On uniqueness of multi-bubble blow-up solutions and multi-solitons to $L^2$-critical nonlinear Schr\"odinger equations

TL;DR

This paper proves the uniqueness of multi-bubble blow-up solutions and multi-solitons for the focusing $L^2$-critical nonlinear Schrödinger equations in low dimensions, with detailed convergence rates and enlarged classes.

Contribution

It establishes the uniqueness of multi-bubble blow-up solutions and multi-solitons in broader classes with various convergence rates for the $L^2$-critical NLS.

Findings

Uniqueness of multi-bubble blow-up solutions with low convergence rate.

Uniqueness of multi-solitons with convergence rate $(1/t)^{2+}$.

Extended uniqueness class including solutions with even lower convergence rate.

Abstract

We are concerned with the focusing $L^2$-critical nonlinear Schr\"odinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $t\to T$, $1\leq K<\infty$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of $K$ solitary waves with convergence rate $(1/t)^{2+}$, as $t\to \infty$. The uniqueness class is further enlarged to contain the multi-solitons with even lower convergence rate $(1/t)^{\frac 12+}$ in the pseudo-conformal space. The proof is mainly based on the pseudo-conformal invariance and the monotonicity properties of several functionals adapted to the multi-bubble case, the latter is crucial towards the upgradation of the convergence to the fast exponential decay rate.