On uniqueness of multi-bubble blow-up solutions and multi-solitons to $L^2$-critical nonlinear Schrödinger equations

Abstract

We are concerned with the focusing $L^2$-critical nonlinear Schrödinger 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 $t\to T$, $1\le K<\infty$. 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 $(1/t{)}^{\frac{1}{2}+}$ 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.