### Abstract

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg - de Vries equation ut + a(t) (u^{3})_{x} + 1/3U_{xxx} = O,(t,r) ϵR x R, with initial data u(0,z) = q1(z),z ϵ R.. We assume that the coefficient a(t) ϵ C^{1}(R) is real, bounded and slowly varying function, such that /a'(t)≤C(l + /t/)-^{7/6}. We suppose that the iriitial dah are real ~ valued and small enough, belonging to the weighted Sobolev space. We prove the time decay estimates of the solutims.We also fiud the asympt,otics for large time of the solution in the neighborhood of the self-similar solution.

Original language | English |
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Title of host publication | International Seminar |

Subtitle of host publication | Day on Diffraction - Proceedings |

Editors | V.E. Grikuro, V.M. Babic, I.V. Androno, V.S. Buldyrev, A.P. Kiselev |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 146-156 |

Number of pages | 11 |

ISBN (Electronic) | 5799701569, 9785799701567 |

DOIs | |

Publication status | Published - 1999 Jan 1 |

Externally published | Yes |

Event | International Seminar: Day on Diffraction, IS-DoD 1999 - St. Petersburg, Russian Federation Duration: 1999 Jun 1 → 1999 Jun 3 |

### Publication series

Name | International Seminar: Day on Diffraction - Proceedings |
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### Conference

Conference | International Seminar: Day on Diffraction, IS-DoD 1999 |
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Country | Russian Federation |

City | St. Petersburg |

Period | 99/6/1 → 99/6/3 |

### ASJC Scopus subject areas

- Radiation
- Acoustics and Ultrasonics
- Atomic and Molecular Physics, and Optics

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## Cite this

Hayashi, N., & Naumkin, P. (1999). On the modified Korteweg - de Vries equation. In V. E. Grikuro, V. M. Babic, I. V. Androno, V. S. Buldyrev, & A. P. Kiselev (Eds.),

*International Seminar: Day on Diffraction - Proceedings*(pp. 146-156). [816195] (International Seminar: Day on Diffraction - Proceedings). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/DD.1999.816195