Ground states to bi-harmonic nonlinear Schrödinger equations with competing dispersion
Abstract
We study ground-state solutions of the 4th order nonlinear Schrödinger equation with mixed dispersion via the elliptic profile equation Δ2 Q + 2a ΔQ + bQ − |Q|αQ=0, focusing on symmetry breaking of least-action ground states in 2D in the regime a=1, b=1+ε, 0<ε≪ 1, where nonradiality was proved by Lenzmann and Weth [26]. We establish a quotient-to-mass relation converting small-ε asymptotics of the Weinstein quotient into mass asymptotics, in any dimension and symmetry class. In 1D we prove the sharp small-ε rate for the quotient for every α>0 and deduce the rates for the mass, action and potential norm. In two and higher dimensions the same relation, combined with the quotient asymptotics of Lenzmann-Weth [26] and Mandel-Oliveira e Silva [27], gives the mass rates for the Knapp-type nonradial and the radial ground states. The resulting threshold αK(d)=8/(d+1) coincides for d≤3 with the existence of normalized minimizers threshold of Fernández-Jeanjean-Mandel-Mariş [16], connecting the least-action and mass-constrained formulations. We complement this with a detailed numerical study. For the cubic nonlinearity we construct a nonradial branch from a Knapp-type example, compare it with radial and angular-mode branches, and verify the predicted rates. For the quartic nonlinearity, turning points and branching are present in the mass-energy diagrams in radial or two- and four-peak nonradial solutions. Here, although least-action ground states are nonradial for small ε, the normalized ground states of the same mass appear radial, a consequence of branching. For α≥4 we find only radial ground states, consistent with the conjectured sharp form of the Stein-Tomas inequality on 𝕊1, equivalent to the radial and nonradial masses having the same leading coefficient.
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BibTeX
@misc{klein2026ground,
title = {{Ground states to bi-harmonic nonlinear Schrödinger equations with competing dispersion}},
author = {Christian Klein and Svetlana Roudenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13783},
eprint = {2609.13783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13783},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ground states to bi-harmonic nonlinear Schrödinger equations with competing dispersion
AU - Christian Klein
AU - Svetlana Roudenko
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.13783
UR - https://arxiv.org/abs/2609.13783
AB - We study ground-state solutions of the 4th order nonlinear Schrödinger equation with mixed dispersion via the elliptic profile equation Δ^2 Q + 2a ΔQ + bQ − |Q|^αQ=0, focusing on symmetry breaking of least-action ground states in 2D in the regime a=1, b=1+ε, 0<ε≪ 1, where nonradiality was proved by Lenzmann and Weth [26]. We establish a quotient-to-mass relation converting small-ε asymptotics of the Weinstein quotient into mass asymptotics, in any dimension and symmetry class. In 1D we prove the sharp small-ε rate for the quotient for every α>0 and deduce the rates for the mass, action and potential norm. In two and higher dimensions the same relation, combined with the quotient asymptotics of Lenzmann-Weth [26] and Mandel-Oliveira e Silva [27], gives the mass rates for the Knapp-type nonradial and the radial ground states. The resulting threshold α_K(d)=8/(d+1) coincides for d≤3 with the existence of normalized minimizers threshold of Fernández-Jeanjean-Mandel-Mariş [16], connecting the least-action and mass-constrained formulations. We complement this with a detailed numerical study. For the cubic nonlinearity we construct a nonradial branch from a Knapp-type example, compare it with radial and angular-mode branches, and verify the predicted rates. For the quartic nonlinearity, turning points and branching are present in the mass-energy diagrams in radial or two- and four-peak nonradial solutions. Here, although least-action ground states are nonradial for small ε, the normalized ground states of the same mass appear radial, a consequence of branching. For α≥4 we find only radial ground states, consistent with the conjectured sharp form of the Stein-Tomas inequality on 𝕊^1, equivalent to the radial and nonradial masses having the same leading coefficient.
ER -