Remarks on some systems of nonlinear Schrödinger equations
The author studies systems of nonautonomous nonlinear Schrödinger equations which are linearly coupled. The first problem studied is \[ \begin{aligned} &-u_1'' + u_1 = (1 + \epsilon b_1(x))u_1^3 + a u_2,\\ &-u_2'' + u_2 = (1 + \epsilon b_2(x))u_2^3 + a u_1. \end{aligned} \] Using perturbation methods, it is shown that, for \(\epsilon\) sufficiently small, this problem has a solution near the manifold \[ \{(u_{1-a}(x+y),u_{1-a}(x+y)) : y \in \mathbb{R}\}, \] where \((u_{1-a}(x),u_{1-a}(x))\) is solution of the unperturbed problem(\(\epsilon=0\)). The second problem studied is \[ \begin{aligned} &-\epsilon^2 u_1'' + u_1 + V_1(x)u_1 = u_1^3 + a u_2,\\ &-\epsilon^2u_2'' + u_2 + V_2(x)u_2 = u_2^3 + a u_1. \end{aligned} \] Using perturbation methods, it is shown that this problem has solutions concentrating, as \(\epsilon\to 0\), at nondegenarate stationary points of \(V_1+V_2\).
scientific article; zbMATH DE number 5508654