Pattern formation via mixed interactions for coupled Schrödinger equations under Neumann boundary condition
This interesting paper is devoted to the asymptotic behavior of positive least energy vector solutions to nonlinear Schrödinger systems with mixed couplings. These systems arise from models in Bose-Einstein condensates and nonlinear optics. The basic problem investigated here is \[ -\triangle u_i+\lambda_iu_i=\mu_iu_{i}^{3}+ \sum\limits_{j\neq i}^{3}\beta_{ij}u_iu_{j}^{2} \;\;\text{in} \;\;\Omega \] under homogeneous Neumann boundary condition \(\partial u_i/\partial\nu =0\) on \(\partial\Omega \) (\(i=1,2,3\)) for the case of mixed couplings. Here \(\Omega \) is a bounded domain with a smooth boundary in \(\mathbb{R}^n\) for \(n\leq 3\), \(\nu \) is the outward unit normal vector on \(\partial \Omega \), \(\lambda_i > 0\), \(\mu_i > 0\) (\(i = 1, 2, 3\)), and \(\beta_{ij} = \beta_{ji} \in \mathbb{R}\). The quantities \(\mu_j\) and \(\beta_{ij}\) are the intraspecies and interspecies scattering lengths, respectively. The sign of \(\beta_{ij}\) determines whether the interactions of states are repulsive or attractive. Here the case \(\beta_{12}>0\), \(\beta_{13}<0\), \(\beta_{23}<0\) is considered. The asymptotic behavior of the system, when both the attractive and the repulsive couplings tend asymptotically to \(\infty \), is investigated. The authors identify the limiting equations and establish energy expansion in this process. A stronger convergence to a limiting system is shown. The existence of a positive least energy vector solution of the considered system satisfying some energy inequality is proved. It turns out that there exists a least energy solution in the class \(H^1(\Omega )\) which attains a solution of a given system of two nonlinear equations, and the radially symmetric positive least energy solution belongs to \(H^1(\mathbb{R}^n)\times H^1(\mathbb{R}^n)\). The least energy solutions exhibit component-wise pattern formations. In particular, co-existence of partial synchronization and segregation is observed.
scientific article; zbMATH DE number 6706633
scientific article