Global dynamics below the standing waves for the focusing semilinear Schrödinger equation with a repulsive Dirac delta potential

Threshold scattering for the focusing NLS with a repulsive Dirac delta potential

“…The effect of the δ(x)-potential on the dynamics of the nonlinear Schrödinger equation has been studied intensively in later years. The Cauchy problem, existence of ground states and their stability/instability, long time dynamics (scattering and global existence, blow-up) to (1.1) with data below the ground state threshold, etc., have been studied in recent years; see for example [2,3,11,12,14,16] for more details.…”

“…The behavior of solutions below the ground state level are now well understood for equation (1.1). Indeed, in [14], M. Ikeda and T. Inui found a necessary and sufficient condition on the data below the ground state to determine the global behavior (i.e., scattering/blow-up) of the solution. The dichotomy in behaviour of solutions below the ground state is dependent upon the sign of the functional…”

“…where a = 2(p−1)(p+1) p+3 and r = p + 1. Note also that if u 0 = 0 satisfies the condition (1.5), then by [14,Proposition 2.18] the corresponding solution u(t) to (1.1) with initial data u 0 obeys the uniform bound P γ (u(t)) Q,u0 1 for all t ∈ R.…”

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

“…In the attractive regime, the orbital stability and instability of standing waves have been studied by [34,36,41], the asymptotic stability has been done in [23,46,47], and the strong instability has been done in [51,29]. In the repulsive regime, the orbital stability and instability have been studied by [31,45], and the global dynamics below ground states were studied in [40].…”

Singular Hartree equation in fractional perturbed Sobolev spaces

“…In the last dozen years a systematic analysis was carried out of the non-linear Schrödinger equation in one dimension, mainly with local non-linearity, of the form i∂ t u = − d 2 dx 2 + δ(x) u + α|u| γ−1 u , or the analogous equation with δ ′ -interaction instead of δ-interaction, initially motivated by phenomenological models of short-range obstacles in non-linear transport [29]. This includes local and global well-posedness in operator domain and energy space and blow-up phenomena [3,1,2], weak L p -solutions [6], scattering [8], solitons [19,21], as well as more recent modifications of the non-linearity [7]. None of such works has a three-dimensional counterpart.…”