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

Abstract

A bstract . 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.