A complete classification of ground-states for a coupled nonlinear Schrödinger system

Abstract

In this paper, we establish the existence of nontrivial ground-state solutions for a coupled nonlinear Schrödinger system (Equation Presented) where n = 1, 2, 3, m ≥ 2 and bij are positive constants satisfying bij = bji. By nontrivial we mean a solution that has all components non-zero. Due to possible systems collapsing it is important to classify ground state solutions. For m = 3, we get a complete picture that describes whether nontrivial groundstate solutions exist or not for all possible cases according to some algebraic conditions of the matrix B = (bij). In particular, there is a nontrivial groundstate solution provided that all coupling constants bij, i ≠ j are sufficiently large as opposed to cases in which any ground-state solution has at least a zero component when bij, i ≠ j are all sufficiently small. Moreover, we prove that any ground-state solution is synchronized when matrix B = (bij) is positive semi-definite.