Segregated and synchronized vector solutions to linearly coupled systems of Schrödinger equations

In this paper, we study the following linearly coupled system [Formula: see text], where ε > 0 is a small parameter, P (x) are positive potentials, and λ = λ > 0 (i ≠ j) are coupling constants for i, j = 1, …, N. We investigate the effect of potentials to the structure of the solutions. More precisely, we construct multi-spikes solutions concentrating near the local maximum point [Formula: see text] of P (x). When [Formula: see text], [Formula: see text], the components have spikes clustering at the same point as ε → 0⁺. When [Formula: see text], the components have spikes clustering at the different points as ε → 0⁺.