A numerical study of stability for solitary waves of a quasi-linear Schrödinger equation

A numerical study of stability for solitary waves of a quasi-linear Schrödinger equation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A numerical study of stability for solitary waves of a quasi-linear Schrödinger equation Meriem Bahhi, Christian Klein, Jonas Lampart, Simona Rota Nodari This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7511157/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 03 Mar, 2026 Read the published version in Journal of Nonlinear Science → Version 1 posted 9 You are reading this latest preprint version Abstract We discuss the (in)stability of solitary waves for a quasi-linear Schrödinger equation. The equation contains a quasi-linear term, responsible for a saturation effect, as well as a power nonlinearity. For different exponents of the nonlinearity, we determine analytically the asymptotic behavior of the $L^2$-mass of the solution as a function of the frequency close to the critical frequencies, which leads to natural conjectures concerning their stability. Depending on the exponent and the dimension, we expect all solitary waves to be stable, or the emergence of both a stable and an unstable branch of solutions. We investigate our conjectures numerically, and find compatible results both for the mass-energy relation and the dynamics. We observe that perturbations of solitary waves on the unstable branch may converge dynamically to the stable solution of a similar mass, or disperse. More general initial conditions show a similar behavior. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 03 Mar, 2026 Read the published version in Journal of Nonlinear Science → Version 1 posted Editorial decision: Revision requested 15 Nov, 2025 Reviews received at journal 14 Nov, 2025 Reviews received at journal 20 Oct, 2025 Reviewers agreed at journal 30 Sep, 2025 Reviewers agreed at journal 27 Sep, 2025 Reviewers invited by journal 26 Sep, 2025 Editor assigned by journal 12 Sep, 2025 Submission checks completed at journal 12 Sep, 2025 First submitted to journal 01 Sep, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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