On Sirakov's equal-frequency uniqueness conjecture

Let $N\in \{2,3\}$, $0<{\mu }_1\le {\mu }_2$, and $0<\beta <{\mu }_1$. We prove that the equal-frequency two-component cubic Schr"odinger system $$-\mathrm{\Delta }u+u={\mu }_1{u}^3+\beta u{v}^2,-\mathrm{\Delta }v+v={\mu }_2{v}^3+\beta u^{2}v\text{in }{\mathbb{R}}^N$$ 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 $-\mathrm{\Delta }w+w=w^3$ 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.