Stability of the standing waves of the concentrated NLSE in dimension two

An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first focus on the NLSE, both in the subcritical and the critical case, and then on the NLDE, highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit, proving the convergence of bound states.

二维和三维中具有点相互作用的NLSE的散射失效

Failure of scattering for the NLSE with a point interaction in dimension two and three

本文研究了二维和三维中具有幂次非线性和点相互作用的NLS方程，证明低幂次非线性下向自由动力学或驻波的散射失效，并将已有的一维结果扩展至更高维度。

具有点相互作用的NLSE的变号多峰驻波

Sign-changing multi-peak standing waves of the NLSE with a point interaction

本文通过 Lyapunov--Schmidt reduction 方法，证明了在特定参数条件下具有点相互作用的 NLSE 存在变号多峰驻波解。

Asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three. II

We investigate the asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in space dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point $x=0$ obtained considering a contact (or $δ$) interaction with strength $α$, and letting the strength $α$ depend on the wavefunction in a prescribed way. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $Φ_ω$, and the linearization around them admits two imaginary eigenvalues which in principle could corre…

On the standing waves of the Schroedinger equation with concentrated nonlinearity

We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobo…

Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrödinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength $α$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_α$, where the strength $α$ depends on the wavefunction: $i\dot u= H_αu$, $α=α(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $α$ depend on $u$ according to th…