Instability of the solitary waves for the 1d NLS with an attrictive delta potential in the degenerate case

Abstract

In this paper, we show the orbital instability of the solitary waves QΩe iΩt of the 1d NLS with an attractive delta potential (γ > 0)where Ω = Ω(p, γ) > γ 2 4 is the critical oscillation number and determined byThe classical convex method and Grillakis-Shatah-Strauss's stability approach in [2,10] don't work in this degenerate case, and the argument here is motivated by those in [5,15,16,20,21].The main ingredients are to construct the unstable second order approximation near the solitary wave QΩe iΩt on the level set M(QΩ) accoding to the degenerate structure of the Hamiltonian and to construct the refined Virial identity to show the orbital instability of the solitary waves QΩe iΩt in the energy space. Our result is the complement of the results in [8] in the degenerate case.

Stability of small solitary waves for the one-dimensional NLS with an attractive delta potential

“…Here the frequency ω is restricted to (0, 1 2 q 2 ) in the defocusing case and to ( 1 2 q 2 , ∞) in the focusing case, with 1 2 q 2 corresponding to the eigenvalue for the underlying linear operator. The stability properties of these solitary waves have been studied in previous works [13,15,25,30,40]. In the mass-supercritical case p > 4 with defocusing nonlinearity, all solitary waves are orbitally stable.…”

“…The stability properties of these solitary waves were studied in [15,25,40]. In the defocusing case, all solitary waves are orbitally stable [25].…”

“…In the focusing case, the authors of [15] utilized the general theory of [35,36] to prove the existence of Ω such that Q ω is stable for ω ∈ ( 1 2 q 2 , Ω) and unstable for ω ∈ (Ω, ∞). Instability for ω = Ω was subsequently established in [40]. Here Ω is defined as the solution to…”