## Abstract

We consider the large time asymptotic behavior of solutions to the Cauchy problem for the modified Korteweg-de Vries equation u_{t} + a(t)(u^{3})_{x} + 1/3u_{xxx} = 0, (t, x) ε R × R, with initial data u(0, x) = u_{0}(x), x 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〈t〉^{-7/6}, where 〈t〉 = (1 + t^{2})^{1/2}. We suppose that the initial data are real-valued and small enough, belonging to the weighted Sobolev space H^{1,1} = {φ ε L^{2}; || √1+x^{2} √1-∂_{x}^{2}φ|| < ∞}. In comparison with the previous paper (Internat. Res. Notices 8 (1999), 395-418), here we exclude the condition that the integral of the initial data u_{0} is zero. We prove the time decay estimates ^{3}√t^{2} ^{3}√〈t〉||u(t)u_{x}(t)||_{∞} ≤ Cε and 〈t〉^{1/3-1/3β}||u(t)||β ≤ Cε for all t ε R, where 4 < β ≤ ∞. We also find the asymptotics for large time of the solution in the neighborhood of the self-similar solution.

Original language | English |
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Pages (from-to) | 197-227 |

Number of pages | 31 |

Journal | Mathematical Physics, Analysis and Geometry |

Volume | 4 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2001 |

## Keywords

- Large time asymptotics
- Modified korteweg-de vries equation

## ASJC Scopus subject areas

- Mathematical Physics
- Geometry and Topology