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

Abstract

We consider the modified Korteweg–de Vries equation, and prove that given any sum $P$ of solitons and breathers (with distinct velocities), there exists a solution $p$ such that $p(t)-P(t)\to 0$ when $t\to +\infty$, 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 when all the velocities are positive (without any hypothesis on the convergence rate).