## Abstract

In this paper, we consider the classical Riemann problem for a generalized Burgers equation,

u

with a spatially dependent, nonlinear sound speed, h

u

_{t}+ h_{α}(x) uu_{x }= u_{xx},with a spatially dependent, nonlinear sound speed, h

_{α}(x) ≡ (1 + x^{2})^{-α}with α > 0, which decays algebraically with increasing distance from a fixed spatial origin. When α = 0, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as t → ∞, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at α = ½.Original language | English |
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Pages (from-to) | 963-995 |

Number of pages | 33 |

Journal | Studies in Applied Mathematics |

Volume | 150 |

Issue number | 4 |

Early online date | 17 Jan 2023 |

DOIs | |

Publication status | Published - 27 Apr 2023 |

## Keywords

- generalized Burgers equation
- large-time structure
- Riemann problem
- spatially decaying sound speed

## ASJC Scopus subject areas

- Applied Mathematics