Instability theory of stationary solutions for nonlinear Schrödinger equations
Abstract
We study the orbital instability of algebraically decaying stationary solutions of nonlinear Schrödinger equations in dimensions d≥2, taking competing double-power nonlinearities as a canonical model. At zero frequency, the essential spectrum of the linearized operator touches zero, and the usual H1 coercivity is lost. We construct the zero-frequency limit η0 of the frequency derivatives ∂ωϕω of standing-wave profiles and determine its sharp asymptotic behavior. For a stationary ground state ϕ0∈ H1(ℝd), we prove instability if ⟨ x⟩ ϕ0∉ L2(ℝd) or, when ⟨ x⟩ ϕ0∈ L2(ℝd), if the duality pairing ⟨η0,ϕ0⟩ is negative. This criterion is a zero-frequency counterpart of the negative slope condition in the Grillakis--Shatah--Strauss theory. The proof combines variation of parameters for the linearized ODE with a shooting analysis that determines the global behavior of the homogeneous solution regular at the origin. We then localize η0 to construct an unstable direction and establish instability through a Lyapunov functional argument based on the variational characterization of ground states. The results extend with essentially the same proofs to a class of multiple-power nonlinearities.
Related
- Uniqueness of positive solutions of double-power nonlinear stationary Schrödinger equations: the fixed frequency case and the fixed mass case
- Oscillatory instability of subcritical stationary solutions in the parametrically driven, damped nonlinear Dirac equation
- Sharp Thresholds of Instability for the Biharmonic Nonlinear Schrödinger Equations
- Pullback dynamics and statistical solutions for semi-dissipative nonlinear Schrödinger equations
- Liouville theorems and symmetry of positive solutions for partially confined nonlinear Schrödinger equations
BibTeX
@misc{fukaya2026instability,
title = {{Instability theory of stationary solutions for nonlinear Schrödinger equations}},
author = {Noriyoshi Fukaya and Masayuki Hayashi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39830},
eprint = {2610.39830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39830},
}
RIS
TY - UNPB
TI - Instability theory of stationary solutions for nonlinear Schrödinger equations
AU - Noriyoshi Fukaya
AU - Masayuki Hayashi
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39830
UR - https://arxiv.org/abs/2610.39830
AB - We study the orbital instability of algebraically decaying stationary solutions of nonlinear Schrödinger equations in dimensions d≥2, taking competing double-power nonlinearities as a canonical model. At zero frequency, the essential spectrum of the linearized operator touches zero, and the usual H^1 coercivity is lost. We construct the zero-frequency limit η_0 of the frequency derivatives ∂_ωϕ_ω of standing-wave profiles and determine its sharp asymptotic behavior. For a stationary ground state ϕ_0∈ H^1(ℝ^d), we prove instability if ⟨ x⟩ ϕ_0∉ L^2(ℝ^d) or, when ⟨ x⟩ ϕ_0∈ L^2(ℝ^d), if the duality pairing ⟨η_0,ϕ_0⟩ is negative. This criterion is a zero-frequency counterpart of the negative slope condition in the Grillakis--Shatah--Strauss theory. The proof combines variation of parameters for the linearized ODE with a shooting analysis that determines the global behavior of the homogeneous solution regular at the origin. We then localize η_0 to construct an unstable direction and establish instability through a Lyapunov functional argument based on the variational characterization of ground states. The results extend with essentially the same proofs to a class of multiple-power nonlinearities.
ER -