Coupled nonlinear Schrödinger equations with point interaction: existence and asymptotic behaviour
Yuki Osada, Alessio Pomponio
TL;DR
This work analyzes a weakly coupled two‑component nonlinear Schrödinger system in $ bb^2$ with a point interaction represented by the self‑adjoint extension $- abla_lpha$. Through a variational framework on the energy space $oldH_lpha$, it proves the existence of ground states and classifies them as scalar or vector and as regular or singular depending on the coupling parameter $eta$ and comparison energies $d(omega)$, $d^0( ilde{omega})$, $c_eta^0$, and related levels. The paper shows a regime where all minimizers are scalar and another where all minimizers become vector and singular, with a sharp threshold $eta^*$ delineating the transition. In the asymptotic regime $eta o ty$, it derives $c_eta= ilde{c}_ ty/eta+o(1/eta)$ and proves convergence, after suitable rescaling, to a limit problem whose ground state $(w_0,z_0)$ captures the asymptotic profile of the original system, including a singular component via $w_0=psi_{mbda,0}+r_0G_mbda$.
Abstract
In this paper we deal with the following weakly coupled nonlinear Schrödinger system \begin{align*} \begin{cases} - Δ_αu + ωu = |u|^2 u + βu |v|^2&\quad \mathrm{in}\ \mathbb{R}^2,\\ - Δv + \tildeω v = |v|^2 v + β|u|^2 v&\quad \mathrm{in}\ \mathbb{R}^2, \end{cases} %\tag{$\mathcal{P}_β$} \end{align*} where $-Δ_α$ denotes the Laplacian operator with a point interaction, $ω$ greater then a suitable positive constant, $\tildeω>0$, and $β\ge 0$. For any $β\ge 0$ this system admits the existence of a ground state solution which can have only one nontrivial component or two nontrivial components and which could be regular or singular. We analyse this phenomenon showing how this depends strongly on the parameters. Moreover we study the asymptotic behaviour of ground state solutions whenever $β\to \infty$.