2022/12/01 by Satoshi Masaki, Masaki, Satoshi · 1 citation
Mathematics · Physics and Astronomy · #35J50 (Primary) #35Q55 #37K40 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Nonlinear Photonic Systems
paper · pdf · doi:10.48550/arxiv.2212.00754
openalex publication_date 2022/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we study the standing-wave solutions to a class of systems of nonlinear Schrödinger equations. Our target is all the standard forms of the NLS systems, with two unknowns, that have a common linear part and cubic gauge-invariant nonlinearities and that yield a Hamiltonian with a coercive kinetic-energy part. We give a necessary and sufficient condition on the existence of the ground state. Further, we give a characterization of the shape of the ground state. It will turn out that the ground states are scalar-type, i.e., multiples of a constant vector and a scalar function. We further give a sufficient condition on the existence of excited states of the same form. The stability and the instability of the ground states are also studied. To this end, we introduce an abstract treatment on the study of scalar-type standing-wave solution that applies to a wide class of NLS systems with homogeneous energy-subcritical nonlinearity. By the argument, some previous results are reproduced.