On Uniqueness of Multi-bubble Blow-Up Solutions and Multi-solitons to L^2 -Critical Nonlinear Schrödinger Equations

We are concerned with the focusing L^2 -critical nonlinear Schrödinger equations in ℝ^d for dimensions 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 the low rate (T-t)^0+ , as t→ T , 1≦ K<∞ . 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→∞ . The uniqueness class is further enlarged to contain the multi-solitons with even lower convergence rate (1/t)^1/2+ in the pseudo-conformal space. Our proof is mainly based on several upgrading procedures of the convergence of remainder in the geometrical decomposition, in which the key ingredients are several monotone functionals constructed particularly in the multi-bubble case.