{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/numerical-simulations-of-the-nonlinear","title":"Numerical Simulations of the Nonlinear Quantum Vacuum in the Heisenberg-Euler Weak-Field Expansion","arxiv_id":"2109.08121","date":"2021-09-16","proceeding":null,"authors":["Andreas Lindner","Baris Ölmez","Hartmut Ruhl"],"abstract":"The Heisenberg-Euler theory of the quantum vacuum supplements Maxwell's theory of electromagnetism with nonlinear light-light interactions. These originate in vacuum fluctuations, a key prediction of quantum theory, and can be triggered by high-intensity laser pulses, causing a variety of intriguing phenomena. A highly accurate numerical scheme for solving the nonlinear equations due to the leading orders of the Heisenberg-Euler weak-field expansion is presented. The algorithm possesses an almost linear vacuum dispersion relation even for comparably small wavelengths and incorporates a nonphysical modes filter. The implemented solver is tested in one spatial dimension against a set of known analytical results for vacuum birefringence and harmonic generation. More complex scenarios for harmonic generation are demonstrated in two and three spatial dimensions.","url_abs":"https://arxiv.org/abs/2109.08121v5","url_pdf":"https://arxiv.org/pdf/2109.08121v5.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"numerical-simulations-of-the-nonlinear","repo_url":"https://gitlab.physik.uni-muenchen.de/ls-ruhl/hewes","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}