Linear‐Coupling Effects in the 1D Gross–Pitaevskii System: Exact Solutions and Eigenvalue Problems

ABSTRACT We examine a linear coupling in the one‐dimensional Gross–Pitaevskii system with Neumann boundary conditions. Exact solutions for this system are derived, and the dependence of these solutions on the linear coupling is thoroughly examined. In addition, we analyze the corresponding linearized eigenvalue problem, classifying both the eigenvalues and eigenfunctions. We also explore how the linear coupling influences the behavior of the eigenvalues.