DOI: 10.1515/dema-2025-0253 ISSN: 2391-4661
Classification of solutions to a coupled nonlinear Schrödinger system
Luyan Zhou Abstract
We are concerned with the classification of the general solutions to the following coupled nonlinear Schrödinger system
−
u
″
+
λ
u
=
μ
u
3
+
β
v
2
u
in
R
,
−
v
″
+
λ
v
=
μ
v
3
+
β
u
2
v
in
R
.
$$\left\{\begin{aligned}\hfill & -{u}^{{\prime\prime}}+\lambda u=\mu {u}^{3}+\beta {v}^{2}u\quad \hfill & \hfill \text{in} \mathbb{R},\\ \hfill & -{v}^{{\prime\prime}}+\lambda v=\mu {v}^{3}+\beta {u}^{2}v\quad \hfill & \hfill \text{in} \mathbb{R}.\end{aligned}\right.$$
The system arises in nonlinear optics and in the Hartree–Fock theory for a double condensate. We concentrate our attention on the case
β
= 3
μ
> 0. Firstly, we prove that the system admits only finitely many nontrivial solutions up to translations, which is significantly different from the high-dimensional cases where there exist infinitely many sign-changing solutions. In particular, the specific formulas for constant-sign solutions and semi-nodal solutions (i.e., one component changes sign and the other one does not change sign) are explicitly provided. Secondly, we prove that the system admits exactly two distinct classes of normalized solutions satisfying the total mass constraint
‖
u
‖
L
2
(
R
)
2
+
‖
v
‖
L
2
(
R
)
2
=
m
${\Vert}u{{\Vert}}_{{L}^{2}\left(\mathbb{R}\right)}^{2}+{\Vert}v{{\Vert}}_{{L}^{2}\left(\mathbb{R}\right)}^{2}=m$
. This result stands in sharp contrast to the uniqueness of normalized solutions when
β
=
μ
, as proved by [Frank, Gontier, and Lewin, Comm. Math. Phys., 2021]. Furthermore, we explicitly derive the value of the Lagrange multiplier
λ
. Finally, we establish the uniqueness of least energy normalized solutions for the system satisfying the total mass constraint.