AI 中文总结
本文在N=2或3、弱耦合范围下，证明了等频两分量三次薛定谔系统的正解在同时平移下唯一，解决了西拉科夫的等频唯一性猜想，通过构造加权泛函等方法排除非同步径向解。
AI 中文摘要
设N∈{2,3}，0<μ₁≤μ₂，且0<β<μ₁。我们证明，等频两分量三次薛定谔系统：-Δu+u=μ₁u³+βuv²，-Δv+v=μ₂v³+βu²v（在ℝᴺ中），在H¹(ℝᴺ)×H¹(ℝᴺ)中恰有一个正解，该解在同时平移下等价。更准确地说，每个正解都是由ℝᴺ中-Δw+w=w³的唯一正径向解构造的同步态的同时平移。这在弱耦合范围内解决了西拉科夫的等频唯一性猜想。证明中的主要困难是排除那些归一化分量比值非恒定的径向解。归一化后，两个分量满足具有共同势的标量方程。我们为该系统构造了带修正项的加权波霍扎耶夫（Pohozaev）泛函，并证明修正后的泛函和相关加权泛函均严格为正。将这些符号性质与径向通量恒等式及与分量比值相关的辅助商结合，可强制实现同步。
英文摘要
Let $N\in\{2,3\}$, $0<μ_1\leqμ_2$, and $0<β<μ_1$. We prove that the equal-frequency two-component cubic Schrödinger system \[ -Δu+u=μ_1u^3+βuv^2, \qquad -Δv+v=μ_2v^3+βu^2v \quad\text{in }\mathbb{R}^N \] has exactly one positive solution in $H^1(\mathbb{R}^N)\times H^1(\mathbb{R}^N)$ modulo simultaneous translations. More precisely, every positive solution is a simultaneous translate of the synchronized state constructed from the unique positive radial solution of $-Δw+w=w^3$ in $\mathbb{R}^N$. This settles Sirakov's equal-frequency uniqueness conjecture throughout the weak-coupling range. The main difficulty in the proof is to exclude radial solutions for which the ratio of the normalized components is nonconstant. After normalization, the two components satisfy scalar equations with a common potential. We construct a weighted Pohozaev functional for the system together with a correction term and prove that both the corrected functional and the associated weighted functional are strictly positive. Combining these sign properties with a radial flux identity and an auxiliary quotient associated with the component ratio forces synchronization.
Comments22 pages
