### Abstract

A classification of special 'nonlinear' front solutions for certain one-time and one-space reaction-diffusion equations is presented, using the method of Weiss, Tabor and Carnevale (WTC). These results are related to known stability criteria, in particular the steepness criterion of van Saarloos (1989). The WTC method is shown to be equivalent to a special first-order reduction, and both of these methods are shown to work for reaction-diffusion equations with special nonlinearities. Of particular interest is the fact that the special first-order reduction is shown to give separatrices in appropriate phase spaces. An example of reaction-diffusion equation is presented without these special nonlinearities. While this equation is shown to have a special 'nonlinear' connection and resulting stability properties, it is intractable for either a singular manifold expansion of a first-order reduction. A Lie symmetry analysis is carried out, and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques. However, the 'rescaling ansatz' of Cariello and Tabor (1991) suggests that some symmetries are present.

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

Article number | 025 |

Pages (from-to) | 3773-3796 |

Number of pages | 24 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 25 |

Issue number | 13 |

DOIs | |

State | Published - 1992 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

### Cite this

*Journal of Physics A: Mathematical and General*,

*25*(13), 3773-3796. [025]. https://doi.org/10.1088/0305-4470/25/13/025

**Non-generic connections corresponding to front solutions.** / Powell, J.; Tabor, Michael.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 25, no. 13, 025, pp. 3773-3796. https://doi.org/10.1088/0305-4470/25/13/025

}

TY - JOUR

T1 - Non-generic connections corresponding to front solutions

AU - Powell, J.

AU - Tabor, Michael

PY - 1992

Y1 - 1992

N2 - A classification of special 'nonlinear' front solutions for certain one-time and one-space reaction-diffusion equations is presented, using the method of Weiss, Tabor and Carnevale (WTC). These results are related to known stability criteria, in particular the steepness criterion of van Saarloos (1989). The WTC method is shown to be equivalent to a special first-order reduction, and both of these methods are shown to work for reaction-diffusion equations with special nonlinearities. Of particular interest is the fact that the special first-order reduction is shown to give separatrices in appropriate phase spaces. An example of reaction-diffusion equation is presented without these special nonlinearities. While this equation is shown to have a special 'nonlinear' connection and resulting stability properties, it is intractable for either a singular manifold expansion of a first-order reduction. A Lie symmetry analysis is carried out, and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques. However, the 'rescaling ansatz' of Cariello and Tabor (1991) suggests that some symmetries are present.

AB - A classification of special 'nonlinear' front solutions for certain one-time and one-space reaction-diffusion equations is presented, using the method of Weiss, Tabor and Carnevale (WTC). These results are related to known stability criteria, in particular the steepness criterion of van Saarloos (1989). The WTC method is shown to be equivalent to a special first-order reduction, and both of these methods are shown to work for reaction-diffusion equations with special nonlinearities. Of particular interest is the fact that the special first-order reduction is shown to give separatrices in appropriate phase spaces. An example of reaction-diffusion equation is presented without these special nonlinearities. While this equation is shown to have a special 'nonlinear' connection and resulting stability properties, it is intractable for either a singular manifold expansion of a first-order reduction. A Lie symmetry analysis is carried out, and it is shown that equations with continuous groups other than translational invariance are only a subclass of equations which are amenable to the special solution techniques. However, the 'rescaling ansatz' of Cariello and Tabor (1991) suggests that some symmetries are present.

UR - http://www.scopus.com/inward/record.url?scp=0042318807&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042318807&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/25/13/025

DO - 10.1088/0305-4470/25/13/025

M3 - Article

AN - SCOPUS:0042318807

VL - 25

SP - 3773

EP - 3796

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 13

M1 - 025

ER -