Stability and instability of standing waves for nonlinear Schrödinger equations

Analytic theory of narrow lattice solitons

“…For example, stability of lattice solitons was studied in [35,[54][55][56] using the GSS theory. In addition, after this paper was submitted, we found out that the GSS theory was applied to narrow lattice solitons in the critical case by Lin and Wei [34].…”

Coupled nonlinear Schrödinger equations with harmonic potential

“…For p = 1 + 4 N , there exists a sharp condition [31] to the global existence; moreover, the standing waves are stable under some sufficient conditions [12]. When p > 1 + 4 N , the solution blows up in a finite time for a class of sufficiently large data and globally exists for a class of sufficiently small data [7,8,28]; moreover, the standing waves are unstable under suitable assumptions [13].…”

Existence of stable standing waves and instability of standing waves to a class of quasilinear Schrödinger equations with potential

“…Also when 1 + 4 N < p < 2 * − 1, Fukuizumi et al [13,14] have proven that there is µ * > 0 such that for any µ > µ * , the standing wave e iµt u(x) is orbital unstable. It is also proved that if (N 2 + 4 + 4 √ N 2 + 1)/N 2 < p < 2 * − 1, then the standing wave e iµt u(x) is unstable for all µ ∈ (0, +∞), see [13].…”