In dimensions 2 and 3, every positive H¹ solution of the equal-frequency cubic Schrödinger system with weak coupling is a simultaneous translate of the unique synchronized ground state.
Ikoma,Uniqueness of positive solutions for a nonlinear elliptic system, Nonlinear Differential Equations Appl.16(2009), 555–567
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On Sirakov's equal-frequency uniqueness conjecture
In dimensions 2 and 3, every positive H¹ solution of the equal-frequency cubic Schrödinger system with weak coupling is a simultaneous translate of the unique synchronized ground state.