### Abstract

This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let ε be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is O(ε^{1/3}) as ε goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out the Lyapunov-Schmidt reduction.

Original language | English |
---|---|

Pages (from-to) | 21-44 |

Number of pages | 24 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 3 |

Issue number | 1 |

Publication status | Published - Feb 2003 |

Externally published | Yes |

### Fingerprint

### Keywords

- Planar traveling waves
- Stability

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

**Instability of planar traveling waves in bistable reaction-diffusion systems.** / Taniguchi, Masaharu.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems - Series B*, vol. 3, no. 1, pp. 21-44.

}

TY - JOUR

T1 - Instability of planar traveling waves in bistable reaction-diffusion systems

AU - Taniguchi, Masaharu

PY - 2003/2

Y1 - 2003/2

N2 - This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let ε be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is O(ε1/3) as ε goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out the Lyapunov-Schmidt reduction.

AB - This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let ε be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is O(ε1/3) as ε goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out the Lyapunov-Schmidt reduction.

KW - Planar traveling waves

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=0042762788&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042762788&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0042762788

VL - 3

SP - 21

EP - 44

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

SN - 1531-3492

IS - 1

ER -