We are concerned with the focusing \(L^2\)-critical nonlinear Schr？dinger equations in \({{\mathbb {R}}}^d\) 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 \(t\rightarrow T\), \(1\leqq K<{\infty }\). 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 \(t\rightarrow {\infty }\). 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. 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. 

　　Publication: 

　　Archive for Rational Mechanics and Analysis volume 247, Article number: 4 (2023)  

　　DOI: 10.1007/s00205-022-01832-x 

　　Author: 

　　Daomin Cao 

　　Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China 

　　University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China 

　　Email address: dmcao@amt.ac.cn 

　　Yiming Su 

　　Department of Mathematics, Zhejiang University of Technology, Zhejiang, 310014, People’s Republic of China 

　　Deng Zhang 

　　School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China