Uniqueness and stability of ground states for some nonlinear Schrödinger equations

Abstract

We discuss the orbital stability of standing waves of a class of nonlinear Schrödinger equations in one space dimension. The crucial feature for our treatment is the presence of a nonconstant linear potential that is even and decreasing away from the origin in space. This enables us to establish the orbital stability of all ground states over the whole range of frequencies for which such solutions exist.

Lectures on the Orbital Stability of Standing Waves and Application to the Nonlinear Schrödinger Equation

“…Furthermore, 0 = inf σ(L 2 λ ) by the positivity of ϕ λ , as before. It is proved in Lemma 3.4 of ( [30]) that, for all λ < Λ, 0 / ∈ σ(L 1 λ ) and L 1 λ has exactly one negative eigenvalue.…”

Existence and stability of TE modes in a stratified non-linear dielectric

“…Note that (E2)-(E4) can be satisfied by a function g of the type (7.14). Theorem 7.1 gives a complete description of all positive solutions of (7.12) and, as is shown in Stuart (2006), these solutions have a variational characterization as 'ground states'. For the homogeneous case (7.14), it is easy to show that solutions of (7.12) cannot change sign.…”

“…In Stuart (2006), it is shown that the conditions introduced in Grillakis et al (1987) to justify the V-K criterion are also satisfied.…”

“…See McLeod et al (2003) and Stuart (2006). EXAMPLE 3 The function defined by (7.15) satisfies all the conditions of this theorem if 0 < σ < 2 and 0 α 1 − σ 2 .…”

Stability and instability of standing waves to a system of Schrödinger equations with combined power-type nonlinearities

“…Some authors considered the stability of the standing wave solutions to a Schrödinger equation (see [4,11,12,15,21]); some authors studied the existence of ground states or bounded states of a Schrödinger equation (see [1,3,5,8,14,25]); and others are interested in the stability of standing wave solutions to a Schrödinger equation with potential (see [9,26,29] and the references therein). In this paper, we will deal with not only the existence of ground states but also the stability of standing wave solutions to (1.1).…”