2023/10/12 by Chen, Pan, Yanheng Ding, Ding, Yanheng +4 · 3 citations
Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.2310.08478
openalex publication_date 2023/10/12 · openalex created_date 2023/10/14 · openalex updated_date 2026/07/28
In this paper, we investigate the nonrelativistic limit of normalized solutions to a nonlinear Dirac equation as given below: \begincases amp;-i c∑k=13αk∂k u +mc2 βu- Γ* (K |u|κ) K|u|κ-2u- P |u|s-2u=ωu,
amp;∫ℝ3\vert u \vert2 dx =1. \endcases Here, c>0 represents the speed of light, m > 0 is the mass of the Dirac particle, ω∈ℝ emerges as an indeterminate Lagrange multiplier, Γ, K, P are real-valued function defined on ℝ3, also known as potential functions. Our research first confirms the presence of normalized solutions to the Dirac equation under high-speed light conditions. We then illustrate that these solutions progress to become the ground states of a system of nonlinear Schrödinger equations with a normalized constraint, exhibiting uniform boundedness and exponential decay irrespective of the light speed. Our results form the first discussion on nonrelativistic limit of normalized solutions to nonlinear Dirac equations. This not only aids in the study of normalized solutions of the nonlinear Schrödinger equations, but also physically explains that the normalized ground states of high-speed particles and low-speed motion particles are consistent.