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On the uniqueness of multi-breathers of the modified Korteweg-de Vries equation

Abstract : We consider the modified Korteweg-de Vries equation (mKdV) and prove that given any sum P of solitons and breathers of (mKdV) (with distinct velocities), there exists a solution p of (mKdV) such that p(t) − P(t) → 0 when t → +∞, which we call multi-breather. In order to do this, we work at the H^2 level (even if usually solitons are considered at the H^1 level). We will show that this convergence takes place in any H^s space and that this convergence is exponentially fast in time. We also show that the constructed multi-breather is unique in two cases: in the class of solutions which converge to the profile P faster than the inverse of a polynomial of a large enough degree in time (we will call this a super polynomial convergence), or (without hypothesis on the convergence rate), when all the velocities are positive.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03337323
Contributor : Alexander Semenov <>
Submitted on : Thursday, September 9, 2021 - 6:49:02 PM
Last modification on : Wednesday, September 15, 2021 - 10:10:01 AM

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multi-breathers-mKdV-14.pdf
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  • HAL Id : hal-03337323, version 2
  • ARXIV : 2109.03584

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Alexander Semenov. On the uniqueness of multi-breathers of the modified Korteweg-de Vries equation. 2021. ⟨hal-03337323v2⟩

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