BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jonathan Jaquette (Boston University) DTSTART:20200630T140000Z DTEND:20200630T150000Z DTSTAMP:20260423T005720Z UID:CRM-CAMP/2 DESCRIPTION:Title: An overabundance of breathers in a nonlinear Schrödinger equation withou t gauge invariance\nby Jonathan Jaquette (Boston University) as part o f CRM CAMP (Computer-Assisted Mathematical Proofs) in Nonlinear Analysis\n \n\nAbstract\nIn this talk we study the nonlinear Schrödinger equation $i u_t + \\triangle u + u^2 = 0$ posed on the 1-torus. Based on their numeri cs\, Cho\, Okamoto\, & Shōji conjectured in their 2016 paper that: (C1) a ny singularity in the complex plane of time arising from inhomogeneous ini tial data is a branch singularity\, and (C2) real initial data will exist globally in real time. If true\, Conjecture 1 would suggest a strong incom patibility with the Painlevé property\, a property closely associated wit h integrable systems. While Masuda proved (C1) in 1983 for close-to-consta nt initial data\, a generalization to other initial data is not known. Usi ng computer assisted proofs we establish a branch singularity in the compl ex plane of time for specific\, large initial data which is not close-to-c onstant.\n\nConcerning (C2)\, we demonstrate an open set of initial data w hich is homoclinic to the 0-homogeneous-equilibrium\, proving (C2) for clo se-to-constant initial data. This proof is then extended to a broader clas s of nonlinear Schrödinger equation without gauge invariance\, and then u sed to prove the non-existence of any real-analytic conserved quantities. Indeed\, while numerical evidence suggests that most initial data is homoc linic to the 0-equilibrium\, there is more than meets the eye. Using compu ter assisted proofs\, we establish an infinite family of unstable nonhomog eneous equilibria\, as well as heteroclinic orbits traveling between these nonhomogeneous equilibria and the 0-equilibrium.\n LOCATION:https://researchseminars.org/talk/CRM-CAMP/2/ END:VEVENT END:VCALENDAR