### Abstract

We study the final state problem for the Korteweg-de Vries type equations: ut -1ρ ∫ x ∫ρ -1 ux =λ u2 ux, (t,x) R+ ×R,u(t)- FS (t) L2 →0 as t→∞, where λR, the function FS (t) we call a final state, defined by the final data u+. We show that there does not exist a nontrivial solution of this equation in the case of FS (t)=U(t) u+, where U(t) is the free evolution group of this equation. We construct the modified wave operator for the Korteweg-de Vries type equations under the conditions that the final data u+ arc real-valued functions and the Fourier transform u + () vanishes at the origin.

Original language | English |
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Article number | 123501 |

Journal | Journal of Mathematical Physics |

Volume | 47 |

Issue number | 12 |

DOIs | |

Publication status | Published - 2006 Dec 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Hayashi, N., & Naumkin, P. I. (2006). Final state problem for Korteweg-de Vries type equations.

*Journal of Mathematical Physics*,*47*(12), [123501]. https://doi.org/10.1063/1.2374883