### Abstract

It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.

Original language | English (US) |
---|---|

Pages (from-to) | 679-702 |

Number of pages | 24 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 60 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2000 |

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### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

**Stability of pole solutions for planar propagating flames : I. Exact eigenvalues and eigenfunctions.** / Vaynblat, Dimitri; Matalon, Moshe.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Mathematics*, vol. 60, no. 2, pp. 679-702. https://doi.org/10.1137/S0036139998346439

}

TY - JOUR

T1 - Stability of pole solutions for planar propagating flames

T2 - I. Exact eigenvalues and eigenfunctions

AU - Vaynblat, Dimitri

AU - Matalon, Moshe

PY - 2000/1/1

Y1 - 2000/1/1

N2 - It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.

AB - It is well known that the nonlinear PDE describing the dynamics of a hydrodynamically unstable planar flame front admits exact pole solutions as equilibrium states. Such a solution corresponds to a steadily propagating cusp-like structure commonly observed in experiments. In this work we investigate the linear stability of these equilibrium states - the steady coalescent pole solutions. In previous similar studies, either a truncated linear system was numerically solved for the eigenvalues or the initial value problem for the linearized PDE was numerically integrated in order to examine the evolution of initially small disturbances in time. In contrast, our results are based on the exact analytical expressions for the eigenvalues and corresponding eigenfunctions. In this paper we derive the expressions for the eigenvalues and eigenfunctions. Their properties and the implication on the stability of pole solutions is discussed in a paper which will appear later.

UR - http://www.scopus.com/inward/record.url?scp=0034565121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034565121&partnerID=8YFLogxK

U2 - 10.1137/S0036139998346439

DO - 10.1137/S0036139998346439

M3 - Article

AN - SCOPUS:0034565121

VL - 60

SP - 679

EP - 702

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 2

ER -