Classification and Nondegeneracy of Cubic Nonlinear Schr\"{o}dinger System in $\mathbb{R}$

TL;DR

This paper classifies solutions of a one-dimensional cubic nonlinear Schrödinger system for three components, revealing different solution classes based on parameters and establishing nondegeneracy of solutions, extending prior results from two components.

Contribution

It provides a complete classification of solutions for the three-component case and proves their nondegeneracy, addressing open questions for N=3 and conjecturing about higher N.

Findings

Solutions can be completely classified for N=3.

Existence of two distinct classes of normalized solutions depending on parameters.

The linearized operator at any nontrivial solution is non-degenerate.

Abstract

We study the following one-dimensional cubic nonlinear Schr\"{o}dinger system: \[ u_i''+2\Big(\sum_{k=1}^Nu_k^2\Big)u_i=-\mu_iu_i \ \,\ \mbox{in}\, \ \mathbb{R} , \ \ i=1, 2, \cdots, N, \] where $\mu_1\leq\mu_2\leq\cdots\leq\mu_N<0$ 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 $\mu_1\leq\mu_2\leq\mu_3<0$, there exist two different classes of normalized solutions $u=(u_1, u_2, u_3)$ satisfying $\int _{R}u_i^2dx=1$ 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 $N=2$.