Bifurcation and noise-induced transitions in weakly dissipative geophysical KdV equation

Bifurcation and noise-induced transitions in weakly dissipative geophysical KdV 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 Bifurcation and noise-induced transitions in weakly dissipative geophysical KdV equation Mairembam Kelvin Singh, R.K. Brojen Singh, Moirangthem Shubhakanta Singh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7883114/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract We investigate the dynamics of the geophysical Korteweg-de Vries (KdV) equation under deterministic, dissipative, and stochastic settings. The geophysical KdV equation is solved using a Fourier pseudo-spectral method combined with an exponential time-differencing fourth-order Runge-Kutta scheme (ETDRK4), which ensures high accuracy for dispersive operators. Traveling-wave reduction transforms the KdV equation into a two-dimensional autonomous system, enabling phase-plane and bifurcation analysis. A transcritical-like bifurcation is identified, where equilibrium stability exchanges between a center and a saddle. Extending the equation with a weak dissipative term and additive Gaussian white noise, we analyze the noise-induced transition from the stable focus towards the separatrix using the concept of Mean First Passage Time (MFPT). The MFPT is approximated via Kramers' formula and validated numerically through stochastic simulations, showing exponential sensitivity to noise amplitude. The results highlight how weak dissipation regularizes oscillatory modes into damped spirals while noise triggers transition events over effective potential barriers. This combined framework of spectral theory, bifurcation theory, and stochastic analysis provides new insights into the stability and transition mechanisms of geophysical dispersive waves. Bifurcation Noise-induced transition Stochastic system Dissipation Dispersive waves Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7883114","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":535737686,"identity":"186e33f4-de0e-4858-bb7b-8c4f55abf265","order_by":0,"name":"Mairembam Kelvin Singh","email":"","orcid":"","institution":"Manipur University","correspondingAuthor":false,"prefix":"","firstName":"Mairembam","middleName":"Kelvin","lastName":"Singh","suffix":""},{"id":535737687,"identity":"57b56929-2d73-4d2a-b353-444daa7420fa","order_by":1,"name":"R.K. 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The geophysical KdV equation is solved using a Fourier pseudo-spectral method combined with an exponential time-differencing fourth-order Runge-Kutta scheme (ETDRK4), which ensures high accuracy for dispersive operators. Traveling-wave reduction transforms the KdV equation into a two-dimensional autonomous system, enabling phase-plane and bifurcation analysis. A transcritical-like bifurcation is identified, where equilibrium stability exchanges between a center and a saddle. Extending the equation with a weak dissipative term and additive Gaussian white noise, we analyze the noise-induced transition from the stable focus towards the separatrix using the concept of Mean First Passage Time (MFPT). The MFPT is approximated via Kramers' formula and validated numerically through stochastic simulations, showing exponential sensitivity to noise amplitude. The results highlight how weak dissipation regularizes oscillatory modes into damped spirals while noise triggers transition events over effective potential barriers. 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