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arXiv 2109.03584math.AP
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修正的Korteweg-de Vries方程多呼吸子的唯一性

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

Alexander Semenov

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摘要：

本文考虑修正的Korteweg-de Vries方程（mKdV），证明了：给定（mKdV）的孤子和呼吸子的任意和P（速度不同），存在（mKdV）的解p，使得当t $\rightarrow$ + $\infty$时，p（t）- P（t）$\rightarrow$ 0，我们称之为多呼吸子.为了做到这一点，我们在H^2水平上工作（即使通常在H^1水平上考虑孤子）。我们将证明这种收敛发生在任何H^s空间中，并且这种收敛在时间上是指数快速的。我们还表明，所构造的多呼吸是唯一的，在两种情况下：在类的解决方案收敛到配置文件P比逆的多项式的足够大的程度在时间（我们将称之为一个超多项式收敛），或（没有假设的收敛速度），当所有的速度是积极的。

英文摘要：

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) $\rightarrow$ 0 when t $\rightarrow$ +$\infty$, 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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