables kept at equilibrium values) of the Hodgkin-Huxley nerve conduction equations and the Field-Noyes model of the Belousov-Zhabotinskii chemical reaction. We prove the existence and uniqueness of a traveling wave front solution. Secondly, we demonstrate the stability of the traveling wave solution for a general class of initial data. We investigate the behavior of solutions of the problem $\frac{\partial x}{\partial t} = F(x, y) + \frac{D \partial^2x}{\partial \zeta^2},\quad \frac{\partial y}{\partial t} = G(x, y),$ $x(\zeta, 0) = \varphi (\zeta), \quad y(\zeta, 0) = \psi (\zeta),$ where t ≥ 0 and $-\infty < \zeta < \infty$. Under appropriate assumptions on F and G this system is a model similar to the degenerate forms (i.e., recovery variables kept at equilibrium values) of the Hodgkin-Huxley nerve conduction equations and the Field-Noyes model of the Belousov-Zhabotinskii chemical reaction. We prove the existence and uniqueness of a traveling wave front solution. Secondly, we demonstrate the stability of the traveling wave solution for a general class of initial data. |
| Author | Klaasen, Gene A. Troy, William C. |
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| CitedBy_id | crossref_primary_10_1103_PhysRevLett_82_2991 crossref_primary_10_1137_0516087 crossref_primary_10_1137_0524059 crossref_primary_10_1007_s00285_006_0381_7 crossref_primary_10_1007_BF03167885 crossref_primary_10_1137_040618291 crossref_primary_10_1007_s00285_006_0057_3 crossref_primary_10_1016_S0034_4877_15_30004_5 crossref_primary_10_1016_j_nonrwa_2018_10_003 crossref_primary_10_1017_S0004972700012107 crossref_primary_10_1090_S0002_9947_1984_0760971_6 crossref_primary_10_1016_0362_546X_95_00142_I crossref_primary_10_1016_0022_0396_81_90021_8 crossref_primary_10_1016_0025_5564_82_90006_2 crossref_primary_10_1063_1_5094351 |
| Cites_doi | 10.1007/BF00280826 10.1038/225535b0 10.1007/BF00280442 10.1021/ja00780a001 10.1063/1.1682542 10.1137/0137042 10.1016/0022-247X(77)90204-9 10.1085/jgp.43.5.867 10.1007/BFb0070595 10.1113/jphysiol.1952.sp004764 10.1007/BF00250432 10.1007/BF01789258 10.1007/978-3-642-86405-6 10.1063/1.1681288 10.1111/j.1469-1809.1937.tb02153.x |
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| Copyright | Copyright 1981 Society for Industrial and Applied Mathematics Copyright © 1980 Society for Industrial and Applied Mathematics [Copyright] © 1980 Society for Industrial and Applied Mathematics |
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| References | R3 R4 R5 R6 R7 R8 Belousov B. P. (R2) 1959 Kolmogoroff A. N. (R13) 1937; 17 Friedman Avner (R9) 1964 Kanel Y. (R12) 1962; 59 R10 R21 R20 R11 R14 Protter M. H. (R15) 1967 R18 R17 R19 R1 |
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| Snippet | We investigate the behavior of solutions of the problem $\frac{\partial x}{\partial t} = F(x, y) + \frac{D \partial^2x}{\partial \zeta^2},\quad \frac{\partial... We investigate the behavior of solutions of the problem\[\begin{gathered} \frac{{\partial x}} {{\partial t}} = F( x,y ) + \frac{{D\partial ^2x }} {{\partial... |
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| SubjectTerms | Applied mathematics Axons Chemical reactions College mathematics Eigenvectors Mathematical functions Nerves Neurosciences Polynomials Population genetics Roots of functions Traveling waves Uniqueness Values |
| Title | The Stability of Traveling Wave Front Solutions of a Reaction-Diffusion System |
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