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arXiv · 2609.16613

On the internal modes of the ground state for 3D cubic--quintic equations

Gong Chen·Zhaojie Yang
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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 $Q_ω$ 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.

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Gong Chen, Zhaojie Yang. 2026-09-15. On the internal modes of the ground state for 3D cubic--quintic equations. https://arxiv.org/abs/2609.16613

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