{"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/constructive-proofs-of-existence-and","title":"Constructive proofs of existence and stability of solitary waves in the Whitham and capillary-gravity Whitham equations","arxiv_id":"2403.18718","date":"2024-03-27","proceeding":null,"authors":["Matthieu Cadiot"],"abstract":"In this manuscript, we present a method to prove constructively the existence and spectral stability of solitary waves in both the Whitham and the capillary-gravity Whitham equations. By employing Fourier series analysis and computer-aided techniques, we successfully approximate the Fourier multiplier operator in this equation, allowing the construction of an approximate inverse for the linearization around an approximate solution $u_0$. Then, using a Newton-Kantorovich approach, we provide a sufficient condition under which the existence of a unique solitary wave $\\tilde{u}$ in a ball centered at $u_0$ is obtained. The verification of such a condition is established combining analytic techniques and rigorous numerical computations. Moreover, we derive a methodology to control the spectrum of the linearization around $\\tilde{u}$, enabling the study of spectral stability of the solution. As an illustration, we provide a (constructive) computer-assisted proof of existence of stable {solitary waves} in both the case with capillary effects ($T>0$) and without capillary effects ($T=0$). Moreover, we provide an existence proof for a branch of solitary waves in the case $T=0$ via a rigorous continuation in the wave velocity. The methodology presented in this paper can be generalized and provides a new approach for addressing the existence and spectral stability of solitary waves in nonlocal nonlinear equations. All computer-assisted proofs, including the requisite codes, are accessible on GitHub at \\cite{julia_cadiot}.","url_abs":"https://arxiv.org/abs/2403.18718v3","url_pdf":"https://arxiv.org/pdf/2403.18718v3.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":"constructive-proofs-of-existence-and","repo_url":"https://github.com/matthieucadiot/whithamsoliton.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}