Complete Classification and Nondegeneracy of $N$-Component Cubic Nonlinear Schrödinger System in $\mathbb{R}$

Abstract

We study the one-dimensional cubic nonlinear Schrödinger system \[ u_i''+2\left(\sum_{k=1}^N u_k^2\right)u_i=-μ_i u_i \quad \mbox{in } \ \mathbb R,\ \ i=1,2,\cdots,N, \] where $u=(u_1,\cdots ,u_N)\in (H^1(\mathbb{R}){)}^N$, ${\mu }_1\le {\mu }_2\le \cdots \le {\mu }_N<0$, and $N\ge 2$ is arbitrary. In this paper, we prove the following results for any $N\ge 2$: (i). All nontrivial solutions of the system can be completely classified; (ii). The linearized operator at any nontrivial solution of the system is non-degenerate; (iii). For all $i=1,2,\cdots ,N$, the exact $L^2$-mass identity of $u_i$ is derived in terms of $2\sqrt{|{\mu }_i|}$, which yields a complete characterization of normalized solutions satisfying ${\int }_{\mathbb{R}}{u}_i^{2}dx=1$. These settle some conjectures of [R. Frank, D. Gontier and M. Lewin, CMP, 2021] and [Y. Guo, Y. Luo and J. Wei, APDE, 2026], where the system was addressed specially for $N=2$ and $N=3$, respectively.

35 pages; any comment is welcome