On the internal modes of the ground state for 3D cubic--quintic equations
Abstract
Motivated by the problem of asymptotic stability for ground states of the Klein--Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic--quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic--quintic Klein--Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let Qω be the positive radial ground state of the three-dimensional cubic--quintic elliptic equation −ΔQω+ωQω−Qω3+Qω5=0. We study two related, but logically distinct, spectral problems as ω approaches the endpoint 3/16 of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian L+=−Δ+ω−3Qω2+5Qω4. This operator is also the exact linearized spatial operator around the static state Qω for the real scalar cubic--quintic Klein--Gordon equation with mass parameter ω. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic--quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to (3/16−ω)−1. %The rotating complex Klein--Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic--quintic models.
Related
- Ground state solutions for fractional p-Laplace equations with mixed nonlinearities in the critical and subcritical cases
- Ground states for pseudo-relativistic Schrödinger equations with logarithm-law: qualitative properties and nonrelativistic limit
- Simultaneous recovery of multiple parameters in nonlocal diffusion equations from internal measurements
- Qualitative analysis of energy ground states for magnetic focusing Gross-Pitaevskii equations
- A Boson-informed neural network for Ground States of Many-Body Schrödinger equations
BibTeX
@misc{chen2026internal,
title = {{On the internal modes of the ground state for 3D cubic--quintic equations}},
author = {Gong Chen and Zhaojie Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16613},
eprint = {2609.16613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16613},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - On the internal modes of the ground state for 3D cubic--quintic equations
AU - Gong Chen
AU - Zhaojie Yang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16613
UR - https://arxiv.org/abs/2609.16613
AB - Motivated by the problem of asymptotic stability for ground states of the Klein--Gordon and Schrödinger equations in three spatial dimensions in the presence of internal modes, we study the corresponding models with cubic--quintic nonlinearities. These equations arise in several physical contexts; in particular, the cubic--quintic Klein--Gordon equation appears naturally in the study of spin-$0$ particles in quantum field theory. More precisely, let Q_ω be the positive radial ground state of the three-dimensional cubic--quintic elliptic equation −ΔQ_ω+ωQ_ω−Q_ω^3+Q_ω^5=0. We study two related, but logically distinct, spectral problems as ω approaches the endpoint 3/16 of the ground-state branch. First, we determine the complete discrete spectrum of the scalar Hessian L_+=−Δ+ω−3Q_ω^2+5Q_ω^4. This operator is also the exact linearized spatial operator around the static state Q_ω for the real scalar cubic--quintic Klein--Gordon equation with mass parameter ω. We prove that the number of its discrete angular-momentum sectors tends to infinity, and obtain sharp asymptotics for their locations and multiplicities. Second, for the cubic--quintic nonlinear Schrödinger equation, we analyze the full Hamiltonian matrix linearization and prove the existence of internal modes in precisely an interval of angular-momentum sectors of length comparable to (3/16−ω)^−1. %The rotating complex Klein--Gordon problem leads %to a different gyroscopically coupled pencil; since that pencil is not %analyzed here, no identification of its internal spectrum is asserted. These results provide a rigorous linear foundation for future nonlinear stability and radiation-damping analysis in three-dimensional cubic--quintic models.
ER -