### Abstract

Chapter 1 begins with some examples of partial differential equations in science and engineering and their linearization and dispersion equations. The concepts of well-posedness, regularity, and solution operator for systems of partial differential equations (PDE's) are discussed. Instabilities can arise from both numerical methods and from real physical instabilities. Some physical instabilities are described, including: (a) the distinction between convective and absolute instabilities, (b) the Rayleigh-Taylor and Kelvin-Helmholtz instabilities in fluids, (c) wave breaking and gradient catastrophe in gas dynamics and in conservation laws, (d) modulational or Benjamin Feir instabilities and nonlinear Schrödinger related equations, (e) three-wave resonant interactions and explosive instabilities associated with negative energy waves. Basic wave concepts are described (e.g. wave-number surfaces, group velocity, wave action, wave diffraction, and wave energy equations). A project from semiconductor transport modeling is described.

Original language | English (US) |
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Title of host publication | Mathematics in Science and Engineering |

Publisher | Elsevier |

Pages | 1-57 |

Number of pages | 57 |

Edition | C |

DOIs | |

State | Published - Jan 1 2010 |

Externally published | Yes |

### Publication series

Name | Mathematics in Science and Engineering |
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Number | C |

Volume | 213 |

ISSN (Print) | 0076-5392 |

### Keywords

- Absolute and convective instabilities
- Advection
- Airy
- Diffraction
- Dispersion relation
- Group velocity
- Heat
- Linear and nonlinear resonant wave interaction
- Modulational instability
- Partial differential equations
- Rayleigh-Taylor and Kevin-Helmholtz instabilities
- Regularity
- Schrödinger
- Shocks and traveling waves
- Solution operator
- Telegrapher equations
- Wave
- Wave breaking
- Wave packets
- Well-posedness

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

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## Cite this

*Mathematics in Science and Engineering*(C ed., pp. 1-57). (Mathematics in Science and Engineering; Vol. 213, No. C). Elsevier. https://doi.org/10.1016/S0076-5392(10)21306-1