## Abstract

The complex dynamics that arise in certain nonlinear partial differential equations in time and in one space dimension are studied. In the general case considered, the equation admits a solitary wave in the form of a pulse tailing off exponentially, fore and aft, with possibly oscillatory character. Complicated solutions are described by a superposition of many such solitary structures in interaction. The description is asymptotic in terms of a parameter that becomes exponentially small as the ratio of typical pulse separation to pulse width becomes large. The outcome is a set of dynamical equations for the motion of the individual pulses with nearest neighbor interactions. This system of ordinary differential equations (ODEs) admits a wide range of patterns, both regular and chaotic. The stability theory of such patterns is sketched and the continuum limit of the lattice-dynamical equations of the pulses is given.

Original language | English |
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Pages (from-to) | 490-503 |

Number of pages | 14 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1990 |

Externally published | Yes |

## ASJC Scopus subject areas

- Applied Mathematics