Classification and Nondegeneracy of Cubic Nonlinear Schrödinger System in $\mathbb{R}$

We study the following one-dimensional cubic nonlinear Schrödinger system: $$u_i^{\mathrm{\prime }\mathrm{\prime }}+2\left(\sum _{k=1}^N{u}_k^2\right)u_i=-{\mu }_i{u}_{i}in\mathbb{R},i=1,2,\cdots ,N,$$ where ${\mu }_1\le {\mu }_2\le \cdots \le {\mu }_N<0$ and $N\ge 2$. In this paper, we mainly focus on the case $N=3$ and prove the following results: (i). The solutions of the system can be completely classified; (ii). Depending on the explicit values of ${\mu }_1\le {\mu }_2\le {\mu }_3<0$, there exist two different classes of normalized solutions $u=(u_1,u_2,u_3)$ satisfying ${\int }_R{u}_i^{2}dx=1$ for all $i=1,2,3$, which are completely different from the case $N=2$; (iii). The linearized operator at any nontrivial solution of the system is non-degenerate. The conjectures on the explicit classification and nondegeneracy of solutions for the system are also given for the case $N>3$. These address the questions of [R. Frank, D. Gontier and M. Lewin, CMP, 2021], where the complete classification and uniqueness results for the system were already proved for the case $N=2$.