Orbital stability and instability of standing waves for an inhomogeneous logarithmic Schrödinger equation

Abstract

We study the inhomogeneous logarithmic Schrödinger equation i∂_tu+Δu+|x|⁻¹ulog|u|²=0, (t,x)∈R×R^N, N≥2. For every initial datum in H¹(R^N), we establish global existence of weak solutions. We also construct explicit positive stationary solutions and solve the associated Nehari minimization problem. For ω≥0, the minimizer is unique up to {a constant} phase and is given by e^{(N−1)/2(1+√(1+ω))} e^{−(1+√(1+ω))|x|}, whereas for ω<0 the infimum is zero and is not attained. Finally, we prove that the standing waves associated with these profiles are orbitally stable for ω>0 by a variational argument, and orbitally unstable for −1<ω<−1+(N−1)⁻² by combining spectral analysis, localized energy and mass estimates, and phase modulation.

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BibTeX RIS

BibTeX

@misc{ardila2026orbital,
  title = {{Orbital stability and instability of standing waves for an inhomogeneous logarithmic Schrödinger equation}},
  author = {Alex H. Ardila},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.07167},
  eprint = {2611.07167},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.07167},
}

RIS

TY  - UNPB
TI  - Orbital stability and instability of standing waves for an inhomogeneous logarithmic Schrödinger equation
AU  - Alex H. Ardila
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.07167
UR  - https://arxiv.org/abs/2611.07167
AB  - We study the inhomogeneous logarithmic Schrödinger equation i∂_tu+Δu+|x|^−1ulog|u|^2=0, (t,x)∈R×R^N, N≥2. For every initial datum in H^1(R^N), we establish global existence of weak solutions. We also construct explicit positive stationary solutions and solve the associated Nehari minimization problem. For ω≥0, the minimizer is unique up to {a constant} phase and is given by e^(N−1)/2(1+√(1+ω)) e^−(1+√(1+ω))|x|, whereas for ω<0 the infimum is zero and is not attained. Finally, we prove that the standing waves associated with these profiles are orbitally stable for ω>0 by a variational argument, and orbitally unstable for −1<ω<−1+(N−1)^−2 by combining spectral analysis, localized energy and mass estimates, and phase modulation.
ER  -

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