## Abstract

The Lie point symmetries of the (1+2)D Zabolotskaya-Khokholov (ZK) equation for a gas with nonzero viscosity at rest are calculated. They form an infinite-dimensional Lie algebra which can be written as the direct sum of a 2D Lie algebra and an infinite-dimensional Lie algebra. The symmetry structure of the ZK equation is used to construct its physically relevant solutions from those of the (1 + 1)D Burgers equation. In particular, travelling wave solutions (one-solitons) as well as the multi-soliton solutions of the ZK equation are obtained. Also, we obtain the N-waves, the periodic waves and the solution to some Cauchy problems for the ZK equation. We emphasize that our method can be utilized to generate a wider class of solutions of the ZK equation.

Original language | English |
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Pages (from-to) | 219-232 |

Number of pages | 14 |

Journal | IMA Journal of Applied Mathematics |

Volume | 69 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jun 2004 |

Externally published | Yes |

## Keywords

- Lie group method
- Travelling wave solutions
- Zabolotskaya-Khokholov (ZK) equation

## ASJC Scopus subject areas

- Applied Mathematics